A020477 -id:A020477 - cp's OEIS frontend

A048698 Nonprime numbers k such that sum of aliquot divisors of k is a cube.

1, 10, 49, 56, 69, 76, 122, 133, 568, 578, 1001, 1018, 1243, 1324, 1431, 1611, 1685, 1819, 1994, 2296, 2323, 3344, 3403, 3627, 3641, 3763, 3981, 4336, 5482, 8186, 9077, 9641, 10113, 10688, 13471, 14188, 14509, 14727, 15940, 16697, 17141, 17619, 19241, 19637

Offset: 1

Enoch Haga

The sum of the aliquot divisors of a prime is exactly 1. - Martin Renner, Oct 13 2011

			a(4) = 56: the aliquot divisors 1,2,4,7,8,14,28 sum to 64, a cube.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Donovan Johnson)

Cf. A020477, A048699, A194948.

a := []; for n from 1 to 1000 do if sigma(n) <> n+1 and type( simplify((sigma(n)-n)^(1/3)), `integer`) then a := [op(a), n]; fi; od: a;

Select[Range[20000], !PrimeQ[#] && IntegerQ @ Surd[DivisorSigma[1, #] - #, 3] &] (* Amiram Eldar, Feb 23 2020 *)

c=0; for(n=1, 13127239, if(isprime(n)==0, if(ispower(sigma(n)-n, 3), c++; write("b048698.txt", c " " n)))) /* Donovan Johnson, Mar 10 2013 */

A187771 Numbers whose sum of divisors is the cube of the sum of its prime divisors.

Original entry on oeis.org

245180, 612408, 639198, 1698862, 1721182, 5154168, 7824284, 15817596, 20441848, 25969788, 27688078, 28404862, 35860609, 67149432, 77378782, 91397838, 96462862, 179302264, 191550135, 289772221, 306901244, 311657084, 392802179, 441839706, 572673855, 652117774, 988918364

Offset: 1

Manuel Valdivia, Jan 04 2013

This sequence and A187824 and A187761 are winners in the contest held at the 2013 AMS/MAA Joint Mathematics Meetings. - T. D. Noe, Jan 14 2013
The identity sigma(k) = (sopf(k))^m only occurs for m = 3 (this sequence) in the given range, however it is likely that it also occurs for other powers m in higher numbers.
The smallest k such that sigma(k) = sopf(k)^m, for m=4,5,6 are 1056331752 (A221262), 213556659624 (A221263) and 45770980141656, respectively. - Giovanni Resta, Jan 07 2013
Prime divisors are taken without multiplicity. - Harvey P. Dale, Dec 17 2016

			a(13) = 35860609 = 41 * 71 * 97 * 127, then sigma(35860609) = 37933056 = (41 + 71 + 97 + 127)^3.

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 38.

Donovan Johnson and Robert Gerbicz, Table of n, a(n) for n = 1..1105 (first 100 terms from _Donovan Johnson_)
W. Sierpinski, Number Of Divisors And Their Sum, Elementary theory of numbers, Warszawa, 1964.

Cf. A000203, A008472, A020477, A070222.

Cf. A221262 (sigma(k)=sopf(k)^4), A221263 (sigma(k)=sopf(k)^5).

d[n_]:= If[Plus@@Divisors[n]-Power[Plus@@Select[Divisors[n], PrimeQ], 3]==0, n]; Select[Range[2,10^9], #==d[#]&]
Select[Range[2, 10^9],DivisorSigma[1,#]==Total[FactorInteger[#][[All, 1]]]^3&] (* Harvey P. Dale, Dec 17 2016 *)

is(n)=my(f=factor(n));sum(i=1,#f~,f[i,1])^3==sigma(n) \\ Charles R Greathouse IV, Jun 29 2013

a(n) = k if sigma(k) = (sopf(k))^3, where sigma(k) = A000203(k) and sopf(k) = A008472(k).

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

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

A194948 Numbers k such that sum of aliquot divisors of k, sigma(k) - k, is a cube.

Original entry on oeis.org

1, 2, 3, 5, 7, 10, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49, 53, 56, 59, 61, 67, 69, 71, 73, 76, 79, 83, 89, 97, 101, 103, 107, 109, 113, 122, 127, 131, 133, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233

Offset: 1

Martin Renner, Oct 13 2011

nonn

For prime numbers, the sum of their aliquot divisors is exactly 1 = 1^3.

			a(6) = 10, since the sum of aliquot divisors 1 + 2 + 5 = 8 = 2^3.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..2500 from Harvey P. Dale)

Union of A000040 and A048698.

