A020477 -id:A020477 - cp's OEIS frontend

A006532 Numbers whose sum of divisors is a square.

1, 3, 22, 66, 70, 81, 94, 115, 119, 170, 210, 214, 217, 265, 282, 310, 322, 343, 345, 357, 364, 382, 385, 400, 472, 497, 510, 517, 527, 642, 651, 679, 710, 742, 745, 782, 795, 820, 862, 884, 889, 930, 935, 966, 970, 1004, 1029, 1066, 1080, 1092, 1146

Offset: 1

N. J. A. Sloane, Simon Plouffe

If a and b are in the sequence and relatively prime, then a*b is also in the sequence. - Franklin T. Adams-Watters, Jan 12 2009
Apart from a(2), all terms are composite. Bunyakovsky's conjecture implies that this sequence is infinite, since then (e.g.) there are infinitely many primes of the form p = 3k^2 - 1, whence sigma(2p) = 3p + 3 = 9k^2. - Charles R Greathouse IV, May 12 2011
See the Beukers, Luca and Oort link for a proof that the sequence is infinite. - Robert Israel, Oct 15 2017

			3 is in the sequence because its divisors are 1 and 3, which add up to 4 = 2^2.
22 is in the sequence because its divisors are 1, 2, 11, 22, which add up to 36 = 6^2.
32 is not in the sequence, because its divisors, 1, 2, 4, 8, 16, 32, add up to 63, which is one short of 8^2.

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 8.
J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 94, p. 33, Ellipses, Paris 2008.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Abraham Verghese, Cutting for Stone: A Novel. New York: Alfred A. Knopf, 2009, p.361, p. 528 large-print edition.
David Wells, Curious and interesting numbers, Penguin Books, p. 111.

T. D. Noe, Table of n, a(n) for n = 1..10000
Frits Beukers, Florian Luca and Frans Oort, Power Values of Divisor Sums, The American Mathematical Monthly, Vol. 119, No. 5 (May 2012), pp. 373-380.
J. Meeus & N. J. A. Sloane, Correspondence, 1974-1975
Index entries for sequences related to sums of divisors

Cf. A074385, A000203, A020477.

a006532 n = a006532_list !! (n-1)
a006532_list = filter ((== 1) . a010052 . a000203) [1..]
-- Reinhard Zumkeller, Jun 09 2013

[n: n in [1..2000] | IsSquare(&+(Divisors(n)))]; // Vincenzo Librandi, May 31 2015

for i from 1 to 1000 do if issqr(sigma(i)) then print(i); fi; od;

Select[ Range[ 1150 ], IntegerQ[ Sqrt[ DivisorSigma[ 1, # ] ] ]& ]

is(n)=issquare(sigma(n)) \\ Charles R Greathouse IV, Jun 05 2013

[n for n in (1..1000) if sigma(n).is_square()] # Giuseppe Coppoletta, Dec 16 2014

A010052(A000203(a(n))) = 1. - Reinhard Zumkeller, Jun 09 2013

a(42)-a(51) from Enoch Haga, circa 1999

A048251 a(n) is the smallest number whose sum of divisors is 6^n.

Original entry on oeis.org

1, 5, 22, 102, 510, 3210, 17490, 112890, 600270, 3466470, 20205570, 118879530, 697118730, 3949737330, 24217298490, 143487592710, 841422307110, 4973562896610, 29520886859310, 180254162529210, 1052751138726210, 6301225298627490, 37854941354933010, 224270177470178070

Offset: 0

Labos Elemer

nonn

Terms of this sequence are products of distinct terms in A005105. - Ray Chandler, Sep 01 2010

			sigma(k) = 1296 = 6^4 for each k in {510, 642, 710, 742, 782, 795, 862, 935, 1177, 1207, 1219}; the smallest of these is a(4)=510.

