A130698 Numbers n such that the sum of the sum-of-divisors function of all integers up to n is a square.

1, 2, 53, 174, 299, 1377, 12695, 44469, 423922, 2068248, 12084331, 46270468, 1330157705, 5319721272, 144083207084, 178358383042, 212557463865, 2125965021380, 6323865789846, 7277308988873

Offset

1,2

Comments

It is not known if this sequence is infinite.

a(15) > 10^11. - Donovan Johnson, Mar 17 2011

The standard heuristic suggests this sequence is infinite with logarithmic density. - Charles R Greathouse IV, Feb 04 2013

Links

Table of n, a(n) for n=1..20.

Formula

n such that sigma(1)+sigma(2)+...+sigma(n) = x^2 with some integer x.

Example

a(2) = 2 since sigma(1) = 1^2 and sigma(1)+sigma(2) = 2^2.

Mathematica

L=0; For[j = 1, j < 1000000, j++, L = L + DivisorSigma[1, j]; If[L == Floor[Sqrt[L]]^2, Print[j, " ", Floor[Sqrt[L]]]]]

Prog

(PARI) s=0; for(n=1, 1e6, if(issquare(s+=sigma(n)), print1(n", "))) \\ Charles R Greathouse IV, Feb 04 2013

Crossrefs

Cf. A024916, A062407 (for the resulting squares).

Sequence in context: A249661 A249662 A249663 * A364080 A248988 A248989

Adjacent sequences: A130695 A130696 A130697 * A130699 A130700 A130701

Keyword

nonn,more

Author

Florian Luca (fluca(AT)matmor.unam.mx), Jul 11 2007

Extensions

a(10)-a(13) from Donovan Johnson, Dec 02 2009

a(14) from Donovan Johnson, Mar 17 2011

Name revised by Charles R Greathouse IV, Feb 04 2013

a(15)-a(20) from Giovanni Resta, Nov 22 2019

Status

approved