A006532 -id:A006532 - cp's OEIS frontend

A038688 Squares that are the sum of the divisors of some number.

1, 4, 36, 121, 144, 256, 324, 400, 576, 784, 900, 961, 1024, 1296, 1600, 1764, 1936, 2304, 2704, 2916, 3136, 3600, 3844, 4096, 4356, 4624, 4900, 5184, 5776, 6084, 6400, 7056, 7744, 8100, 9216, 9604, 10000, 10404, 10816, 11664, 12544, 12996, 14400

Offset: 1

David W. Wilson

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems: Inversion of Multiplicative Functions (invphi.gp).

Intersection of A000290 and A002191.

Cf. A000203, A006532, A228061.

nn = 14400; t = Select[Union[DivisorSigma[1, Range[nn]]], IntegerQ[Sqrt[#]] &]; t = Select[t, # <= nn &] (* T. D. Noe, Sep 04 2013 *)

lista(kmax) = for(k = 1, kmax, if(invsigmaNum(k^2) > 0, print1(k^2, ", "))); \\ Amiram Eldar, Aug 12 2024, using Max Alekseyev's invphi.gp

a(n) = A228061(n)^2. - Amiram Eldar, Aug 12 2024

A048253 a(n) is the number of integers whose sum of divisors is 6^n.

Original entry on oeis.org

1, 1, 1, 5, 11, 18, 30, 48, 85, 148, 250, 415, 669, 1066, 1697, 2635, 4036, 6111, 9137, 13540, 19930, 29098, 42184, 60655, 86598, 122821, 173314, 243469, 340329, 473221, 654779, 901741, 1236668, 1689322, 2298592, 3115200, 4206016, 5658677, 7588039

Offset: 0

Labos Elemer

nonn

			For n=3, sigma(1,k) = 6^3 = 216 for each of 5 integers: 102, 110, 142, 159, and 187, so a(3) = 5.

Ray Chandler, Table of n, a(n) for n = 0..1000

Cf. A006532, A020477, A019422, A019423, A018427, A048251-A048256.

With[{s = Array[DivisorSigma[1, #] &, 6^8]}, Array[Count[s, 6^#] &, Log[6, Length@ s] + 1, 0]] (* Michael De Vlieger, May 14 2018 *)

a(n) = sum(k=1, 6^n, sigma(k)==6^n); \\ Michel Marcus, May 14 2018

a(9)-a(14) from Donovan Johnson, Sep 02 2008

Edited and extended by Ray Chandler, Sep 01 2010

A126028 Perfect square roots: numbers n such that (sopfr(n)*d(n))^2 = sigma(n) where sopfr = sum of prime factors with multiplicity (A001414), d(n) = number of divisors of n, sigma(n) = sum of divisors of n.

Original entry on oeis.org

22446139, 26116291, 28097023, 30236557, 31090489, 31124341, 49941589, 61137673, 62224039, 66960589, 71334867, 71585139, 82266591, 83045869, 88658031, 92346023, 92837591, 105183961, 114762567, 123563821, 129616270, 130399138, 131494219, 134156197

Offset: 1

Fred Schneider, Dec 14 2006

nonn

			n = 22446139 = 31*67*101*107. sopfr(n) = 31+67+101+107 = 306, d(n) = 2^4 = 16, sigma(n) = (31+1)*(67+1)*(101+1)*(107+1) = 23970816, (sopfr(n)*d(n))^2 = (306*16)^2 = 23970816 = sigma(n).

Donovan Johnson, Table of n, a(n) for n = 1..1000
Mersenne Forum, Perfect roots

Subsequence of A006532. Cf. A126029, A008472, A000005, A000203, A001414, A226479, A226480.

Clarified and extended by Charles R Greathouse IV, Oct 11 2009

Clarified by Donovan Johnson, Jun 09 2013

A226479 Numbers n such that (sopf(n)*d(n))^2 = sigma(n) where sopf(n) = sum of distinct prime factors of n (A008472) and d(n) = number of divisors of n.

Original entry on oeis.org

22446139, 26116291, 28097023, 30236557, 31090489, 31124341, 39618558, 41628195, 49941589, 51777957, 61137673, 62224039, 66960589, 71096795, 71334867, 71585139, 72304400, 82266591, 83045869, 92346023, 92837591, 105183961, 114762567, 117908994, 123563821

Offset: 1

Donovan Johnson, Jun 09 2013

nonn

Suggested by N. J. A. Sloane.

			n = 22446139 = 31*67*101*107. sopf(n) = 31+67+101+107 = 306. d(n) = 16. (sopf(n)*d(n))^2 = (306*16)^2 = 23970816 = sigma(n).