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

Cf. A006532, A020477, A019422, A019423, A018427, A048252-A048256.

a(n) = {my(k=1); while (sigma(k) != 6^n, k++); k;} \\ Michel Marcus, May 14 2018

a(n) = Min{k : A000203(k) = 6^n}.

a(9)-a(14) from Donovan Johnson, Sep 02 2008
a(15)-a(24) from Walter Kehowski, Aug 22 2010
Edited and extended by Ray Chandler, Sep 01 2010
Error in sequence corrected by N. J. A. Sloane, Oct 04 2010

A019423 Numbers whose sum of divisors is a fifth power.

Original entry on oeis.org

1, 21, 31, 651, 889, 3210, 3498, 3710, 3882, 3910, 4310, 4922, 4982, 5182, 5457, 5885, 6035, 6095, 6307, 6797, 7117, 7327, 7597, 24573, 27559, 71193, 82110, 90510, 94981, 97410, 98671, 99301, 99510, 100110, 103362, 104622, 107778, 108438, 108822

Offset: 1

David W. Wilson

nonn

Marius A. Burtea, Table of n, a(n) for n = 1..3648 (terms 1..1000 from Donovan Johnson; a(1211) re-indexed and duplicate a(2311) removed by _Georg Fischer_, Mar 21 2022).
Frits Beukers, Florian Luca and Frans Oort, Power Values of Divisor Sums, The American Mathematical Monthly, Vol. 119, No. 5 (May 2012), pp. 373-380.

Cf. A000203, A006532, A019422, A019424, A020477, A048257, A048258.

[n:n in [1..10000]| IsPower(SumOfDivisors(n),5)]; // Marius A. Burtea, Apr 17 2019

lista(nn) = {for (i=1, nn, s = sigma(i); if (s == 1 || ispower(s, 5), print1(i, ", ")););} \\ Michel Marcus, Jun 12 2013

A048699 Nonprime numbers whose sum of aliquot divisors (A001065) is a perfect square.

Original entry on oeis.org

1, 9, 12, 15, 24, 26, 56, 75, 76, 90, 95, 119, 122, 124, 140, 143, 147, 153, 176, 194, 215, 243, 287, 332, 363, 386, 407, 477, 495, 507, 511, 524, 527, 536, 551, 575, 688, 738, 791, 794, 815, 867, 871, 892, 924, 935, 963, 992, 1075, 1083, 1159, 1196, 1199, 1295, 1304

Offset: 1

Enoch Haga

The sum of aliquot divisors of prime numbers is 1.

If a^2 is an odd square for which a^2-1 = p + q with p,q primes, then p*q is a term. If m = 2^k-1 is a Mersenne prime then m*(2^k) (twice an even perfect number) is a term. If b = 2^j is a square and b-7 = 3s is a semiprime then 4s is a term. - Metin Sariyar, Apr 02 2020

			a(3)=15; aliquot divisors are 1,3,5; sum of aliquot divisors = 9 and 3^2=9.

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

Cf. A001065, A006532, A020477, A048698, A073040 (includes primes).

a := []; for n from 1 to 2000 do if sigma(n) <> n+1 and issqr(sigma(n)-n) then a := [op(a), n]; fi; od: a;

nn=1400;Select[Complement[Range[nn],Prime[Range[PrimePi[nn]]]],IntegerQ[ Sqrt[DivisorSigma[1,#]-#]]&] (* Harvey P. Dale, Apr 25 2011 *)

isok(k) = !ispseudoprime(k) && issquare(sigma(k) - k); \\ Michel Marcus, May 13 2025

A019424 Numbers whose sum of divisors is a sixth power.

Original entry on oeis.org

1, 2667, 3937, 17490, 19410, 22578, 24610, 24910, 25466, 25910, 26554, 26818, 27285, 29342, 29733, 29762, 31102, 31535, 32043, 32997, 33985, 35585, 36635, 37985, 39697, 41393, 41837, 42347, 44047, 45637, 45739, 45937, 46117, 172011, 253921, 640737

Offset: 1

David W. Wilson

nonn

			sigma(2667) = 1+3+7+21+127+381+889+2667 = 4096 = 4^6.
sigma(3937) = 1+31+127+3937 = 4096 = 4^6.