Cf. A020477, A073040.

for n do s:=numtheory[sigma](n)-n; if root(s,3)=trunc(root(s,3)) then print(n); fi; od:

Select[Range[250],IntegerQ[Power[DivisorSigma[1,#]-#, (3)^-1]]&] (* Harvey P. Dale, Nov 25 2011 *)

A334339 Least positive integer m such that sigma(m * n) is a cube, where sigma(k) is the sum of the divisors of k.

Original entry on oeis.org

1, 51, 34, 291, 22, 17, 1, 1347, 597, 11, 10, 97, 892, 51, 46, 1758, 6, 3540, 343, 1649, 34, 5, 30, 449, 2928, 446, 199, 291, 472, 23, 34, 879, 235, 3, 22, 1770, 8661, 356, 3007, 1593, 884, 17, 241, 298, 1416, 15, 22, 586, 133, 1464, 2, 223, 3, 1180, 2, 1347, 711, 236, 232, 1062, 1200, 17, 597, 96771, 586, 265, 577, 485, 10, 11

Offset: 1

Zhi-Wei Sun, Apr 23 2020

nonn

Conjecture: a(n) exists for any n > 0. In other words, for any positive integer n, there is a positive integer m with sigma(m * n) equal to a cube.
The author's conjecture in A259915 implies that for each n = 1, 2, 3, ... there is a positive integer m with sigma(m * n) equal to a square.
See also A334337 for a similar conjecture.

			a(2) = 51 with sigma(2*51) = 216 = 6^3.
a(4) = 291 with sigma(4*291) = 2744 = 14^3.
a(578) = 34312749 with sigma(578*34312749) = 42144192000 = 3480^3.
a(673) = 49061802 with sigma(673*49061802) = 66135317184 = 4044^3.

Zhi-Wei Sun, Table of n, a(n) for n = 1..1000
Zhi-Wei Sun, Conjectures on representations involving primes, in: M. Nathanson (ed.), Combinatorial and Additive Number Theory II, Springer Proc. in Math. & Stat., Vol. 220, Springer, Cham, 2017, pp. 279-310. See also arXiv, arXiv:1211.1588 [math.NT], 2012-2017. (Cf. Conjecture 4.5.)

Cf. A000203, A000578, A020477, A259915, A334337.

cubeQ[n_] := cubeQ[n] = IntegerQ[n^(1/3)];
sigma[n_] := sigma[n] = DivisorSigma[1, n];
tab = {}; Do[m = 0; Label[aa]; m = m + 1; If[cubeQ[sigma[m * n]], tab = Append[tab, m], Goto[aa]], {n, 70}]; tab
lpi[n_]:=Module[{k=1},While[!IntegerQ[Surd[DivisorSigma[1,n*k],3]],k++]; k]; Array[lpi,70] (* Harvey P. Dale, Nov 05 2020 *)

a(n) = my(m=1); while (!ispower(sigma(n*m), 3), m++); m; \\ Michel Marcus, Apr 23 2020

Corrected and extended by Harvey P. Dale, Nov 05 2020

A048255 Integers whose sum of divisors is 6^5 = 7776.

Original entry on oeis.org

3210, 3498, 3710, 3882, 3910, 4310, 4922, 4982, 5182, 5457, 5885, 6035, 6095, 6307, 6797, 7117, 7327, 7597

Offset: 1

Labos Elemer

Sequence has A048253(5)=18 terms from A048251(5)=3210 to A048252(5)=7597. - Ray Chandler

			Divisors of 7597 are {1,71,107,7597}, whose sum is 7776, so 7597 is in the sequence.

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

Select[Range[7600],DivisorSigma[1,#]==7776&] (* Harvey P. Dale, Jun 04 2016 *)

for(i=1,t=6^5, sigma(i)==t & print1(i",")) \\ M. F. Hasler, Dec 09 2009

A048255 = { n | A000203(n)=6^5 }. - M. F. Hasler, Dec 09 2009

Minor edits, keywords added, and values checked with given PARI code by M. F. Hasler, Dec 09 2009

A244428 Sum of divisors of n and product of divisors of n are both perfect cubes.

Original entry on oeis.org

1, 1164, 8148, 11596, 12028, 28128, 32980, 34144, 34528, 36244, 38764, 39916, 41164, 41516, 73200, 75252, 81172, 84196, 94023, 100348, 181948, 182430, 192175, 193380, 193612, 194044, 195780, 196896, 200574, 204180, 208416, 211620, 214176, 217668, 220116, 225696, 230860, 235716

Offset: 1

Derek Orr, Jun 27 2014

nonn

This is also the intersection of A020477 and A048944.