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A000005, A000203, A006532, A008472, A126028, A226480.

A226480 Squarefree numbers n such that (sopf(n)*d(n))^2 = sigma(n) where sopf(n) = sum of prime factors of n and d(n) = number of divisors of n.

Original entry on oeis.org

22446139, 26116291, 28097023, 30236557, 31090489, 31124341, 49941589, 61137673, 62224039, 66960589, 71334867, 71585139, 82266591, 83045869, 92346023, 92837591, 105183961, 114762567, 123563821, 130399138, 131494219, 134156197, 134867722, 135095767, 136026037

Offset: 1

Donovan Johnson, Jun 09 2013

nonn

Suggested by N. J. A. Sloane.

			n = 22446139 = 31*67*101*107. sopf(n) = 31+67+101+107 = 306. d(n) = 16. (sopf(n)*d(n))^2 = (306*16)^2 = 23970816 = sigma(n).

Donovan Johnson, Table of n, a(n) for n = 1..500

Cf. A000005, A000203, A006532, A008472, A126028, A226479.

A272405 Numbers n such that sum of the divisors of n is not of the form x^2 + y^2 + z^2 where x, y, z are integers.

Original entry on oeis.org

4, 8, 12, 16, 18, 24, 25, 32, 38, 48, 59, 64, 75, 91, 96, 99, 114, 125, 128, 130, 135, 158, 166, 169, 177, 192, 196, 203, 205, 209, 221, 239, 242, 251, 256, 268, 273, 283, 290, 315, 324, 347, 358, 365, 367, 375, 378, 379, 384, 387, 390, 392, 403, 422, 423, 427, 443, 445, 460, 474, 476, 493

Offset: 1

Altug Alkan, Apr 29 2016

Numbers n such that sum of the positive divisors of n is the sum of 4 but no fewer nonzero squares.
Prime terms of this sequence are 59, 239, 251, 283, 347, 367, 379, 443, 571, ...
A006532 is a subsequence of complement of this sequence.
Pollack (2011) proved that the complementary sequence has asymptotic density 7/8. Therefore the asymptotic density of this sequence is 1/8. - Amiram Eldar, Apr 09 2020

			1 is not a term since sigma(1) = 1 = 0^2 + 0^2 + 1^2 is the sum of 3 squares.
4 is a term since sigma(4) = 7 is not the sum of 3 squares.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Paul Pollack, Values of the Euler and Carmichael functions which are sums of three squares, Integers, Vol. 11 (2011), pp. 145-161.

Cf. A000203, A004215, A006532.

Select[Range@ 500, ! SquaresR[3, DivisorSigma[1, #]] > 0 &] (* Michael De Vlieger, Apr 29 2016 *)

isA004215(n) = {n\4^valuation(n, 4)%8==7}
lista(nn) = for(n=1, nn, if(isA004215(sigma(n)), print1(n, ", ")));

from itertools import count, islice
from sympy import divisor_sigma
def A272405_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:not (m:=(~(s:=int(divisor_sigma(n)))&s-1).bit_length())&1 and (s>>m)&7==7,count(max(startvalue,1)))
A272405_list = list(islice(A272405_gen(),30)) # Chai Wah Wu, Jul 09 2022

{n: A000203(n) in A004215}. - R. J. Mathar, May 02 2016

A331752 Numbers k such that squarefree part of sigma(k) is equal to squarefree part of 2*k.

Original entry on oeis.org

6, 28, 468, 496, 775, 2268, 3780, 4655, 7448, 8128, 9000, 10880, 10976, 25137, 40131, 40176, 58752, 62775, 66960, 91000, 137541, 137940, 140800, 160930, 167400, 173600, 195938, 224450, 307125, 377055, 399360, 406224, 417477, 494832, 569184, 603288, 634725, 639158, 658368, 773175, 869022, 881280, 889056, 1005480

Offset: 1

Antti Karttunen, Feb 06 2020

nonn

Numbers k such that A007913(sigma(k)) is equal to A007913(2*k), thus numbers for which sigma(k) has the same set of distinct prime factors with an odd exponent as 2*k.
Among the first 257 terms, these four are also in A228058:
  46277101  = 61 * 13^2 * 67^2,
  49889853  = 13 * 3^2 * 653^2,
  106706925 = 13 * 3^2 * 5^2 * 191^2,
  676830973 = 37 * 7^2 * 13^2 * 47^2.

