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
Keywords
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
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
Comments