Houston Journal of Mathematics c 2050 University of Houston Volume 76, No. 1, Communicated by George Washington

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1 Houston Journal of Mathematics c 2050 University of Houston Volume 76, No., 2050 SUMS OF PRIME DIVISORS AND MERSENNE NUMBERS WILLIAM D. BANKS AND FLORIAN LUCA Communicated by George Washington Abstract. In this note, we study those positive integers n with the property that the sum of the distinct prime factors of n divides the n-th Mersenne number.. Introduction For any integer n, let βn) denote the sum of the prime divisors of n: βn) = p n p. The study of the function βn) originated in the paper of Nelson, Penney, and Pomerance [7], where the question was raised as to whether the set of Ruth-Aaron numbers i.e., natural numbers n for which βn) = βn + )) has zero density in the set of all positive integers. This question was answered in the affirmative by Erdős and Pomerance [5], and the main result of [5] was later improved by Pomerance [0]. More recently, De Koninck and Luca [3] studied positive composite integers n which are divisible by the sum of their prime divisors; they showed that, for suitable positive constants c and c 2, the inequalities xe c log x log log x # { composite n x : βn) n } xe c2 log x log log x hold for all sufficiently large values of x Mathematics Subject Classification. A4; A05. Key words and phrases. Mersenne numbers, prime divisors. The second author was supported in part by the grant SEP-CONACyT and by a Guggenheim Fellowship.

2 2 WILLIAM D. BANKS AND FLORIAN LUCA In this note, we extend the ideas of [3] and derive upper and lower bounds for the number of positive integers n x for which βn) is a divisor of 2 n. Theorem. There exist positive constants c 3 and c 4 such that the inequalities x c3 log log log x/log log x # { n x : βn) 2 n } c 4 xlog log x log x hold for all sufficiently large values of x. Throughout the paper, the letter p is always used to denote a prime number, and we put πx) = #{p x} as usual. For an integer n >, we use Pn) to denote the largest prime factor of n, ωn) the number of distinct prime divisors of n, and Ωn) the total number of prime factors of n, counted with multiplicity; we also put P) =, and ω) = Ω) = 0. The Euler function is denoted by ϕn). For any real number x > 0 and an integer k 2, log k x denotes the k-th iterate of the function log x = max{lnx, }, where ln ) is the natural logarithm. Finally, we use the Vinogradov symbols and, as well as the Landau symbols O and o, with their usual meanings. 2. Proof of Theorem 2.. The Upper Bound. For an odd prime p, let tp) denote the multiplicative order of 2 modulo p. Let x be a large real number and put ) log xlog3 x y = yx) = exp 2 log 2 x and Let u = ux) = log x log y = 2 log 2 x log 3 x. Bx) = { n x : βn) 2 n }, and consider the following subsets of Bx): B x) = { n Bx) : Pn) y }, B 2 x) = { n Bx) \ B x) : Pn) 2 n }, B 3 x) = { n Bx) \ B x) B 2 x)) : ωn) > 0 log 2 x }, B 4 x) = { n Bx) \ 3 j=b j x) ) : βn) 2Pn) }, B 5 x) = { n Bx) \ 4 j= B jx) ) : Ωβn)) > 0 log 2 x }, B 6 x) = { n Bx) \ 5 j= B jx) ) : tp) > log 2 x for some p βn) }, B 7 x) = Bx) \ 6 j= B ix) ).

3 SUMS OF PRIME DIVISORS AND MERSENNE NUMBERS 3 As Bx) is the union of the sets B j x), j =,...,7, it suffices to find an appropriate bound for the cardinality of each set B j x). Using the following well known estimate for smooth numbers see [2]): #{n x : Pn) y} = xu u+ou), which holds as x with our choice of y and u, we derive that x ) #B x) xexp 2 + o))log 2 x) = x ). log x) 2+o) Next, for every integer n B 2 x) there exists a prime p > y such that p 2 n. For a fixed prime p, the number of positive integers n x with the latter property is precisely x/p 2 ; therefore, 2) #B 2 x) p>y x p 2 x k>y k 2 x ) x y = o log x x ). To estimate #B 3 x), let A = {n : Ωn) > 0 log 2 n}, and put At) = A [, t] for all t. A result of Nicolas [8] implies that the bound 3) #At) t log t log 2 t 2 0log 2 t = t log t) α+o) holds as t, where α = 0 ln2 > 3. Since B 3 x) Ax), it follows that the inequality 4) #B 3 x) x log 3 x holds if x is sufficiently large. Next, let n B 4 x). Write n = Pm with P = Pn), and put Q = Pm); then P > max{y, Q}. Since βm) + P = βn) 2P, we have the estimate P βm) ωm)q 0Q log 2 x, or P Q 0 log 2 x. Noting that PQ n, for fixed values of P and Q the number of such integers n B 4 x) does not exceed x/pq). Summing up over all possible choices for P and Q and using Mertens estimate p x p = log 2 x + b + O ) log x