Numbers m such that sigma(m) is a cube and (m is a cube or number of divisors of m is a multiple of 3). - Chai Wah Wu, Mar 10 2016

			The divisors of 1164 are {1, 2, 3, 4, 6, 12, 97, 194, 291, 388, 582, 1164}. 1*2*3*4*6*12*97*194*291*388*582*1164 = 2487241979165915136 = 1354896^3 = (1164^2)^3. 1+2+3+4+6+12+97+194+291+388+582+1164 = 2744 = 14^3. Thus, since both the sum of divisors and the product of divisors are perfect cubes, 1164 is a member of this sequence.

Chai Wah Wu, Table of n, a(n) for n = 1..500

Cf. A064710, A007955, A000203, A048944, A020477.

Select[Range[236000],AllTrue[{CubeRoot[DivisorSigma[1,#]],CubeRoot[Times@@Divisors[#]]},IntegerQ]&] (* Harvey P. Dale, Nov 26 2024 *)

for(n=1,10^6,d=divisors(n);s=sum(i=1,#d,d[i]);p=prod(j=1,#d,d[j]);if(ispower(s,3)&&ispower(p,3),print1(n,", ")))

from gmpy2 import iroot
from sympy import divisor_sigma
A244428_list = [i for i in range(1,10**4) if (iroot(i,3)[1] or not divisor_sigma(i,0) % 3) and iroot(int(divisor_sigma(i,1)),3)[1]] # Chai Wah Wu, Mar 10 2016

A284284 Let x be the sum of the divisors d_i of k such that d_i | sigma(k). Sequence lists the numbers k for which x^3 = sigma(k).

Original entry on oeis.org

1, 690, 714, 75432, 81172, 81192, 81624, 82248, 84196, 305320, 312040, 315880, 619542, 639198, 646758, 665874, 684342, 737694, 743958, 750114, 751626, 761454, 762966, 763614, 4349280, 4651680, 4789920, 4939680, 4981920, 5259936, 5325216, 5428896, 5474976

Offset: 1

Paolo P. Lava, Mar 24 2017

nonn

Subset of A020477.

			Divisors of 690 are 1, 2, 3, 5, 6, 10, 15, 23, 30, 46, 69, 115, 138, 230, 345, 690 and sigma(690) = 1728. Then:
1728 / 1 = 1728, 1728 / 2 = 864, 1728 /  3 = 576, 1728 / 6 = 288 and (1 + 2 + 3 + 6)^2 = 12^3 = 1728.

Giovanni Resta, Table of n, a(n) for n = 1..200

Cf. A000203, A020477, A284283.

with(numtheory): P:=proc(q) local a,k,n,x;
for n from 1 to q do a:=sort([op(divisors(n))]); x:=0;
for k from 1 to nops(a)-1 do if type(sigma(n)/a[k],integer) then x:=x+a[k]; fi; od;
if x^3=sigma(n) then print(n); fi; od; end: P(10^6);

Select[Range[10^5], (d = DivisorSigma[1, #]; IntegerQ[ d^(1/3)] && d == DivisorSigma[1, GCD[d, #]]^3) &] (* Giovanni Resta, Mar 28 2017 *)

a(1), a(25)-a(33) from Giovanni Resta, Mar 28 2017

A329239 Numbers k such that the sum of unitary divisors of k is a cube.

Original entry on oeis.org

1, 7, 102, 110, 120, 142, 159, 184, 187, 381, 588, 684, 690, 714, 770, 796, 840, 931, 994, 1034, 1054, 1065, 1113, 1128, 1173, 1240, 1265, 1288, 1293, 1309, 1528, 1633, 1643, 2619, 2667, 3638, 3937, 4280, 4505, 4664, 4788, 4830, 4855, 5176, 5572, 5671, 5730

Offset: 1

Amiram Eldar, Mar 13 2020

nonn

			7 is a term since usigma(7) = 8 = 2^3, where usigma is A034448.
102 is a term since usigma(102) = 216 = 6^3.

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

The unitary version of A020477.

Cf. A000578, A034448, A063937, A333258.

usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); Select[Range[10000], IntegerQ @ Surd[usigma [#], 3] &]

isok(m) = ispower(sumdiv(m, d, if(gcd(d, m/d)==1, d)), 3); \\ Michel Marcus, Mar 15 2020

cp's OEIS Frontend

A048698 Nonprime numbers k such that sum of aliquot divisors of k is a cube.

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A187771 Numbers whose sum of divisors is the cube of the sum of its prime divisors.

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

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

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A194948 Numbers k such that sum of aliquot divisors of k, sigma(k) - k, is a cube.

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A334339 Least positive integer m such that sigma(m * n) is a cube, where sigma(k) is the sum of the divisors of k.

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A048255 Integers whose sum of divisors is 6^5 = 7776.

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