A377201 - OEIS

%I #32 Nov 18 2024 22:30:59

%S 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,

%T 73,79,83,85,89,91,97,101,103,105,107,109,113,120,121,127,131,132,137,

%U 139,149,151,157,163,167,173,179,181,191,193,197,199,211,223,227,229,231,233,239,241

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

%C From _Jianing Song_, Nov 17 2024: (Start)

%C Includes all primes.

%C 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.

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

%H Amiram Eldar, <a href="/A377201/b377201.txt">Table of n, a(n) for n = 1..10000</a>

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

%t Select[Range[300], PowerMod[DivisorSigma[0, #], #, #] == Mod[DivisorSigma[0, #], #] &] (* _Stefano Spezia_, Oct 20 2024 *)

%o (Magma) [k: k in [1..300] | #Divisors(k)^k mod k eq #Divisors(k) mod k];

%o (PARI) isA377201(k) = my(d = numdiv(k)); Mod(d,k)^k == d

%Y Cf. A000005, A014127.

%K nonn,easy,changed

%O 1,2

%A _Juri-Stepan Gerasimov_, Oct 19 2024

%E Terms 1, 2 prepended by _Jianing Song_, Nov 17 2024