A333909 -id:A333909 - cp's OEIS frontend

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

Amiram Eldar, Apr 09 2020

nonn

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

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.

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

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

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

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).

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

Original entry on oeis.org

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

Amiram Eldar, Apr 09 2020

nonn

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

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.

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

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

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

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).

A333912 Numbers k such that phi(k) is not the sum of 3 squares, where phi is the Euler totient function (A000010).

Original entry on oeis.org

29, 58, 61, 77, 93, 99, 113, 122, 124, 141, 145, 154, 157, 169, 186, 188, 198, 226, 232, 237, 241, 253, 282, 287, 290, 301, 305, 314, 316, 317, 325, 338, 348, 349, 363, 369, 381, 385, 387, 413, 429, 441, 449, 465, 474, 482, 484, 488, 493, 495, 496, 506, 508, 509

Offset: 1

Amiram Eldar, Apr 09 2020

nonn

Pollack (2011) proved that the complementary sequence has asymptotic density 7/8. Therefore the asymptotic density of this sequence is 1/8.

			1 is not a term since phi(1) = 1 = 0^2 + 0^2 + 1^2 is the sum of 3 squares.
29 is a term since phi(29) = 28 is not the sum of 3 squares.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Paul Pollack, Values of the Euler and Carmichael functions which are sums of three squares, Integers, Vol. 11 (2011), pp. 145-161.

Cf. A000010, A004215, A039770, A272405, A333909, A333913.

Select[Range[500], SquaresR[3, EulerPhi[#]] == 0 &]

cp's OEIS Frontend

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

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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).

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A333912 Numbers k such that phi(k) is not the sum of 3 squares, where phi is the Euler totient function (A000010).

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