			For n = 46277101 = 61 * 13^2 * 67^2, sigma(46277101) = 51703722 = 2 * 3^2 * 7^2 * 31^2 * 61, with A007913(sigma(46277101)) = 2*61 = A007913(2*46277101), thus 46277101 is included in this sequence.

Antti Karttunen, Table of n, a(n) for n = 1..257
Index entries for sequences where any odd perfect numbers must occur

Cf. A000203, A006532, A007913, A228058, A331751, A332208.

Cf. A000396 (a subsequence).

Select[Range[10^6], SameQ @@ Map[Sqrt[#] /. (c_: 1)*a_^(b_: 0) :> (c*a^b)^2 &, {DivisorSigma[1, #], 2 #}] &] (* Michael De Vlieger, Feb 08 2020, after Bill Gosper at A007913 *)

isA331752(n) = (core(2*n)==core(sigma(n)));

A063869 Least k such that sigma(k)=m^n for some m>1.

Original entry on oeis.org

2, 3, 7, 217, 21, 2667, 93, 217, 381, 651, 2752491, 2667, 8191, 11811, 24573, 57337, 82677, 172011, 393213, 761763, 1572861, 2752491, 5332341, 11010027, 21845397, 48758691, 85327221, 199753347, 341310837, 677207307, 1398273429, 3220807683

Offset: 1

Labos Elemer, Aug 27 2001

nonn

For n=2 to 20 sigma(a(n)) = m^n with m=2 or m=4. Computed terms are products of Mersenne primes (A000668). Is this true for larger n? Validity of a(11) was tested individually.
The Nagell-Ljunggren conjecture implies that sigma(x) is never 3^n for n>1. If this is true, then m=2 and m=4 are the smallest possible solutions. When A063883(n)>0, we can take m=2 and, as explained by Brown, find k to be a product of Mersenne primes (i.e. one of the numbers in A046528). When A063883(n)=0, which is true for the n in A078426, then m=4 and we have a(n)=a(2n) because 4=2^2. - T. D. Noe, Oct 18 2006
Sierpiński says that he proved sigma(x) is never 3^r for r>1. Hence m=2 and m=4 are the smallest possible solutions. When A063883(n)>0, we can take m=2 and, as explained by Brown, find k to be a product of Mersenne primes (i.e. one of the numbers in A046528). When A063883(n)=0, which is true for the n in A078426, then m=4 and we have a(n)=a(2n) because 4=2^2. - T. D. Noe, Oct 18 2006

			For n = 11, sigma(a(n)) = sigma(2752491) = sigma(3 * 7 * 131071) = 4^11.

T. D. Noe, Table of n, a(n) for n=1..500
K. S. Brown, Sum of Divisors Equals a Power of 2
W. Sierpiński, Elementary Theory of Numbers, Warszawa 1964, page 165.

Cf. A006532, A020477, A019422, A019423, A019424, A048257, A048258, A000668, A046528.

d={2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253}; nn=3700; t=Table[Infinity, {nn}]; t[[1]]=2; u={0}; k=1; While[2+d[[k]]<=nn, mer=2^d[[k]]-1; Do[a=u[[i]]+d[[k]]; If[a<=nn, If[u[[i]]==0, t[[a]]=Min[t[[a]], mer], t[[a]]=Min[t[[a]], t[[u[[i]]]]*mer]]], {i, Length[u]}]; u=Union[u, u+d[[k]]]; k++ ]; Do[If[t[[i]]==Infinity, t[[i]]=t[[2i]]], {i, nn}]; t (* T. D. Noe, Oct 13 2006 *)
c[] = 0; c[1] = 2; r = 1; Do[S = If[# > 1, Rest@ Divisors@ #, 0] &[GCD @@ FactorInteger[DivisorSigma[1, i]][[All, -1]]]; If[Length[S] > 0, Map[If[c[#] == 0, Set[c[#], i]] &, S]; If[# > r, r = #] &@ Max@ S], {i, 2^22}]; TakeWhile[Array[c, r], # > 0 &] (* _Michael De Vlieger, May 23 2022 *)

a(n) = my(k=2); while (!ispower(sigma(k), n), k++); k; \\ Michel Marcus, May 23 2022

a(n) = Min{x : A000203(x)=m^n} for some m.

a(24) corrected by T. D. Noe, Oct 15 2006

A074386 Numbers k such that sigma(k) is the square of a prime.

