A160763 Least number having n divisors such that every sum of two or more divisors is composite.

1, 3, 49, 87, 130321, 4753, 7212549413161, 285541, 7890946561, 834472284661, 174913992535407978606601, 19699251391, 23205949656945057666311162427422570380321

Offset

1,2

Comments

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, (10^7)^22<a(23)<=892371481^22, ...

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. [T. D. Noe, Jun 01 2009]

If a(14) < 2311^6*50821, then a(14) = p^6*q with primes p,q such that 139<=p<1000 and p^6 in A093893. - Hagen von Eitzen, Jun 03 2009

If a(14) < 2311^6*50821, then a(14) = p^6*q with p in {139,151,181,211,241} and q being prime. - Max Alekseyev, Sep 24 2015

Links

Table of n, a(n) for n=1..13.

Mathematica

(* 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

Crossrefs

Cf. A093893, A000005.

Sequence in context: A033494 A188380 A252171 * A041523 A054206 A063777

Adjacent sequences: A160760 A160761 A160762 * A160764 A160765 A160766

Keyword

nonn

Author

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

Extensions

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

Term a(9) corrected, a(10)-a(13) and more upper bounds added by Max Alekseyev and Hagen von Eitzen, May 30 2009

a(19) from M. F. Hasler, May 31 2009

Edited by Max Alekseyev, Sep 25 2009

a(1)=1 prepended by Max Alekseyev, Mar 31 2015

Status

approved