A007691 -id:A007691 - cp's OEIS frontend

A232354 Numbers k that divide sigma(k^2) where sigma is the sum of divisors function (A000203).

1, 39, 793, 2379, 7137, 13167, 76921, 78507, 230763, 238887, 549549, 692289, 863577, 1491633, 1672209, 2076867, 4317885, 7615179, 8329831, 10441431, 23402223, 24989493, 37776123, 53306253, 53695813, 55871145, 74968479, 83766969, 133854435, 144688401, 161087439, 189437391

Offset: 1

Alex Ratushnyak, Nov 22 2013

nonn

Squarefree terms are: 1, 39, 793, 2379, 76921, 230763, 8329831, 24989493, 53695813, 161087439, ... Quotients are: 1, 61, 873, 3783, 11737, 26543, 85563, 141911, 370773, 417263, 1155561, ... - Michel Marcus, Nov 23 2013

Many terms are also in sequence A069520, cf. A232067 for the intersection of these two sequences. - M. F. Hasler, Nov 24 2013

Donovan Johnson, Table of n, a(n) for n = 1..200
Jose Arnaldo Bebita Dris, A new approach to odd perfect numbers via GCDs, arXiv:2202.08116 [math.NT], 2022.

Cf. A000203, A007691, A065764, A069520, A227302, A232067, A232703, A232704.

Select[Range[10^5], Divisible[DivisorSigma[1, #^2], #] &] (* Alonso del Arte, Dec 06 2013 *)

isok(n) = (sigma(n^2) % n) == 0; \\ Michel Marcus, Nov 23 2013

A065764(a(n)) mod a(n) = 0.

A323652 Numbers m having at least one divisor d such that m divides sigma(d).

Original entry on oeis.org

1, 6, 12, 28, 56, 120, 360, 496, 672, 992, 2016, 8128, 16256, 30240, 32760, 60480, 65520, 120960, 131040, 523776, 1571328, 2178540, 4357080, 8714160, 23569920, 33550336, 45532800, 47139840, 67100672, 91065600, 94279680, 142990848, 182131200, 285981696

Offset: 1

Jaroslav Krizek, Jan 21 2019

nonn

Generalization of multiperfect numbers (A007691).
Multiperfect numbers (A007691) are terms. If m is a k-multiperfect number and d divides k (for k > 1 and d > 1), then d*m is also a term.
Number 1379454720 is the smallest number with two divisors d with this property (459818240 and 1379454720). Another such number is 153003540480 with divisors 51001180160 and 153003540480. Is there a number with three divisors d with this property?
Supersequence of A081756.

			12 is a term because 6 divides 12 and simultaneously 12 divides sigma(6) = 12.

Cf. A000203, A007691, A081756, A323653.

[n: n in [1..10000] | #[d: d in Divisors(n) | SumOfDivisors(d) mod n eq 0] gt 0];

Select[Range[530000],AnyTrue[DivisorSigma[1,Divisors[#]]/#,IntegerQ]&] (* The program generates the first 20 terms of the sequence. To generate more, increase the Range constant, but the program may take a long time to run. *) (* Harvey P. Dale, Jan 17 2022 *)

isok(n) = {fordiv(n, d, if (!(sigma(d) % n), return (1));); return (0);} \\ Michel Marcus, Jan 21 2019

A332318 Real part of Gaussian norm-multiply-perfect numbers, in order of increasing norm.

Original entry on oeis.org

1, 2, 1, 3, 6, 5, 6, 10, 2, 8, 16, 12, 21, 18, 26, 30, 13, 43, 28, 6, 52, 60, 20, 10, 72, 56, 26, 28, 85, 95, 20, 100, 58, 80, 40, 36, 108, 120, 48, 190, 144, 176, 204, 38, 106, 214, 4, 84, 232, 276, 136, 216, 300, 174, 181, 263, 216, 312, 340, 140, 464, 20, 380

