A332511 -id:A332511 - cp's OEIS frontend

A332512 Numbers k such that phi(k) == 0 (mod 12), where phi is the Euler totient function (A000010).

13, 21, 26, 28, 35, 36, 37, 39, 42, 45, 52, 56, 57, 61, 63, 65, 70, 72, 73, 74, 76, 77, 78, 84, 90, 91, 93, 95, 97, 99, 104, 105, 108, 109, 111, 112, 114, 117, 119, 122, 124, 126, 129, 130, 133, 135, 140, 143, 144, 146, 147, 148, 152, 153, 154, 155, 156, 157, 161

Offset: 1

Amiram Eldar, Feb 14 2020

nonn

Dence and Pomerance showed that the asymptotic number of the terms below x is ~ x.

			13 is a term since phi(13) = 12 == 0 (mod 12).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Thomas Dence and Carl Pomerance, Euler's function in residue classes, in: K. Alladi, P. D. T. A. Elliott, A. Granville and G. Tenenbaum (eds.), Analytic and Elementary Number Theory, Developments in Mathematics, Vol. 1, Springer, Boston, MA, 1998, pp. 7-20, alternative link.

Cf. A000010, A008594, A332511, A332513, A332514, A332515, A332516.

[k:k in [1..170]| EulerPhi(k) mod 12 eq 0]; // Marius A. Burtea, Feb 14 2020

Select[Range[200], Divisible[EulerPhi[#], 12] &]

A332513 Numbers k such that phi(k) == 4 (mod 12), where phi is the Euler totient function (A000010).

Original entry on oeis.org

5, 8, 10, 12, 17, 29, 32, 34, 40, 41, 48, 53, 55, 58, 60, 75, 82, 85, 88, 89, 100, 101, 106, 110, 113, 115, 125, 128, 132, 136, 137, 145, 149, 150, 160, 170, 173, 178, 184, 187, 192, 197, 202, 204, 205, 226, 230, 232, 233, 235, 240, 250, 253, 257, 265, 269, 274

Offset: 1

Amiram Eldar, Feb 14 2020

nonn

Dence and Pomerance showed that the asymptotic number of the terms below x is ~ c1 * x/sqrt(log(x)), where c1 = (sqrt(2*sqrt(3))/(3*Pi)) * c3^(-1/2) * (2*c3 + c4) = 0.6109136202..., c3 = Product_{primes p == 2 (mod 3)} (1 + 1/(p^2-1)), and c4 = Product_{primes p == 2 (mod 3)} (1 - 1/(p+1)^2).

			17 is a term since phi(17) = 16 == 4 (mod 12).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Thomas Dence and Carl Pomerance, Euler's function in residue classes, in: K. Alladi, P. D. T. A. Elliott, A. Granville and G. Tenenbaum (eds.), Analytic and Elementary Number Theory, Developments in Mathematics, Vol. 1, Springer, Boston, MA, 1998, pp. 7-20, alternative link.

Cf. A000010, A017569, A175646, A332511, A332512, A332514, A332515, A332516.

[k:k in [1..300]| EulerPhi(k) mod 12 eq 4]; // Marius A. Burtea, Feb 14 2020

Select[Range[300], Mod[EulerPhi[#], 12] == 4 &]

A332514 Numbers k such that phi(k) == 6 (mod 12), where phi is the Euler totient function (A000010).

Original entry on oeis.org

7, 9, 14, 18, 19, 27, 31, 38, 43, 49, 54, 62, 67, 79, 81, 86, 98, 103, 127, 134, 139, 151, 158, 162, 163, 199, 206, 211, 223, 243, 254, 271, 278, 283, 302, 307, 326, 331, 343, 361, 367, 379, 398, 422, 439, 446, 463, 486, 487, 499, 523, 542, 547, 566, 571, 607, 614

Offset: 1

Amiram Eldar, Feb 14 2020

nonn

Dence and Pomerance showed that the asymptotic number of the terms below x is ~ (3/8) * x/log(x).

			19 is a term since phi(19) = 18 == 6 (mod 12).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Thomas Dence and Carl Pomerance, Euler's function in residue classes, in: K. Alladi, P. D. T. A. Elliott, A. Granville and G. Tenenbaum (eds.), Analytic and Elementary Number Theory, Developments in Mathematics, Vol. 1, Springer, Boston, MA, 1998, pp. 7-20, alternative link.

