A008849 Numbers n such that the sum of divisors of n^3 is a square.

1, 7, 751530, 4730879, 5260710, 33116153, 37200735, 187062910, 226141311, 259109835, 260405145, 370049418, 522409465, 836308083, 1105725765, 1309440370, 1343713507, 1582989177, 1609505430, 1813768845, 2590345926, 3039492538, 3656866255

Offset

1,2

Comments

In 1657 Fermat challenged the world to find such numbers. [Dickson, Vol. 1, p. 54]

If n is a term and n is not divisible by 7, then 7*n is a term. - Don Dechman (dondechman_2000(AT)yahoo.com), Mar 26 2008

References

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 9.

L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 1, p. 54.

Julian Havil, Gamma, Exploring Euler's Constant, Princeton University Press, Princeton and Oxford, 2003, page 92.

I. Kaplansky, The challenges of Fermat, Wallis and Ozanam (and several related challenges): I. Fermat's first challenge, Preprint, 2002.

Links

Donovan Johnson, Table of n, a(n) for n = 1..40 (terms < 10^11)

Donovan Johnson, 73688 terms > 10^11

Eric Weisstein's World of Mathematics, Fermat's Divisor Problem.

Mathematica

max = 10^11; primes = {5, 7, 11, 13, 17, 19, 23, 31, 41, 43, 47, 83, 89, 191, 193, 239, 307, 443, 463, 499, 557, 701, 743, 1087, 1487, 2309, 3583, 4373, 5087, 5507, 5807, 44179}; subs = Select[ Times @@@ Subsets[primes, 7], # < max &] // Sort; f[e2_, e3_, p_] := If[n = 2^e2*3^e3*p; IntegerQ[ Sqrt[ DivisorSigma[1, n^3]]], Print[{2^e2, 3^e3, p}]; Sow[n]]; r = Reap[ Scan[ ((f[0, 0, #]; f[0, 1, #]; f[0, 3, #]; f[1, 0, #]; f[1, 1, #]; f[1, 3, #]; f[3, 0, #]; f[3, 1, #]; f[3, 3, #])& ), subs]][[2, 1]]; Select[r, # < max &] // Union (* Jean-François Alcover, Sep 07 2012, after Donovan Johnson *)

Prog

(PARI) is(n)=issquare(sigma(n^3)) \\ Charles R Greathouse IV, Jun 20 2013
(Python)
from functools import reduce
from operator import mul
from sympy import factorint, integer_nthroot
A008849_list, n = [], 1
while n < 10**7:
    fs = factorint(n)
    if integer_nthroot(reduce(mul, ((p**(3*fs[p]+1)-1)//(p-1) for p in fs), 1), 2)[1]:
        A008849_list.append(n)
    n += 1 # Chai Wah Wu, Apr 05 2021

Crossrefs

Cf. A046872, A008850, A048948.

Sequence in context: A138878 A050938 A246562 * A358812 A076914 A362152

Adjacent sequences: A008846 A008847 A008848 * A008850 A008851 A008852

Keyword

nonn,nice

Author

N. J. A. Sloane

Extensions

David W. Wilson has supplied terms a(4) = 4730879 and beyond and verified completeness up to a(3) = 751530

I. Kaplansky and Will Jagy have verified that there are no other terms below 3.8*10^9

3656866255 added by Don Dechman, Mar 26 2008

Status

approved