Offset: 1

Amiram Eldar, Feb 09 2020

nonn

Norm-multiply-perfect numbers where defined by Spira as Gaussian integers z such that norm(sigma(z)) is divisible by norm(z), where sigma(z) is the sum of the divisors of z = a + b*i and norm(z) = z * conj(z) = a^2 + b^2.
If the ratio norm(sigma(z))/norm(z) is 2 then the number is called a norm-perfect. The norm-perfect numbers correspond to n = 1, 13, 26, ... They are 2 + i, 21 + 22*i, 56 + 64*i, ...
Only nonnegative terms are included, since if z = a + b*i is a norm-multiply-perfect number then z*i, -z and -z*i are also norm-multiply-perfect numbers and one of the four has nonnegative a and b. Terms with the same norm are ordered by their real parts.
The corresponding imaginary parts are in A332319.
The corresponding norms are 1, 5, 10, 10, 40, 50, 52, 200, 260, 260, 260, 468, ...
The corresponding ratios norm(sigma(k))/k are 1, 2, 5, 4, 5, 8, 5, 10, 9, 10, ...

			2 + i is a norm-perfect number since sigma(2 + i) = 3 + i, and (3^2 + 1^2) = 10 = 2 * 5 = 2 * (2^1 + 1^2).
1 + 3*i is a norm-multiply-perfect number since sigma(1 + 3*i) = 5 + 5*i, and (5^2 + 5^2) = 50 = 5 * 10 = 5 * (1^2 + 3^2).

Amiram Eldar, Table of n, a(n) for n = 1..182
Wayne L. McDaniel, Perfect Gaussian integers, Acta Arithmetica, Vol. 2, No. 25 (1974), pp. 137-144.
Robert Spira, The Complex Sum of Divisors, The American Mathematical Monthly, Vol. 68, No. 2 (1961), pp. 120-124.

Cf. A000396, A007691, A100884, A100885, A100889, A100890, A102506, A102507, A102531, A102532, A103228, A103229, A103230, A332319.

csigma[z_] := DivisorSigma[1, z, GaussianIntegers -> True]; normultPerf[z_] := Divisible[Abs[csigma[z]]^2, Abs[z]^2]; seq = {}; max = 10^2; Do[z = a + b*I; If[Abs[z] <= max && normultPerf[z], AppendTo[seq, {Abs[z]^2, z}]], {a, 1, max}, {b, 0, max}]; Re[Transpose[Sort[seq]][[2]]] (* after T. D. Noe at A102531 *)

A332319 Imaginary part of Gaussian norm-multiply-perfect numbers, in order of increasing norm.

Original entry on oeis.org

0, 1, 3, 1, 2, 5, 4, 10, 16, 14, 2, 18, 22, 26, 18, 20, 41, 1, 36, 48, 24, 12, 65, 70, 24, 64, 82, 88, 45, 15, 100, 20, 94, 100, 130, 132, 84, 120, 184, 30, 148, 108, 32, 216, 192, 48, 228, 212, 51, 24, 252, 188, 60, 282, 283, 209, 312, 216, 198, 440, 102, 490

Offset: 1

Amiram Eldar, Feb 09 2020

nonn

See A332318 (the corresponding real parts) for the definition of Gaussian norm-multiply-perfect numbers.

Amiram Eldar, Table of n, a(n) for n = 1..182

Cf. A000396, A007691, A100884, A100885, A100889, A100890, A102506, A102507, A102531, A102532, A103228, A103229, A103230, A332318.

csigma[z_] := DivisorSigma[1, z, GaussianIntegers -> True]; normultPerf[z_] := Divisible[Abs[csigma[z]]^2, Abs[z]^2]; seq = {}; max = 10^2; Do[z = a + b*I; If[Abs[z] <= max && normultPerf[z], AppendTo[seq, {Abs[z]^2, z}]], {a, 1, max}, {b, 0, max}]; Im[Transpose[Sort[seq]][[2]]] (* after T. D. Noe at A102532 *)

A348601 Nonexponential multiply-perfect numbers: numbers k such that k | A160135(k).

Original entry on oeis.org

1, 6, 40, 234, 588, 89376, 10805558400

Offset: 1

Amiram Eldar, Oct 25 2021

The corresponding quotients A160135(k)/k are 1, 1, 1, 1, 1, 2, 3, ...

a(8) > 1.5*10^10, if it exists.

