A332512 -id:A332512 - cp's OEIS frontend

A332511 Numbers k such that phi(k) == 2 (mod 12), where phi is the Euler totient function (A000010).

3, 4, 6, 121, 242, 529, 1058, 2209, 3481, 4418, 5041, 6889, 6962, 10082, 11449, 13778, 14641, 17161, 22898, 27889, 29282, 32041, 34322, 36481, 51529, 55778, 57121, 63001, 64082, 69169, 72962, 96721, 103058, 114242, 120409, 126002, 128881, 138338, 146689, 175561

Offset: 1

Amiram Eldar, Feb 14 2020

nonn

Dence and Dence noted that the values of phi(k) congruent to 2 (mod 12) are sparse compared to the other possible even values. For example, for k <= 10^4 there only 10 values of phi(k) congruent to 2 (mod 12), compared to 6114, 1650, 511, 1233, and 476 values congruent to 0, 4, 6, 8, and 10 (mod 12), respectively. They proved that the asymptotic density of this sequence is 0 by showing that the only terms above 6 are of the form p^e and 2*p^e with p == 11 (mod 12) a prime and e even.

Dence and Pomerance showed that the asymptotic number of the terms below x is ~ (1/2 + 1/(2*sqrt(2)))*sqrt(x)/log(x).

			121 is a term since phi(121) = 110 == 2 (mod 12).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Joseph B. Dence and Thomas P. Dence, A Surprise Regarding the Equation phi(x) = 2(6n + 1), The College Mathematics Journal, Vol. 26, No. 4 (1995), pp. 297-301.
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, A017545, A201488 (coefficient in asymptotic formula), A332512, A332513, A332514, A332515, A332516.

[k:k in [1..180000]| EulerPhi(k) mod 12 eq 2]; // Marius A. Burtea, Feb 14 2020

Select[Range[2*10^5], Mod[EulerPhi[#], 12] == 2 &]

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 &]

A358043 Numbers k such that phi(k) is a multiple of 8.

Original entry on oeis.org

15, 16, 17, 20, 24, 30, 32, 34, 35, 39, 40, 41, 45, 48, 51, 52, 55, 56, 60, 64, 65, 68, 70, 72, 73, 75, 78, 80, 82, 84, 85, 87, 88, 89, 90, 91, 95, 96, 97, 100, 102, 104, 105, 110, 111, 112, 113, 115, 116, 117, 119, 120, 123, 128, 130, 132, 135, 136, 137, 140, 143

Offset: 1

Darío Clavijo, Oct 26 2022

nonn

Cf. A000010 (phi), A053574 (its 2-adic valuation), A037074 (a subsequence).

Totient multiples: A066498 (3), A172019 (4), A066500 (5), A066502 (7), A332512 (12).

Select[Range[150], Divisible[EulerPhi[#], 8] &] (* Amiram Eldar, Oct 27 2022 *)

isok(k) = Mod(eulerphi(k), 8) == 0; \\ Michel Marcus, Oct 27 2022

from sympy.ntheory import totient
def isok(n): return totient(n) % 8 == 0

A000010(a(n)) == 0 (mod 8).

cp's OEIS Frontend

A332511 Numbers k such that phi(k) == 2 (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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A332516 Numbers k such that phi(k) == 10 (mod 12), where phi is the Euler totient function (A000010).

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A358043 Numbers k such that phi(k) is a multiple of 8.

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