A241481 Numbers such that the GCD of the x's that satisfy sigma(x) = sigma(n) is not equal to 1 while the number of such x's is not 1 either.

48, 68, 75, 80, 82, 104, 116, 122, 144, 156, 160, 189, 196, 212, 225, 237, 242, 273, 279, 291, 309, 328, 342, 356, 364, 403, 490, 513, 524, 531, 592, 597, 614, 640, 651, 679, 684, 688, 712, 784, 788, 804, 808, 810, 822, 833, 889, 898, 903, 922, 925, 927, 954

Offset: 1

Author

Michel Marcus, Apr 23 2014

Keywords

• nonn

Comments

Subsequence of A241480, restricted to those terms that do not belong to A211656.

Is it possible, for each term of A211656, to find a corresponding term in the present sequence such that the corresponding GCD is equal to the initial A211656 term?

The first 11 terms of A211656 are: 2, 3, 4, 5, 7, 8, 9, 12, 13, 18, 19.

For these, we have 68, 48, 104, 12735, 364, 7848, 144, 9984, 273, 1764, 1197 in the present sequence.

For instance for m = 9984, the x's are [9984, 12252], with gcd = 12.

Is it possible to find a term here with corresponding gcd = 22, the 12th term of A211656?

Examples

			48 is in the sequence because sigma(48)=124 and the x's such that sigma(x) = 124 are 48 and 75, with gcd(48, 75) not equal to 1.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Crossrefs

Cf. A000203, A211656, A241479, A241480.

Programs

Maple

M:=1000: # to get all terms <= M
N:= 0:
for n from 1 to M do
  v:= numtheory:-sigma(n);
  N:= max(N,v);
  if assigned(R[v]) then R[v]:= igcd(R[v],n); S[v]:= S[v] union {n}
  else R[v]:= n; S[v]:= {n}
  fi;
od:
for n from M+1 to N do
  v:= numtheory:-sigma(n);
  if assigned(R[v]) then R[v]:= igcd(R[v],n);  S[v]:= S[v] union {n} fi;
od:
A:=
`union`(seq(S[v], v = select(t -> R[t]>1 and nops(S[t])>1, map(op,[indices(R)])))) intersect {$1..M}:
sort(convert(A,list)); # Robert Israel, Oct 24 2019

PARI

sigv(n) =  select(i->sigma(i) == n, vector(n, i, i));
isok(n) = my(v = sigv(sigma(n))); ((gcd(v)!=1) && (#v != 1));

PARI

isok(k) = my(v = invsigma(sigma(k))); #v > 1 && gcd(v) > 1; \\ Amiram Eldar, May 28 2025, using Max Alekseyev's invphi.gp (see links).