A373725 Numbers k such that A011776(k) = A011776(k+1).

1, 2, 3, 4, 8, 9, 15, 27, 63, 195, 728, 1443, 3843, 5475, 6174, 11913, 13376, 24963, 37635, 77283, 98595, 113398, 158403, 178083, 209763, 293763, 294335, 319124, 376995, 406503, 438243, 454275, 538755, 574563, 770883, 996003, 1196835, 1331715, 1444803, 1473795

Offset: 1

Author

Amiram Eldar, Jun 15 2024

Keywords

• nonn

Comments

The corresponding values of A011776 are 1, 1, 1, 1, 2, 2, 3, 4, 10, 16, 60, ... .

All the terms above 3 are composite numbers since A011776(k) = 1 if and only if k = 4 or a prime.

Are there 3 consecutive integers above 8 that have an equal value of A011776? There are none below 10^10.

Conjecture: if p != 3 is a prime such that 2*p-1 is also a prime (p is in A005382 \ {3}), then 4*p^2 - 1 is a term of this sequence.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A005382, A011776.

Programs

Mathematica

s[1] = 1; s[n_] := IntegerExponent[n!, n]; seq[kmax_] := Module[{v = {}, s1 = s[1]}, Do[s2 = s[k]; If[s1 == s2, AppendTo[v, k-1]]; s1 = s2, {k, 2, kmax}]; v]; seq[10^4]

PARI

lista(kmax) = {my(s1 = 1, s2); for(k = 2, kmax, s2 = valuation(k!, k); if(s1 == s2, print1(k-1, ", ")); s1 = s2);}