A272872 -id:A272872 - cp's OEIS frontend

A354711 Numbers k such that the number of divisors of k divides k-1.

1, 3, 4, 5, 7, 11, 13, 16, 17, 19, 21, 23, 25, 29, 31, 33, 37, 41, 43, 47, 49, 53, 57, 59, 61, 64, 65, 67, 69, 71, 73, 77, 79, 81, 83, 85, 89, 93, 97, 100, 101, 103, 105, 107, 109, 113, 121, 125, 127, 129, 131, 133, 137, 139, 141, 145, 149, 151, 157, 161, 163, 167, 169, 173, 175, 177, 179, 181

Offset: 1

Max Alekseyev, Jun 03 2022

nonn

Numbers k such that k == 1 (mod A000005(k)).

Every odd prime is a term. Nonprime terms are listed in A354712.

Cf. A000005, A033950, A272872, A354712, A354714, A354715, A354716.

Select[Range[200], Divisible[# - 1, DivisorSigma[0, #]] &] (* Amiram Eldar, Jun 03 2022 *)

A354715 Numbers k such that the number of divisors of k divides k-2.

Original entry on oeis.org

1, 2, 6, 10, 14, 20, 22, 26, 32, 34, 38, 42, 44, 46, 50, 58, 62, 66, 68, 74, 82, 86, 92, 94, 98, 106, 112, 114, 116, 118, 122, 130, 134, 138, 142, 146, 154, 158, 162, 164, 166, 170, 178, 186, 188, 194, 202, 206, 210, 212, 214, 218, 226, 236, 242, 250, 254, 258, 262, 266, 272, 274, 278, 282

Offset: 1

Max Alekseyev, Jun 03 2022

nonn

Numbers k such that k == 2 (mod A000005(k)).
Odd terms are squares. Next odd term after 1 is 5^16 * 29^6 = 90762835845947265625 (cf. A354716).
The smallest even square is 2^16 * 5^6 = 1024000000. - Jianing Song, Jun 04 2022

Jianing Song, Table of n, a(n) for n = 1..10000
Jianing Song, Possible numbers of divisors of terms in A354715

Cf. A000005, A033950, A272872, A354711, A354712, A354714, A354716, A354079.

Select[Range[300], Divisible[# - 2, DivisorSigma[0, #]] &] (* Amiram Eldar, Jun 03 2022 *)

isA354715(k) = (Mod(k,numdiv(k)) == 2) \\ Jianing Song, Jun 04 2022

from sympy import divisor_count
def ok(n): return n > 0 and (n-2)%divisor_count(n) == 0
print([k for k in range(300) if ok(k)]) # Michael S. Branicky, Jun 04 2022

A354716 Odd numbers k such that the number of divisors of k^2 divides k^2-2.

Original entry on oeis.org

1, 9526953125, 151959765625, 334912109375, 356512890625, 478532421875, 878160546875, 2122945380125, 3078358984375, 4941147265625, 8302317578125, 9788873160125, 11750090234375, 18994970703125, 21265601171875, 87362712109375

Offset: 1

Max Alekseyev, Jun 03 2022

nonn

Numbers k^2 are the odd terms of A354715.

Max Alekseyev, Table of n, a(n) for n = 1..7095

Odd terms of A354079.

Cf. A000005, A033950, A272872, A354711, A354712, A354714, A354715.

A354712 Nonprime numbers k such that the number of divisors of k divides k-1.

Original entry on oeis.org

1, 4, 16, 21, 25, 33, 49, 57, 64, 65, 69, 77, 81, 85, 93, 100, 105, 121, 125, 129, 133, 141, 145, 161, 169, 175, 177, 185, 201, 205, 209, 213, 217, 221, 237, 249, 253, 265, 273, 289, 297, 301, 305, 309, 321, 325, 329, 341, 345, 361, 365, 377, 381, 385, 393, 413, 417, 437, 445, 453, 465

Offset: 1

Max Alekseyev, Jun 03 2022

nonn

Nonprime numbers k such that k == 1 (mod A000005(k)).

Nonprime terms of A354711.

Cf. A000005, A033950, A272872, A354714, A354715, A354716.

Select[Range[500], ! PrimeQ[#] && Divisible[# - 1, DivisorSigma[0, #]] &] (* Amiram Eldar, Jun 03 2022 *)

A354714 Nonprime numbers k such that the number of divisors of k divides k+1.

Original entry on oeis.org

1, 15, 27, 35, 39, 51, 55, 87, 91, 95, 111, 115, 119, 123, 135, 143, 155, 159, 183, 187, 203, 215, 219, 231, 235, 245, 247, 255, 259, 267, 275, 287, 291, 295, 299, 303, 319, 323, 327, 335, 339, 343, 351, 355, 371, 375, 391, 395, 399, 403, 407, 411, 415, 425, 427, 447, 451, 455, 471, 511, 515

Offset: 1

Max Alekseyev, Jun 03 2022

nonn

Nonprime numbers k such that k == -1 (mod A000005(k)).

Nonprime terms of A272872.

Cf. A000005, A033950, A354711, A354712, A354715, A354716.

Select[Range[500], ! PrimeQ[#] && Divisible[# + 1, DivisorSigma[0, #]] &] (* Amiram Eldar, Jun 03 2022 *)

A354079 Numbers k such that the number of divisors of k^2 divides k^2-2.

Original entry on oeis.org

1, 32000, 6243584, 14047232, 99588352, 233644288, 313611008, 575511296, 796706816, 2017433344, 2303721472, 2337108224, 3238230272, 3673320192, 5441006848, 7700539136, 7925540864, 9526953125, 13936624384, 16755411712, 19238770688, 34763966464, 52577024000, 57254027008, 60130588928, 67423928576, 87057508352

Offset: 1

Max Alekseyev, Jun 05 2022

nonn

Max Alekseyev, Table of n, a(n) for n = 1..89375

Numbers k such that k^2 is a term of A354715.
The odd terms form A354716.
Cf. A000005, A033950, A272872, A354711, A354712, A354714.

cp's OEIS Frontend

A354711 Numbers k such that the number of divisors of k divides k-1.

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A354715 Numbers k such that the number of divisors of k divides k-2.

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A354716 Odd numbers k such that the number of divisors of k^2 divides k^2-2.

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A354712 Nonprime numbers k such that the number of divisors of k divides k-1.

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A354714 Nonprime numbers k such that the number of divisors of k divides k+1.

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A354079 Numbers k such that the number of divisors of k^2 divides k^2-2.

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