			6 is a term since its nonexponential divisors are 1, 2 and 3, so A160135(6) = 1 + 2 + 3 = 6 which is divisible by 6.
40 is a term since its nonexponential divisors are 1, 2, 4, 5, 8 and 20, so A160135(40) = 1 + 2 + 4 + 5 + 8 + 20 = 40 which is divisible by 40.

Cf. A160135.

Similar sequences: A007691, A064594, A064595, A189000, A327158.

esigma[n_] := Times @@ (Sum[First[#]^d, {d, Divisors[Last[#]]}] &) /@ FactorInteger[n]; Select[Range[1000], Divisible[DivisorSigma[1, #] - esigma[#], #] &]

A356156 The nearest common ancestor of n and gcd(n, sigma(n)) in the Doudna tree (A005940).

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 1, 1, 1, 2, 1, 2, 1, 2, 3, 1, 1, 2, 1, 2, 1, 2, 1, 12, 1, 2, 1, 28, 1, 6, 1, 1, 3, 2, 1, 1, 1, 2, 1, 10, 1, 3, 1, 2, 3, 2, 1, 2, 1, 1, 3, 2, 1, 2, 1, 2, 1, 2, 1, 6, 1, 2, 1, 1, 1, 3, 1, 2, 3, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 5, 1, 2, 3, 2, 1, 2, 5, 2, 1, 2, 5, 12, 1, 1, 3, 1, 1, 3, 1, 2, 3

Offset: 1

Antti Karttunen, Jul 30 2022

nonn

Cf. A000203, A007691 (fixed points), A009194, A348040, A348041.

Cf. also A347879, A356157, A356158, A356306.

Abincompreflen(n, m) = { my(x=binary(n),y=binary(m),u=min(#x,#y)); for(i=1,u,if(x[i]!=y[i],return(i-1))); (u);};
Abinprefix(n,k) = { my(digs=binary(n)); fromdigits(vector(k,i,digs[i]),2); };
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A156552(n) = {my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res}; \\ From A156552
A348040sq(x,y) = Abincompreflen(A156552(x), A156552(y));
A348041sq(x,y) = A005940(1+Abinprefix(A156552(x),A348040sq(x,y)));
A356156(n) = A348041sq(n,gcd(n, sigma(n)));

a(n) = A348041(n, A009194(n)) = A348041(n, gcd(n, A000203(n))).

A371920 Abundant numbers whose abundance is also an abundant number.

Original entry on oeis.org

24, 30, 42, 54, 60, 66, 78, 84, 90, 96, 102, 112, 114, 120, 126, 132, 138, 140, 150, 156, 168, 174, 176, 180, 186, 198, 204, 208, 210, 216, 222, 224, 228, 234, 240, 246, 252, 258, 264, 270, 276, 280, 282, 294, 304, 306, 308, 312, 318, 330, 336, 342, 348, 354, 360

Offset: 1

Amiram Eldar, Apr 12 2024

First differs from A125639 at n = 12.
Numbers k such that A033880(k) > 0 and A033880(A033880(k)) > 0.
This sequence is infinite: if m is divisible by 6 and coprime to 5, then 5*m is a term.
All the multiply-perfect numbers (A007691) that are not 1 or perfect (A000396), i.e., the terms of A166069, are terms of this sequence.

			24 is a term since A033880(24) = 12 > 0 and A033880(12) = 4 > 0.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A033880 (abundance), A000396, A007691, A125639.
Subsequence of A005101.
Subsequences: A141545, A166069, A223610, A223611, A223613.

ab[n_] := DivisorSigma[1, n] - 2*n; q[n_] := Module[{k = ab[n]}, k > 0 && ab[k] > 0]; Select[Range[360], q]

ab(n) = sigma(n) - 2*n;
is(n) = {my(k = ab(n)); k > 0 && ab(k) > 0;}

A386426 Odd nondeficient numbers k such that sigma(k) has the same powerful part as k, where sigma is the sum of divisors function.

