A019423 -id:A019423 - cp's OEIS frontend

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

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

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), ...

A303995 Numbers whose sum of divisors is the fifth power of one of their divisors.

Original entry on oeis.org

1, 3210, 3498, 3882, 6453804, 7873684, 7943640, 8028120, 8099880, 9112230, 9561990, 10079430, 182626920, 192651480, 196192920, 199939320, 200271960, 201632760, 203289240, 206367480, 206645880, 207815160, 208955160, 210368760, 210406680, 210717720, 211645560

Offset: 1

Paolo P. Lava, May 04 2018

nonn

Subset of A019423.

			Divisors of 3210 are 1, 2, 3, 5, 6, 10, 15, 30, 107, 214, 321, 535, 642, 1070, 1605, 3210 and their sum is 7776 = 6^5.

Cf. A000203, A019423, A303123, A303993, A303994, 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]^5 then print(n); break; fi; od; od; end: P(10^9);

Select[Range[10^4], IntegerQ[t = DivisorSigma[1, #]^(1/5)] && Mod[#, t] == 0 &] (* Giovanni Resta, May 04 2018 *)

isok(n) = (n==1) || (ispower(s=sigma(n), 5) && !(n % sqrtnint(s, 5))); \\ Michel Marcus, May 05 2018

a(13)-a(27) from Giovanni Resta, May 04 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

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

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

cp's OEIS Frontend

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

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A019424 Numbers whose sum of divisors is a sixth power.

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Magma

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

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Mathematica

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

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

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A303995 Numbers whose sum of divisors is the fifth power of one of their divisors.

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Maple

Mathematica

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

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Extensions

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