cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A240667 a(n) is the GCD of the solutions x of sigma(x) = n; sigma(n) = A000203(n) = sum of divisors of n.

Original entry on oeis.org

1, 0, 2, 3, 0, 5, 4, 7, 0, 0, 0, 1, 9, 13, 8, 0, 0, 1, 0, 19, 0, 0, 0, 1, 0, 0, 0, 12, 0, 29, 1, 1, 0, 0, 0, 22, 0, 37, 18, 27, 0, 1, 0, 43, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 49, 0, 0, 1, 0, 61, 32, 0, 0, 0, 0, 67, 0, 0, 0, 1, 0, 73, 0, 0, 0, 45, 0, 1, 0, 0
Offset: 1

Views

Author

Michel Marcus, Apr 10 2014

Keywords

Comments

From n = 1 to 5, the least integers such that a(x) = n, depending on if singletons (see A007370 and A211656) are accepted or not, are 1, 3, 4, 7, 6 or 12, 126, 124, 210, 22152.
Is it possible to find an integer n such that a(n) = 6? Answer: n = A241625(6) = 6187272.

Examples

			There are no integers such that sigma(x) = 2, so a(2) = 0.
There is a single integer, x = 2, such that sigma(x) = 3, so a(3) = 2.
There are 2 integers, x = 6 and 11, such that sigma(x)=12, their gcd is 1, so a(12) = 1.

Links

Crossrefs

Cf. A000203, A007369, A007370, A211656, A241479 (a variant), A241625.

Programs

  • Maple
    A240667 := n -> igcd(op(select(k->sigma(k)=n, [$1..n]))):
seq(A240667(n), n=1..82); # Peter Luschny, Apr 13 2014
  • Mathematica
    a[n_] := GCD @@ Select[Range[n], DivisorSigma[1, #] == n&];
Array[a, 100] (* Jean-François Alcover, Jul 30 2018 *)
  • PARI
    sigv(n) =  select(i->sigma(i) == n, vector(n, i, i));
a(n) = {v = sigv(n); if (#v == 0, 0, gcd(v));}
  • PARI
    a(n) = my(s = invsigma(n)); if(#s, gcd(s), 0); \\ Amiram Eldar, Dec 19 2024, using Max Alekseyev's invphi.gp

Formula

a(A007369(n)) = 0.

A241480 Numbers k such that the GCD of the x's that satisfy sigma(x) = sigma(k) is not equal to 1.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 12, 13, 18, 19, 22, 27, 29, 32, 36, 37, 43, 45, 48, 49, 50, 61, 64, 67, 68, 72, 73, 75, 80, 81, 82, 91, 98, 100, 101, 104, 106, 109, 116, 121, 122, 128, 129, 133, 134, 137, 144, 146, 148, 149, 152, 156, 157, 160, 162, 163, 169, 171, 173
Offset: 1

Views

Author

Michel Marcus, Apr 23 2014

Keywords

Comments

Apart from 1, all terms of A211656 belong here since the solutions to sigma(x)=sigma(n) form a singleton and thus their GCD is n itself.

Links

Crossrefs

Cf. A000203, A211656, A241479.

Programs

  • PARI
    sigv(n) =  select(i->sigma(i) == n, vector(n, i, i));
isok(n) = (gcd(sigv(sigma(n))) != 1);
  • PARI
    is(k) = gcd(invsigma(sigma(k))) > 1; \\ Amiram Eldar, Dec 19 2024, using Max Alekseyev's invphi.gp

Formula

Numbers k such that A241479(k) is not equal to 1.

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.

Original entry on oeis.org

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

Views

Author

Michel Marcus, Apr 23 2014

Keywords

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

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).
Showing 1-3 of 3 results.

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