A300869 Odd numbers m such that sigma(x) = m has more than 1 solution.

31, 399, 403, 1767, 3751, 4123, 5187, 5673, 9517, 11811, 12369, 17143, 22971, 27001, 30783, 33883, 34671, 43617, 48279, 53413, 53599, 54873, 58683, 68859, 69967, 73017, 73749, 80199, 86831, 88753, 109771, 117273, 122493, 123721, 141267, 152019, 153543, 158503, 160797

Offset: 1

Author

M. F. Hasler, following a suggestion from Altug Alkan, Mar 16 2018

Keywords

• nonn

Comments

Goormaghtigh conjecture implies that 31 is the only prime in this sequence. - Jianing Song, Apr 27 2019

Examples

			a(1) = 31 = A123523(2), the smallest odd number m for which sigma(x) = m has (at least, and also exactly) two solutions, x = 16 and x = 25.
a(56) = 347529 = A123523(3) is the smallest odd m for which sigma(x) = m has (at least, and also exactly) three solutions, x = 406^2, x = 2*319^2 and x = 489^2.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..9260
• Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).
• Wikipedia, Goormaghtigh conjecture.

Crossrefs

Odd terms in A159886.

Cf. A000203 (sigma), A002191, A007368.

A123523 is a subsequence, except for the initial 1.

Cf. A331036.

Programs

Maple

N:= 200000: # for terms <= N
Res:= NULL: count:= 0:
for m from 1 to floor(sqrt(N)) by 2 do
sm:= numtheory:-sigma(m^2);
for k from 1 to floor(log[2](N/sm+1)) do
   v:= sm*(2^k-1);
   if v <= N then Res:= Res, v; count:= count+1 fi;
od
od:
B:= sort([Res]):
Dups:= select(t -> B[t+1]=B[t], [$1..nops(B)-1]):
sort(convert(convert(B[Dups],set),list)); # Robert Israel, Jan 15 2020

Mathematica

With[{s = PositionIndex@ Array[DivisorSigma[1, #] &, 10^6]}, Keys@ KeySort@ KeySelect[s, And[OddQ@ #, Length@ Lookup[s, #] > 1] &]] (* Michael De Vlieger, Mar 16 2018 *)

PARI

MAX=1e6; LIM=1e4; b=0; A300869=[]; for(x=1,LIM, for(i=1,2, (s=sigma(i*x^2))>MAX && next(2); bittest(b,s\2) && (setsearch(A300869,s) || S=setunion(A300869,[s])) || b+=1<<(s\2)))

PARI

is(k) = k%2 && invsigmaNum(k) > 1; \\ Amiram Eldar, Dec 16 2024, using Max Alekseyev's invphi.gp