4 4 WILLIAM D. BANKS AND FLORIAN LUCA for some constant b, we derive that #B 4 x) x PQ P>y 0.P/log 2 x Q<P = x log P 2 P log 2 P>y log + 5) = x P>y P x P>y xlog 3 x log y log 3 x P log P = xlog 3 x = 2xlog 2 x log x, P 0 log 2 x ) + O log0 log 2 x) log P log0 log 2 x) t>y dπt) t log t )) log0.p/ log 2 x) )) log P ) + O where we have used the fact that the estimate log0.p/ log 2 x) = log P log ) 3 x + O) log P )) log3 x = log P + O = + o))log P log y holds uniformly for P > y as x. Next, we turn our attention to integers n B 5 x). As before, write n = Pm with P = Pn) and m < x/y. For a fixed choice of m < x/y, the prime P = βn) βm) is determined uniquely by βn) A. We also have βn) < 2P 2x/m. Using estimate 3) again, it follows that the inequality #At) t log 3 t holds whenever t > t 0, for some constant t 0. Now, assuming that x is large enough, we have 2x/m > P > y > t 0, therefore the number of possibilities for P or βn)) for each fixed choice of m is at most #A2x/m) 2x m log 3 2x/m) x m log 3 y xlog3 2 x m log 3 x. Summing over all the possible choices for m, we get that 6) #B 5 x) xlog3 2 x log 3 x m<x/y m xlog3 2 x log 2 x.

5 SUMS OF PRIME DIVISORS AND MERSENNE NUMBERS 5 Next, we study integers n B 6 x). Again, write n = Pm with P = Pn); then x/m P > y. Fix m. For every prime p βn) 2 n, it is clear that tp) n. If P tp), then since tp) p, we see that p mod P). Since P > y > 2 is odd, it follows that p 2P + ; thus, βn) p 2P + ; but this contradicts the fact that n B 4 x). Hence, P tp), and therefore tp) m. Now, let p be a prime factor of βn) for which d = tp) > log 2 x; note that p mod d). We claim that p P. Indeed, if p = P, then P βn), and since βn) < 2P, it follows that βn) = P. But this implies that n = P; hence, P 2 P, which is clearly impossible since 2 P mod P). The congruence βn) 0 mod p) leads to P βm) 0 mod p). Since P x/m, it follows that for fixed m) the number of possibilities for P cannot exceed x/mp)+; since p βn) < 2P 2x/m, we see that the number of such possibilities is x/mp) + 3x/mp). Now, summing over all primes p mod d), then over positive multiples m x of d, and finally, over all integers d > log 2 x, it follows that 7) #B 6 x) 3 Here, we have used the bound d>log 2 x m x p x m 0 mod d) p mod d) x d>log 2 x x d>log 2 x p x p mod d) d k x/d log xlog 2 x dϕd) k p log 2 x ϕd), x mp p x p mod d) xlog 2 x log x. p which holds uniformly for 2 d x see formula 3.) of [4], or Lemma of []), together with Landau s bound [6]: d>t dϕd) t. Finally, we consider integers n = Pm in the set B 7 x). For every prime p βn), we have p 2 d, where d = tp) log 2 x, and d m as before. Thus, every

