A232354 -id:A232354 - cp's OEIS frontend

A069520 Numbers k such that (1/k) * Sum_{d|k} d*sigma(d) is an integer.

1, 39, 793, 1638, 2379, 2394, 7137, 8190, 11970, 14274, 18135, 19530, 30927, 31122, 35685, 36270, 50700, 61854, 71370, 76921, 81900, 92781, 99918, 119700, 154635, 155610, 185562, 195300, 230763, 253890, 269500, 299754, 304038, 309270

Offset: 1

Benoit Cloitre, Apr 16 2002

Sequence A232354 starts in a very similar way, and the two sequences have a common subsequence A232067 = (1, 39, 793, 2379, 7137, 76921, 230763, ...), but neither is a subsequence of the other. - M. F. Hasler, Nov 24 2013

Amiram Eldar, Table of n, a(n) for n = 1..500

Cf. A000203, A232067, A232354.

s[n_] := DivisorSum[n, # * DivisorSigma[1, #] &]; Select[Range[300000], Divisible[s[#], #] &] (* Amiram Eldar, May 14 2022 *)

is_A069520(n)=!(sumdiv(n,d,d*sigma(d))%n) \\ - M. F. Hasler, Nov 24 2013

A232067 Numbers k such that sigma(k^2) and Sum_{d|k} d*sigma(d) are both multiples of k.

Original entry on oeis.org

1, 39, 793, 2379, 7137, 76921, 230763, 692289, 2076867, 8329831, 24989493, 53695813, 74968479, 161087439, 224905437, 243762649, 324863409, 375870691, 483262317, 731287947, 1127612073, 1449786951, 2094136707, 2193863841, 2631094837, 3382836219, 3606816823

Offset: 1

M. F. Hasler, Nov 24 2013

nonn

Intersection of A069520 and A232354.

Can these numbers be characterized as the terms of A232354 which do not have a factor in {11, 1093, ...}? Is this A090814, or (a subsequence of) A126197?

Donovan Johnson, Table of n, a(n) for n = 1..70

Cf. A000203, A069520, A232354.

fQ[n_] := Mod[DivisorSigma[1, n^2], n] == 0 && Mod[DivisorSum[n, #*DivisorSigma[1, #] &], n] == 0; Select[Range[100000], fQ] (* T. D. Noe, Nov 25 2013 *)

A247222 Numbers n such that n = gcd(n^2, sigma(n^2)).

Original entry on oeis.org

1, 39, 793, 7137, 76921, 863577, 2076867, 4317885, 8329831, 23402223, 53695813, 55871145, 224905437, 243762649, 1449786951, 2631094837, 2781581517, 3606816823, 6105766123, 6605555983, 6838433189, 8771312043, 13907907585, 35225161895, 42580779709, 56541691089

Offset: 1

Michel Marcus, Nov 26 2014

nonn

Previous name was: Numbers n such that numerator(sigma(n^2)/n^2)*denominator(sigma(n^2)/n^2) = sigma(n^2).
That is, numbers n, such that A249670(n^2) = A000203(n^2).
Appears to be a subsequence of A232354.

			sigma(39^2)/39^2 = 61/39 = 2379 = sigma(39^2), so 39 is a term.

Giovanni Resta, Table of n, a(n) for n = 1..48 (terms < 10^12)

Cf. A000203, A249670, A249671.

Select[Range[10^6], GCD[#^2, DivisorSigma[1, #^2]] == # &] (* Giovanni Resta, Jun 10 2016 *)

isok(n) = {ab = sigma(n^2)/n^2; numerator(ab)*denominator(ab) == sigma(n^2);}

{isa(n) = if( n<1, 0, n == gcd(n^2, sigma(n^2)))}; /* Michael Somos, Nov 26 2014 */

New name after Michael Somos by Michel Marcus, Nov 27 2014
a(13)-a(14) from Michel Marcus, Nov 27 2014
a(15)-a(26) from Giovanni Resta, Jun 10 2016

cp's OEIS Frontend

A069520 Numbers k such that (1/k) * Sum_{d|k} d*sigma(d) is an integer.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A232067 Numbers k such that sigma(k^2) and Sum_{d|k} d*sigma(d) are both multiples of k.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A247222 Numbers n such that n = gcd(n^2, sigma(n^2)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Extensions