Cf. A000010, A017593, A332511, A332512, A332513, A332515, A332516.

[k:k in [1..650]| EulerPhi(k) mod 12 eq 6]; // Marius A. Burtea, Feb 14 2020

Select[Range[600], Mod[EulerPhi[#], 12] == 6 &]

A332515 Numbers k such that phi(k) == 8 (mod 12), where phi is the Euler totient function (A000010).

Original entry on oeis.org

15, 16, 20, 24, 25, 30, 33, 44, 50, 51, 64, 66, 68, 69, 80, 87, 92, 96, 102, 116, 120, 123, 138, 141, 159, 164, 165, 174, 176, 177, 188, 200, 212, 213, 220, 236, 246, 249, 255, 256, 264, 267, 272, 275, 282, 284, 289, 300, 303, 318, 320, 321, 330, 332, 339, 340

Offset: 1

Amiram Eldar, Feb 14 2020

nonn

Dence and Pomerance showed that the asymptotic number of the terms below x is ~ c2 * x/sqrt(log(x)), where c2 = (sqrt(2*sqrt(3))/(3*Pi)) * c3^(-1/2) * (2*c3 - c4) = 0.3284176245..., c3 = Product_{primes p == 2 (mod 3)} (1 + 1/(p^2-1)), and c4 = Product_{primes p == 2 (mod 3)} (1 - 1/(p+1)^2).

			25 is a term since phi(25) = 20 == 8 (mod 12).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Thomas Dence and Carl Pomerance, Euler's function in residue classes, in: K. Alladi, P. D. T. A. Elliott, A. Granville and G. Tenenbaum (eds.), Analytic and Elementary Number Theory, Developments in Mathematics, Vol. 1, Springer, Boston, MA, 1998, pp. 7-20, alternative link.

Cf. A000010, A017617, A175646, A332511, A332512, A332513, A332514, A332516.

[k:k in [1..350]| EulerPhi(k) mod 12 eq 8]; // Marius A. Burtea, Feb 14 2020

Select[Range[400], Mod[EulerPhi[#], 12] == 8 &]

A332516 Numbers k such that phi(k) == 10 (mod 12), where phi is the Euler totient function (A000010).

Original entry on oeis.org

11, 22, 23, 46, 47, 59, 71, 83, 94, 107, 118, 131, 142, 166, 167, 179, 191, 214, 227, 239, 251, 262, 263, 311, 334, 347, 358, 359, 382, 383, 419, 431, 443, 454, 467, 478, 479, 491, 502, 503, 526, 563, 587, 599, 622, 647, 659, 683, 694, 718, 719, 743, 766, 827

Offset: 1

Amiram Eldar, Feb 14 2020

nonn

Dence and Pomerance showed that the asymptotic number of the terms below x is ~ (3/8) * x/log(x).

			23 is a term since phi(23) = 22 == 10 (mod 12).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Thomas Dence and Carl Pomerance, Euler's function in residue classes, in: K. Alladi, P. D. T. A. Elliott, A. Granville and G. Tenenbaum (eds.), Analytic and Elementary Number Theory, Developments in Mathematics, Vol. 1, Springer, Boston, MA, 1998, pp. 7-20, alternative link.

Cf. A000010, A017641, A332511, A332512, A332513, A332514, A332515.

[k:k in [1..850]| EulerPhi(k) mod 12 eq 10]; // Marius A. Burtea, Feb 14 2020

Select[Range[800], Mod[EulerPhi[#], 12] == 10 &]

cp's OEIS Frontend

A332512 Numbers k such that phi(k) == 0 (mod 12), where phi is the Euler totient function (A000010).

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A332513 Numbers k such that phi(k) == 4 (mod 12), where phi is the Euler totient function (A000010).

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A332514 Numbers k such that phi(k) == 6 (mod 12), where phi is the Euler totient function (A000010).

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A332515 Numbers k such that phi(k) == 8 (mod 12), where phi is the Euler totient function (A000010).

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Magma

Mathematica

A332516 Numbers k such that phi(k) == 10 (mod 12), where phi is the Euler totient function (A000010).

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Magma

Mathematica