Original entry on oeis.org

81022725, 891360225, 992106225, 1863765225, 2349967725, 3322372725, 7211992725, 8670600225, 9156802725, 11101612725, 13208490225, 15477435225, 15963637725, 18394650225, 18880852725, 21311865225, 21960135225, 22446337725, 22932540225, 25687687725, 25849755225, 28280767725, 28604902725, 30711780225, 31035915225

Offset: 1

Antti Karttunen, Aug 17 2025

nonn

Sequence by definition contains also any such hypothetical odd terms of A007691 that are mentioned in the comments of A386425. However, if no such terms exist, then this is a subsequence of A386427.
This sequence contains also the intersection of A001694 and A386425, even though it is probably an empty set. See comments in A386428.
The first three terms not divisible by 25 are: a(191) = 283806508293, a(247) = 371184932349, a(328) = 502252568433.

Intersection of A023196 and A386425.
Conjectured to be a subsequence of A386427.
Cf. A000203, A001694, A003557, A007691, A057521, A386428.
Cf. also A005231.

A057521(n)=my(f=factor(n)); prod(i=1, #f~, if(f[i, 2]>1, f[i, 1]^f[i, 2], 1))
isA386426(n) = if(!(n%2), 0, my(s=sigma(n)); ((s>=(2*n)) && (A057521(s)==A057521(n))));

{k | k is odd, A000203(k) >= 2*k and A003557(A000203(k)) = A003557(k)}.

a(8)-a(25) from Giovanni Resta, Aug 18 2025

A007358 Infinitary multi-perfect numbers.

Original entry on oeis.org

1, 6, 60, 90, 120, 36720, 73440, 12646368, 22276800, 44553600, 126463680, 133660800, 252927360, 758782080, 4201148160, 8402296320, 28770487200, 287704872000, 575409744000

Offset: 1

N. J. A. Sloane

The sequence contains numbers n such that A049417(n) = k*n for some integer k>=1. A007357 is the subsequence with quotient k=2. Cohen lists n=120, 73440, 44553600, 252927360, 575409744000 as entries with k=3, provides seven entries with k=4 and two entries with k=5.

G. L. Cohen, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. L. Cohen, On an integer's infinitary divisors, Math. Comp., 54 (1990), 395-411.
Michel Marcus, Unexhaustive list of terms

Cf. A049417, A007357.

Cf. A007691 (analog for sigma).

a049417(n) = {my(b, f=factorint(n)); prod(k=1, #f[, 2], b = binary(f[k, 2]); prod(j=1, #b, if(b[j], 1+f[k, 1]^(2^(#b-j)), 1)))}
isok(n) = frac(a049417(n)/n) == 0; \\ Michel Marcus, Sep 05 2018

a(10)-a(18) from Donovan Johnson, Nov 21 2013

a(1)=1 prepended by Michel Marcus, Sep 04 2018

A054027 Numbers that do not divide their sum of divisors.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74

Offset: 1

Asher Auel, Jan 19 2000

nonn

Does not contain numbers like 1, 120, 672, 30240, 32760, 523776,.. which are in A132999. - R. J. Mathar, Jun 13 2025

Complement of A007691. Cf. A000203, A054024.

with(numtheory): [seq(`if`(sigma(i) mod i <> 0,i,print( )),i=1..90)];

Select[Range[100],!Divisible[DivisorSigma[1,#],#]&] (* Harvey P. Dale, May 29 2019 *)

isok(m) = (sigma(m) % m) != 0; \\ Michel Marcus, Jun 20 2021

cp's OEIS Frontend

A232354 Numbers k that divide sigma(k^2) where sigma is the sum of divisors function (A000203).

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A323652 Numbers m having at least one divisor d such that m divides sigma(d).

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A332318 Real part of Gaussian norm-multiply-perfect numbers, in order of increasing norm.

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A332319 Imaginary part of Gaussian norm-multiply-perfect numbers, in order of increasing norm.

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A348601 Nonexponential multiply-perfect numbers: numbers k such that k | A160135(k).

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A356156 The nearest common ancestor of n and gcd(n, sigma(n)) in the Doudna tree (A005940).

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A371920 Abundant numbers whose abundance is also an abundant number.

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A386426 Odd nondeficient numbers k such that sigma(k) has the same powerful part as k, where sigma is the sum of divisors function.

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