6 6 WILLIAM D. BANKS AND FLORIAN LUCA prime divisor of βn) also divides M = 2 d ) exp d log 2 x d log 2 x If W denotes the set of distinct prime factors of M, then #W = ωm) log M log 2 M log4 x log 2 x. d = exp Olog 4 x) ). Since n B 5 x), we have Ωβn)) 0 log 2 x. Hence, if m is fixed and E m = {βpm) : n = Pm B 7 x)}, then ) #W + k #E m 0 log k 2 x#w + 0 log 2 x) 0 log 2 x k 0 log 2 x = exp O log 2 x) 2)). Since m x/y, we obtain that #B 7 x) #E m x y exp O log 2 x) 2)) which implies that m x/y 8) #B 7 x) = o = xexp O log 2 x) 2) 2 logxlog ) 3 x, log 2 x ) x log x x ). Combining the estimates ), 2), 4), 5), 6), 7), and 8), we obtain the upper bound stated in Theorem The Lower Bound. Let x be a large real number. Let y = yx) be a function that tends to infinity with x to be determined later), and put z = exp 0 ) log y log 2 y and v = log y log z = log y 0 log 2 y. Let P denote the set of primes p in the interval [z/2, z] with the property that p is square-free; it is known see [9]) that #P = 0.5α+o))πz) as x, where α = p 2 ) = pp )

7 SUMS OF PRIME DIVISORS AND MERSENNE NUMBERS 7 is the Artin constant. In particular, #P > z 0 logz > v if x is sufficiently large. Let M be a fixed square-free positive integer obtained by multiplying together v distinct elements of P. Then, z ) v y = z v z v y M 2 2 v z > y z 2 if x is large enough. We now put R = p z p z 2 and N = M R. Clearly, R = om) as x ; hence, the inequalities 9) y > N > y 2z 2 hold if x is sufficiently large. Next, let K be an integer of size 0) K = z log z + O) such that K satisfies the parity condition K N mod 2). Let I be the interval [ N/3K), N/2K) ]. If x is sufficiently large, then, by 9), we have ) N K > y 2z 2 log z 2z = y log z 4z 3. In particular, N/3K) > z, which shows every prime in I is larger than z. Using 9) and ), we also see that πi), the number of primes in the interval I, satisfies the lower bound πi) = π N/2K) ) π N/3K) ) 2) N 7 K logn/k) > y log z 28z 3 log y > y z 3 if x is large enough, and therefore πi) > 2K. Now, let S be an arbitrary set of K 3 distinct primes in I, and put S = p. It is clear that S p S K 3) N 2K < N 2, and that the numbers S and N have opposite parities. Thus, N S > N/2 is an odd number. We now apply Vinogradov s Three Primes Theorem see, for

8 8 WILLIAM D. BANKS AND FLORIAN LUCA example, []), to conclude that the number V S) of representations of N S of the form 3) N S = p + p 2 + p 3, p < p 2 < p 3, where p, p 2, and p 3 are prime numbers, satisfies the lower bound N S) 2 V S) > c 5 logn S)) 3 > c N 2 6 log N) 3 for suitable absolute) positive constants c 5 and c 6 once x is large enough. Moreover, we can assume that each prime p that appears in 3) satisfies the bound p > c 7 N S) > c 8 N, where c 7 is a positive absolute constant, and c 8 = c 7 /2. Since K tends to infinity with x, it follows that c 8 N > N/2K) if x is large enough; therefore, p j S for j =, 2, 3. The above argument shows that if W is the set of ordered K-tuples of primes q,...,q K ) satisfying the conditions q < < q K ; q j I for j =,...,K 3; q K 2 > N/2K); N = q + + q K ; then the cardinality of W is at least 4) #W S I #S=K 3 V S) c 6 N 2 log N) 3 Let Q = q,...,q K ) be such a K-tuple in W, and put n Q = p z p K q j. j= ) πi). K 3 Since every prime in I is larger than z, it follows that n Q is square-free. Moreover, by unique factorization, the map Q n Q is one-to-one. We claim that each integer n Q satisfies the desired property, namely, βn Q ) 2 nq. To see this, we first observe that 5) βn Q ) = k p + q j = R + N = M. p z j=