Marius A. Burtea, Table of n, a(n) for n = 1..2010 (terms 1..1000 from Donovan Johnson)
Frits Beukers, Florian Luca and Frans Oort, Power Values of Divisor Sums, The American Mathematical Monthly, Vol. 119, No. 5 (May 2012), pp. 373-380.

Cf. A000203, A006532, A019422, A019423, A020477, A048257, A048258.

[n:n in [1..100000]| IsPower(SumOfDivisors(n),6)]; // Marius A. Burtea, Apr 17 2019

Select[Range[700000],IntegerQ[Surd[DivisorSigma[1,#],6]]&] (* Harvey P. Dale, Apr 19 2019 *)

c=0; for(n=1, 306455560, if(ispower(sigma(n), 6), c++; write("b019424.txt", c " " n))) /* Donovan Johnson, Jun 13 2013 */

A048256 Numbers whose sum of divisors is 6^6 = 46656.

Original entry on oeis.org

17490, 19410, 22578, 24610, 24910, 25466, 25910, 26554, 26818, 27285, 29342, 29733, 29762, 31102, 31535, 32043, 32997, 33985, 35585, 36635, 37985, 39697, 41393, 41837, 42347, 44047, 45637, 45739, 45937, 46117

Offset: 1

Labos Elemer

Sequence has A048253(6)=30 terms from A048251(6)=17490 to A048252(6)=46117. - Ray Chandler, Sep 01 2010

			The divisors of 19410 are 1, 2, 3, 5, 6, 10, 15, 30, 647, 1294, 1941, 3235, 3882, 6470, 9705, and 19410; their sum is 46656, so 19410 is in the sequence.

Cf. A006532, A020477, A019422, A019423, A018427, A048251-A048255.

Select[Range[6^6], DivisorSigma[1, # ] == 6^6 &] (* Ray Chandler, Sep 01 2010 *)

A048257 Integers whose sum of divisors is a 7th power.

Original entry on oeis.org

1, 93, 127, 11811, 112890, 120054, 124338, 127330, 132770, 133998, 134090, 137058, 138754, 139962, 146710, 148665, 148810, 149534, 153986, 155510, 160215, 161194, 164985, 167134, 170986, 173098, 183687, 184682, 187143, 191913, 198485, 206823, 206965, 207687

Offset: 1

Labos Elemer

nonn

If m and n are coprime members of the sequence, then m*n is also a member. - Robert Israel, May 10 2018

			Divisors(11811) = {1,3,31,93,127,381,3937,11811} and sigma(11811) = 16384 = 4^7.

Donovan Johnson, Table of n, a(n) for n = 1..1000
Frits Beukers, Florian Luca and Frans Oort, Power Values of Divisor Sums, The American Mathematical Monthly, Vol. 119, No. 5 (May 2012), pp. 373-380.

Cf. A000203, A006532, A020477, A019422, A019423, A019424, A048251-A048256, A048258.

filter:= n -> type(map(t -> t[2]/7, ifactors(numtheory:-sigma(n))[2]),list(integer)):
select(filter, [$1..21*10^4]); # Robert Israel, May 09 2018

Select[Range[210000],IntegerQ[Surd[DivisorSigma[1,#],7]]&] (* Harvey P. Dale, Jun 09 2017 *)

isok(n) = ispower(sigma(n), 7); \\ Michel Marcus, Dec 20 2013

sigma(a(n)) = x^7, where the initial values of x are 1, 2, 4, 6 (48 times), ...

A048258 Integers whose sum of divisors is an 8th power.

