A379723 Possible values of the sum of squares of divisors function (A001157).

1, 5, 10, 21, 26, 50, 85, 91, 122, 130, 170, 210, 250, 260, 290, 341, 362, 455, 500, 530, 546, 610, 651, 820, 842, 850, 962, 1050, 1220, 1300, 1365, 1370, 1450, 1682, 1700, 1810, 1850, 1911, 2210, 2366, 2451, 2500, 2562, 2650, 2810, 2900, 3172, 3255, 3410, 3482

Offset: 1

Author

Amiram Eldar, Jan 03 2025

Keywords

• nonn
• easy

Comments

The distinct values of the sigma_2(n) function, in ascending order.

The asymptotic density of this sequence is 0 (Niven, 1951).

5460 = sigma_2(60) and 5461 = sigma_2(64) are two consecutive integers in this sequence. Are there any other such pairs? There are none below 10^10.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems: Inversion of Multiplicative Functions (invphi.gp).
• Ivan Niven, The asymptotic density of sequences, Bull. Amer. Math. Soc., Vol. 57 (1951), pp. 420-434. See Theorem 3, p. 429.

Crossrefs

Cf. A001157, A002191, A002202.

A066872 is a subsequence.

Subsequence of A211347.

Programs

Mathematica

seq[lim_] := Select[Union[DivisorSigma[2, Range[lim]]], # <= lim &]; seq[3500]

PARI

is(n) = invsigmaNum(n, 2) > 0; \\ Amiram Eldar, Jan 03 2025, using Max Alekseyev's invphi.gp