A333911 Numbers k such that sigma(k) is the sum of 2 squares, where sigma is the sum of divisors function (A000203).

1, 3, 7, 9, 10, 17, 19, 21, 22, 27, 30, 31, 40, 46, 51, 52, 55, 57, 58, 63, 66, 67, 70, 71, 73, 79, 81, 88, 89, 90, 93, 94, 97, 103, 106, 115, 118, 119, 120, 127, 133, 138, 145, 153, 154, 156, 163, 165, 170, 171, 174, 179, 184, 189, 190, 193, 198, 199, 201, 202

Offset

1,2

Links

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

William D. Banks, Florian Luca, Filip Saidak, and Igor E. Shparlinski, Values of arithmetical functions equal to a sum of two squares, Quarterly Journal of Mathematics, Vol. 56, No. 2 (2005), pp. 123-139, alternative link.

Formula

c1 * x/log(x)^(3/2) < N(x) < c2 * x/log(x)^(3/2), where N(x) is the number of terms <= x, and c1 and c2 are two positive constants (Banks et al., 2005).

Example

1 is a term since sigma(1) = 1 = 0^2 + 1^2.

Mathematica

Select[Range[200], SquaresR[2, DivisorSigma[1, #]] > 0 &]

Prog

(Python)
from itertools import count, islice
from collections import Counter
from sympy import factorint
def A333911_gen(): # generator of terms
    return filter(lambda n:all(p & 3 != 3 or e & 1 == 0 for p, e in sum((Counter(factorint((p**(e+1)-1)//(p-1))) for p, e in factorint(n).items()), start=Counter()).items()), count(1))
A333911_list = list(islice(A333911_gen(), 30)) # Chai Wah Wu, Jun 27 2022

Crossrefs

Cf. A000203, A001481, A079545, A272405, A333909, A333910.

Sequence in context: A294571 A083214 A091210 * A276492 A023992 A179197

Adjacent sequences: A333908 A333909 A333910 * A333912 A333913 A333914

Keyword

nonn

Author

Amiram Eldar, Apr 09 2020

Status

approved