Original entry on oeis.org

1, 217, 57337, 600270, 621690, 669990, 685290, 693294, 693770, 699810, 725934, 747670, 769930, 774894, 782598, 805970, 813378, 823938, 835670, 839802, 854930, 865490, 873334, 895594, 918435, 920414, 923410, 931634, 935715, 959565, 965174, 969034, 969206

Offset: 1

Labos Elemer

nonn

			Divisors(217) = {1,7,31,217}, sum = 256 = 2^8.
Divisors(57337) = {1,7,8191,57337}, sum = 65536 = 4^8.
Divisors(1676377) = {1,647,2591,1676377}, sum = 1679616 = 6^8.

Donovan Johnson, Table of n, a(n) for n = 1..1000
Frits Beukers, Florian Luca and Frans Oort, Power Values of Divisor Sums, The American Mathematical Monthly, Vol. 119, No. 5 (May 2012), pp. 373-380.

Cf. A006532, A020477, A019422, A019423, A019424, A048251-A048257.

Select[Range[10^6], IntegerQ@ Surd[DivisorSigma[1, #], 8] &] (* Michael De Vlieger, Aug 01 2017, after Harvey P. Dale at A048257 *)

isok(n) = ispower(sigma(n), 8); \\ Michel Marcus, Dec 30 2013

Sigma(1, a(n)) = x^8, where the initial values of x are 1, 2, 4, 6 (occurs 85 times), ...

A303993 Numbers whose sum of divisors is the cube of one of their divisors.

Original entry on oeis.org

1, 102, 8148, 63720, 66120, 71880, 196896, 446040, 452760, 462840, 471960, 503160, 517320, 544920, 549240, 554280, 559320, 575880, 756400, 1458912, 1499232, 1579872, 1634040, 1659960, 1748520, 5294800, 9740640, 10103520, 11103456, 11438280, 11583264, 11619720, 11915640

Offset: 1

Paolo P. Lava, May 04 2018

Subset of A020477.

			Divisors of 102 are 1, 2, 3, 6, 17, 34, 51, 102 and their sum is 216 = 6^3.

Cf. A000203, A020477, A303123, A303994, A303995, A303996.

with(numtheory): P:=proc(q) local a,k,n;
for n from 1 to q do a:=sort([op(divisors(n))]);
for k from 1 to nops(a) do if sigma(n)=a[k]^3 then print(n); break; fi; od; od; end: P(10^9);

Select[Range[10^6], Mod[#, DivisorSigma[1, #]^(1/3)] == 0 &] (* Michael De Vlieger, May 06 2018 *)

isok(n) = (n==1) || (ispower(s=sigma(n), 3) && !(n % sqrtnint(s, 3))); \\ Michel Marcus, May 05 2018

A048252 Largest number whose sum of divisors is 6^n.

Original entry on oeis.org

1, 5, 22, 187, 1219, 7597, 46117, 278857, 1676377, 10067797, 60450517, 362758177, 2176626817, 13060193977, 78363525817, 470183516857, 2820894903487, 16926601754197, 101559860054047, 609359671998037, 3656158318966357

Offset: 0

Labos Elemer

nonn

Terms of this sequence are products of distinct terms in A005105. - Ray Chandler, Sep 01 2010

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

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

a(n) = {sn = 6^n; forstep(x=sn, 1, -1, if (sigma(x) == sn, return (x)););} \\ Michel Marcus, Dec 15 2013

a(9)-a(14) from Donovan Johnson, Sep 02 2008
a(15)-a(20) from Donovan Johnson, Nov 22 2008
Edited and extended by Ray Chandler, Sep 01 2010

cp's OEIS Frontend

A006532 Numbers whose sum of divisors is a square.

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A048251 a(n) is the smallest number whose sum of divisors is 6^n.

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A019423 Numbers whose sum of divisors is a fifth power.

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A048699 Nonprime numbers whose sum of aliquot divisors (A001065) is a perfect square.

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A019424 Numbers whose sum of divisors is a sixth power.

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A048256 Numbers whose sum of divisors is 6^6 = 46656.

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A048257 Integers whose sum of divisors is a 7th power.

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