Original entry on oeis.org

3, 81, 400

Offset: 1

Labos Elemer, Aug 22 2002

The next term, if it exists, is > 10^11. - Donovan Johnson, Aug 24 2012
a(4), if it exists, satisfies sigma(a(4)) > 10^36.  - Hiroaki Yamanouchi, Sep 10 2014
If n belongs to this sequence, it may have at most two distinct prime divisors. If n=p^k, then sigma(p^k) = (p^(k+1)-1)/(p-1) = r^2 for some prime r. For k=1, it trivially has the only solution n=3; while for k>1 it is a partial case of the Nagell-Ljunggren equation and has the only prime solution r=11 (see Bennett-Levin 2015) corresponding to n=3^4=81. If n=p^k*q^m, then sigma(n) = (p^(k+1)-1)/(p-1) * (q^(m+1)-1)/(q-1) = r^2 for some prime r, implying that (p^(k+1)-1)/(p-1) = (q^(m+1)-1)/(q-1) = r. Here k+1 and m+1 must be odd distinct primes. The Goormaghtigh conjecture would imply that its only solution is n=400 with (p,k,q,m)=(5,2,2,4). - Max Alekseyev, Apr 24 2015

			sigma[{3,81,400}]={4,121,961}.

M. A. Bennett and A. Levin, The Nagell-Ljunggren equation via Runge’s method, Monatshefte für Mathematik 177:1 (2015), 15-31.
Wikipedia, Goormaghtigh conjecture

Cf. A000203, A001248, A008848, A023194, A028982.
Cf. A084738, A065854, A128164.
Subsequence of A006532.

Do[s=DivisorSigma[1, n]; If[PrimeQ[Sqrt[s]], Print[n]], {n, 1, 1000000}] (* Corrected by N. J. A. Sloane, May 26 2008 *)

Definition corrected by Juan Lopez, May 26 2008

Edited by N. J. A. Sloane, May 26 2008

A355928 Squarefree part of the sum of divisors of n.

Original entry on oeis.org

1, 3, 1, 7, 6, 3, 2, 15, 13, 2, 3, 7, 14, 6, 6, 31, 2, 39, 5, 42, 2, 1, 6, 15, 31, 42, 10, 14, 30, 2, 2, 7, 3, 6, 3, 91, 38, 15, 14, 10, 42, 6, 11, 21, 78, 2, 3, 31, 57, 93, 2, 2, 6, 30, 2, 30, 5, 10, 15, 42, 62, 6, 26, 127, 21, 1, 17, 14, 6, 1, 2, 195, 74, 114, 31, 35, 6, 42, 5, 186, 1, 14, 21, 14, 3, 33, 30, 5, 10

Offset: 1

Antti Karttunen, Jul 24 2022

nonn

Not multiplicative.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to sigma(n)

Cf. A000203, A007913, A006532 (positions of 1's), A355929.

Cf. also A080398.

PARI
```
A355928(n) = core(sigma(n));
```

from sympy.ntheory.factor_ import core, divisor_sigma
def A355928(n): return core(divisor_sigma(n)) # Chai Wah Wu, Jul 28 2022

a(n) = A007913(A000203(n)).

a(n) = A355929(n) + A007913(n).

cp's OEIS Frontend

A038688 Squares that are the sum of the divisors of some number.

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A048253 a(n) is the number of integers whose sum of divisors is 6^n.

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A126028 Perfect square roots: numbers n such that (sopfr(n)*d(n))^2 = sigma(n) where sopfr = sum of prime factors with multiplicity (A001414), d(n) = number of divisors of n, sigma(n) = sum of divisors of n.

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A226479 Numbers n such that (sopf(n)*d(n))^2 = sigma(n) where sopf(n) = sum of distinct prime factors of n (A008472) and d(n) = number of divisors of n.

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A226480 Squarefree numbers n such that (sopf(n)*d(n))^2 = sigma(n) where sopf(n) = sum of prime factors of n and d(n) = number of divisors of n.

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A272405 Numbers n such that sum of the divisors of n is not of the form x^2 + y^2 + z^2 where x, y, z are integers.

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A331752 Numbers k such that squarefree part of sigma(k) is equal to squarefree part of 2*k.

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A063869 Least k such that sigma(k)=m^n for some m>1.

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