A333910 Numbers k such that psi(k) is the sum of 2 squares, where psi is the Dedekind psi function (A001615).

1, 3, 7, 10, 17, 18, 19, 20, 21, 22, 27, 30, 31, 36, 40, 44, 45, 46, 50, 51, 55, 57, 58, 60, 66, 67, 70, 71, 72, 73, 79, 80, 88, 89, 92, 93, 94, 97, 99, 100, 103, 106, 115, 116, 118, 119, 120, 126, 127, 132, 133, 138, 140, 144, 145, 150, 154, 160, 162, 163, 165

Offset: 1

Author

Amiram Eldar, Apr 09 2020

Keywords

• nonn

Examples

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

Crossrefs

Cf. A001481, A001615, A079545, A333909, A333911.

Programs

Mathematica

psi[1] = 1; psi[n_] := n * Times @@ (1 + 1/Transpose[FactorInteger[n]][[1]]); Select[Range[200], SquaresR[2, psi[#]] > 0 &]

Python

from itertools import count, islice
from collections import Counter
from sympy import factorint
def A333910_gen(): # generator of terms
    return filter(lambda n:all(p & 3 != 3 or e & 1 == 0 for p, e in sum((Counter(factorint(1+p))+Counter({p:e-1}) for p ,e in factorint(n).items()),start=Counter()).items()),count(1))
A333910_list = list(islice(A333910_gen(),30)) # Chai Wah Wu, Jun 27 2022

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).