A377201 Numbers k such that d(k)^k == d(k) (mod k), where d(k) = A000005(k) is the number of divisors of k.

1, 2, 3, 5, 6, 7, 11, 13, 15, 17, 19, 23, 29, 31, 36, 37, 41, 43, 47, 53, 56, 59, 61, 67, 71, 73, 79, 83, 85, 89, 91, 97, 101, 103, 105, 107, 109, 113, 120, 121, 127, 131, 132, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 231, 233, 239, 241

Offset

1,2

Comments

From Jianing Song, Nov 17 2024: (Start)

Includes all primes.

For a prime p and e with (p,e) != (2,1), p^e is a term if and only if e+1 is an order-e base-p Wieferich prime, i.e., (e+1)^(p-1) == 1 (mod p^e). For example, p^2 is a term if and only if p is in A014127.

Are there other square terms than 1, 36, 121, and 1006003^2? (End)

Links

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

Example

6 is in this sequence because d(6) = 4 and 4^6 = 2^12 = 4096 == 4 (mod 6).

Mathematica

Select[Range[300], PowerMod[DivisorSigma[0, #], #, #] == Mod[DivisorSigma[0, #], #] &] (* Stefano Spezia, Oct 20 2024 *)

Prog

(Magma) [k: k in [1..300] | #Divisors(k)^k mod k eq #Divisors(k) mod k];
(PARI) isA377201(k) = my(d = numdiv(k)); Mod(d, k)^k == d

Crossrefs

Cf. A000005, A014127.

Sequence in context: A359397 A098962 A073485 * A062101 A330597 A352336

Adjacent sequences: A377198 A377199 A377200 * A377202 A377203 A377204

Keyword

nonn,easy,changed

Author

Juri-Stepan Gerasimov, Oct 19 2024

Extensions

Terms 1, 2 prepended by Jianing Song, Nov 17 2024

Status

approved