A333911 -id:A333911 - cp's OEIS frontend

A333909 Numbers k such that phi(k) is the sum of 2 squares, where phi is the Euler totient function (A000010).

1, 2, 3, 4, 5, 6, 8, 10, 11, 12, 15, 16, 17, 19, 20, 22, 24, 25, 27, 30, 32, 33, 34, 37, 38, 40, 41, 44, 48, 50, 51, 53, 54, 55, 57, 59, 60, 63, 64, 66, 68, 73, 74, 75, 76, 80, 82, 83, 85, 88, 91, 95, 96, 100, 101, 102, 106, 107, 108, 110, 111, 114, 117, 118, 120

Offset: 1

Amiram Eldar, Apr 09 2020

nonn

			1 is a term since phi(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. A000010, A001481, A039770, A079545, A333910, A333911, A333912.

Select[Range[120], SquaresR[2, EulerPhi[#]] > 0 &]

from itertools import count, islice
from sympy import factorint, totient
def A333909_gen(): # generator of terms
    return filter(lambda n:all(p & 3 != 3 or e & 1 == 0 for p, e in factorint(totient(n)).items()),count(1))
A333909_list = list(islice(A333909_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).

A363967 Numbers whose divisors can be partitioned into two disjoint sets whose both sums are squares.

Original entry on oeis.org

1, 3, 9, 22, 27, 30, 40, 63, 66, 70, 81, 88, 90, 94, 115, 119, 120, 138, 153, 156, 170, 171, 174, 184, 189, 190, 198, 210, 214, 217, 232, 264, 265, 270, 280, 282, 310, 318, 322, 323, 343, 345, 357, 360, 364, 376, 382, 385, 399, 400, 414, 416, 462, 468, 472, 495, 497

Offset: 1

Amiram Eldar, Jun 30 2023

nonn

If one of the two sets is empty then the term is a number whose sum of divisors is a square (A006532).
If k is a number such that (6*k)^2 is the sum of a twin prime pair (A037073), then (18*k^2)^2 - 1 is a term.
3 is the only prime term.

			9 is a term since its divisors, {1, 3, 9}, can be partitioned into the two disjoint sets, {1, 3} and {9}, whose sums, 1 + 3 = 4 = 2^2 and 9 = 3^2, are both squares.

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

Cf. A000203, A000290, A037073.
Subsequence of A333911.
A006532 is a subsequence.
Similar sequences: A333677, A360694.

sqQ[n_] := IntegerQ[Sqrt[n]]; q[n_] := Module[{d = Divisors[n], s, p}, s = Total[d]; p = Position[Rest @ CoefficientList[Product[1 + x^i, {i, d}], x], _?(# > 0 &)] // Flatten; AnyTrue[p, sqQ[#] && sqQ[s - #] &]]; Select[Range[500], q]

cp's OEIS Frontend

A333909 Numbers k such that phi(k) is the sum of 2 squares, where phi is the Euler totient function (A000010).

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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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Python

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A363967 Numbers whose divisors can be partitioned into two disjoint sets whose both sums are squares.

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Mathematica