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A333910
Numbers k such that psi(k) is the sum of 2 squares, where psi is the Dedekind psi function (A001615).
3
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
1,2
LINKS
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 psi(1) = 1 = 0^2 + 1^2.
MATHEMATICA
psi[1] = 1; psi[n_] := n * Times @@ (1 + 1/Transpose[FactorInteger[n]][[1]]); Select[Range[200], SquaresR[2, psi[#]] > 0 &]
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Amiram Eldar, Apr 09 2020
STATUS
approved