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

%I A373725 #13 Jun 16 2024 02:19:22
%S A373725 1,2,3,4,8,9,15,27,63,195,728,1443,3843,5475,6174,11913,13376,24963,
%T A373725 37635,77283,98595,113398,158403,178083,209763,293763,294335,319124,
%U A373725 376995,406503,438243,454275,538755,574563,770883,996003,1196835,1331715,1444803,1473795
%N A373725 Numbers k such that A011776(k) = A011776(k+1).
%C A373725 The corresponding values of A011776 are 1, 1, 1, 1, 2, 2, 3, 4, 10, 16, 60, ... .
%C A373725 All the terms above 3 are composite numbers since A011776(k) = 1 if and only if k = 4 or a prime.
%C A373725 Are there 3 consecutive integers above 8 that have an equal value of A011776? There are none below 10^10.
%C A373725 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.
%H A373725 Amiram Eldar, <a href="/A373725/b373725.txt">Table of n, a(n) for n = 1..1000</a>
%t A373725 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]
%o A373725 (PARI) lista(kmax) = {my(s1 = 1, s2); for(k = 2, kmax, s2 = valuation(k!, k); if(s1 == s2, print1(k-1, ", ")); s1 = s2);}
%Y A373725 Cf. A005382, A011776.
%K A373725 nonn
%O A373725 1,2
%A A373725 _Amiram Eldar_, Jun 15 2024