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

%I A300869 #32 Dec 16 2024 01:51:40
%S A300869 31,399,403,1767,3751,4123,5187,5673,9517,11811,12369,17143,22971,
%T A300869 27001,30783,33883,34671,43617,48279,53413,53599,54873,58683,68859,
%U A300869 69967,73017,73749,80199,86831,88753,109771,117273,122493,123721,141267,152019,153543,158503,160797
%N A300869 Odd numbers m such that sigma(x) = m has more than 1 solution.
%C A300869 Goormaghtigh conjecture implies that 31 is the only prime in this sequence. - _Jianing Song_, Apr 27 2019
%H A300869 Robert Israel, <a href="/A300869/b300869.txt">Table of n, a(n) for n = 1..9260</a>
%H A300869 Max Alekseyev, <a href="https://oeis.org/wiki/User:Max_Alekseyev/gpscripts">PARI/GP Scripts for Miscellaneous Math Problems</a> (invphi.gp).
%H A300869 Wikipedia, <a href="https://en.wikipedia.org/wiki/Goormaghtigh_conjecture">Goormaghtigh conjecture</a>.
%e A300869 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.
%e A300869 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.
%p A300869 N:= 200000: # for terms <= N
%p A300869 Res:= NULL: count:= 0:
%p A300869 for m from 1 to floor(sqrt(N)) by 2 do
%p A300869 sm:= numtheory:-sigma(m^2);
%p A300869 for k from 1 to floor(log[2](N/sm+1)) do
%p A300869    v:= sm*(2^k-1);
%p A300869    if v <= N then Res:= Res, v; count:= count+1 fi;
%p A300869 od
%p A300869 od:
%p A300869 B:= sort([Res]):
%p A300869 Dups:= select(t -> B[t+1]=B[t], [$1..nops(B)-1]):
%p A300869 sort(convert(convert(B[Dups],set),list)); # _Robert Israel_, Jan 15 2020
%t A300869 With[{s = PositionIndex@ Array[DivisorSigma[1, #] &, 10^6]}, Keys@ KeySort@ KeySelect[s, And[OddQ@ #, Length@ Lookup[s, #] > 1] &]] (* _Michael De Vlieger_, Mar 16 2018 *)
%o A300869 (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)))
%o A300869 (PARI) is(k) = k%2 && invsigmaNum(k) > 1; \\ _Amiram Eldar_, Dec 16 2024, using _Max Alekseyev_'s invphi.gp
%Y A300869 Odd terms in A159886.
%Y A300869 Cf. A000203 (sigma), A002191, A007368.
%Y A300869 A123523 is a subsequence, except for the initial 1.
%Y A300869 Cf. A331036.
%K A300869 nonn
%O A300869 1,1
%A A300869 _M. F. Hasler_, following a suggestion from _Altug Alkan_, Mar 16 2018