A354712 -id:A354712 - cp's OEIS frontend

A272872 Numbers k such that k+1 is divisible by number of divisors of k.

1, 3, 5, 7, 11, 13, 15, 17, 19, 23, 27, 29, 31, 35, 37, 39, 41, 43, 47, 51, 53, 55, 59, 61, 67, 71, 73, 79, 83, 87, 89, 91, 95, 97, 101, 103, 107, 109, 111, 113, 115, 119, 123, 127, 131, 135, 137, 139, 143, 149, 151, 155, 157, 159, 163, 167

Offset: 1

Altug Alkan, May 11 2016

nonn

Inspired by A272353.
All odd primes are obvious members.
Numbers k such that k == -1 (mod A000005(k)). Nonprime terms are listed in A354714. - Max Alekseyev, Jun 04 2022
63 is the least number that is not in this sequence but is a member of A187929.

			15 is a term because A000005(15) = 4 divides 15+1 = 16.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Claudia A. Spiro, How Often Does the Number of Divisors of an Integer Divide Its Successor?, J. London Math. Soc. (1985) s2-31 (1): 30-40.

Cf. A000005, A187929, A272353, A033950, A354711, A354712, A354714, A354715, A354716.

Select[Range@167, Mod[#+1, DivisorSigma[0, #]] == 0 &] (* Giovanni Resta, May 21 2016 *)

lista(nn) = {for(n=1, nn, if((n+1) % numdiv(n) == 0, print1(n, ", ")));}

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

Original entry on oeis.org

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.

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

A272872 Numbers k such that k+1 is divisible by number of divisors of k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

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

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Python

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

Views

Author

Keywords

Comments

Links

Crossrefs

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

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Links

Crossrefs