9 SUMS OF PRIME DIVISORS AND MERSENNE NUMBERS 9 Let λm) be the value of the Carmichael function at M; i.e., the exponent of the multiplicative group Z/MZ). Since M is odd, square-free, and composed solely of primes q from I, we have λm) = lcm q M {q } p n Q. Here, we have used the fact that q is square-free and smaller than z for all primes q M. Since 2 M, it follows that 2 Z/MZ), and since λm) n Q, we have 2 nq mod λm)). Then, in view of 5), we have βn Q ) 2 nq, as claimed. It remains to get a lower bound for the number of integers n Q constructed in this way. First, note that the largest such n Q satisfies the inequality n Q p y 2K p z since we can choose p z ) K < exp 2z + K log y) exp 2z + z log y ) log z + log y, K { } z z, + log z log z to satisfy 0) as well as the parity condition K N mod 2). Therefore, given x, let us now define y hence also z) implicitly via the relation 6) x = exp 2z + z log y ) log z + log y. We remark that, if y is large enough, the right side of 6) is a strictly increasing function of y; therefore, if x is sufficiently large, the value of y is uniquely determined by 6). Using 4), we have for all sufficiently large x: 7) #Bx) #W N 2 log N) 3 N 2 log N) 3 ) πi) K 3 πi) K 3 ) K 3 N 2 K 3 log N) 3 πi) 3 exp K log πi)/k )). By 9) and the fact that N = y +o) as x, it follows that N 2 log N) 3 y2 z 4 log y) 3.

10 0 WILLIAM D. BANKS AND FLORIAN LUCA Using 9) and ), we also have the bound πi) πn/k) N K logn/k) y K log y. Applying the last two estimates and 2) to the bound 7), we derive that where #Bx) K6 y )) y 2 z 4 exp K log Kz 3 = exp K log y K logkz 3 ) + log K 6 /y 2 z 4 ) )) = exp log x) E log x )) E = log x K log y + K logkz 3 ) log K 6 /y 2 z 4 ) ). Using 0) and 6), we immediately deduce that )) log y E = 6z + O = 6 + o))z, logz and therefore 8) #Bx) exp log x) )) 6 + o))z log x x ). Using 6) and the definition of z, we also have 9) log x = + o)) z log y log y = 0. + o)) z log z log 2 y, which implies that or log 2 x = + o))log z = 0 + o)) log y log 2 y, log y = ) o) log2 x) 2 log 3 x Inserting this expression into 9), it follows that z log x = o) ) log 3 x log 2 x x ). x ). Finally, substituting this expression into 8), we obtain a lower bound of the form stated in Theorem, where we can take c 3 = 20.

11 SUMS OF PRIME DIVISORS AND MERSENNE NUMBERS 3. Remarks and Comments Theorem shows that the number of n x for which βn) 2 n is x +o) as x. On the other hand, the upper bound is rather weak; in particular, we cannot say anything about the convergence or the divergence of the sum of the reciprocals of integers n with this property, and we would like to pose this as an open problem for the reader. References [] N. L. Bassily, I. Kátai and M. Wijsmuller, On the prime power divisors of the iterates of the Euler-φ function, Publ. Math. Debrecen ), [2] E. R. Canfield, P. Erdős and C. Pomerance, On a problem of Oppenheim concerning factorisatio numerorum, J. Number Theory 7 983), no., 28. [3] J.-M. De Koninck and F. Luca, Positive integers divisible by the sum of their prime factors, Mathematika, to appear. [4] P. Erdős, A. Granville, C. Pomerance and C. Spiro, On the normal behavior of the iterates of some arithmetic functions, Analytic Number Theory, Birkhäuser, Boston, 990, [5] P. Erdős and C. Pomerance, On the largest prime factors of n and n +, Aequationes Math ), [6] E. Landau, Über die Zahlentheoretische Function ϕn) und ihre Beziehung zum Goldbachschen Satz, Nachr. Königlichen Ges. Wiss. Göttingen, Math.-Phys. Klasse, Göttingen, 900, [7] C. Nelson, D.E. Penney and C. Pomerance, 74 and 75, J. Recreational Mathematics, 7 974), [8] J.-L. Nicolas, Sur la distribution des nombres entiers ayant une quantité fixée de facteurs premiers, Acta Arith ), [9] F. Pappalardi, F. Saidak and I.E. Shparlinski, Square-free values of the Carmichael function, J. Number Theory ), no., [0] C. Pomerance, Ruth-Aaron numbers revisited in Paul Erdős and his mathematics, I Budapest, 999), , Bolyai Soc. Math. Stud.,, János Bolyai Math. Soc., Budapest, [] K. Ramachandra and A. Sankaranarayanan, Vinogradov Three Primes Theorem, Math. Student ), -4 and William D. Banks) Department of Mathematics, University of Missouri, Columbia, MO 652, USA address: bbanks@math.missouri.edu Florian Luca) Instituto de Matemáticas, Universidad Nacional Autónoma de México, C.P , Morelia, Michoacán, México address: fluca@matmor.unam.mx