2,1

First term of A093893 to have n divisors. a(2)=3, a(3)=7^2, a(4)=3*29, a(5)=19^4, a(6)=7^2*97, a(7)=139^6, a(8)=31*61*151, a(9)=211^2*421^2, a(10)=211^4*421, a(11)=211^10, a(12)=211^2*421*1051, a(13)=2311^12, 5.92*10^20<a(14)<=2311^6*50821, a(15)<=120121^4*150151^2, a(16)<=120121*150151*180181*270271, a(17)=120121^16, a(18)<=4084081^2*5105101^2*8168161, a(19)=2312311^18, ...

Proof that a(n) exists for all n: We will show that there is a prime p such that the sums of two or more divisors of p^(n-1) are all composite. Let Q be the product of the primes less than or equal to n. Let p be a prime of the form Qk+1. Observe that the divisors of p^(n-1), which are just powers of p, have the same form Qk+1 (but with different k, of course). Hence a sum of r of these powers will have the form Qk+r (for some k). Due to the way Q is constructed and r <= n, r and Q have a common factor, making Qk+r composite. Furthermore, by Dirichlet's theorem, we know there are an infinite number of primes p that will work for each n. [From T. D. Noe (noe(AT)sspectra.com), Jun 01 2009]

If a(14) < 2311^6*50821, then a(14) = p^6*q with primes p,q such that 139<=p<=1571 and p^6 in A093893. [From Hagen von Eitzen (math(AT)von-eitzen.de), Jun 01 2009]

(* first do *) Needs["Combinatorica`"] (* then *) f[n_] := Block[{d = Divisors@n, k, mx = 1 + 2^DivisorSigma[0, n]}, k = 2 + Length@d; While[k < mx, If[ PrimeQ[Plus @@ NthSubset[k, d]], Break[]]; k++ ]; If[k == mx, Length@d, 0]]; t = Table[0, {20}]; k = 1; While[k < 2*10^7, a = f@k; If[a > 0 && t[[a]] == 0, t[[a]] = k; Print[{a, k}]]; k += 2]; t

Cf. A093893, A000005.

Sequence in context: A094045 A033494 A079837 this_sequence A041523 A054206 A063777

Adjacent sequences: A160760 A160761 A160762 this_sequence A160764 A160765 A160766

nonn

Robert G. Wilson v, May 25 2009, May 29 2009

Definition revised by N. J. A. Sloane, May 30 2009

More terms and upper bounds from Max Alekseyev, May 30 2009

Term a(9) corrected and a(10)..a(13) added by Hagen von Eitzen (math(AT)von-eitzen.de), Jun 01 2009

Edited by Max Alekseyev (maxale(AT)gmail.com), Sep 25 2009