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A345935 Number of divisors d of n for which A002034(d) = A002034(n), where A002034(n) is the smallest positive integer k such that n divides k!.

%I A345935 #16 Jul 04 2021 22:05:27
%S A345935 1,1,1,1,1,2,1,2,1,2,1,2,1,2,2,1,1,2,1,3,2,2,1,4,1,2,1,3,1,4,1,1,2,2,
%T A345935 2,3,1,2,2,4,1,4,1,3,2,2,1,2,1,2,2,3,1,2,2,4,2,2,1,6,1,2,3,2,2,4,1,3,
%U A345935 2,4,1,4,1,2,2,3,2,4,1,2,2,2,1,6,2,2,2,4,1,4,2,3,2,2,2,2,1,2,3,3,1,4,1,4,4
%N A345935 Number of divisors d of n for which A002034(d) = A002034(n), where A002034(n) is the smallest positive integer k such that n divides k!.
%H A345935 Antti Karttunen, <a href="/A345935/b345935.txt">Table of n, a(n) for n = 1..65537</a>
%H A345935 <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>
%F A345935 a(n) = Sum_{d|n} [A002034(d) = A002034(n)], where [ ] is the Iverson bracket.
%F A345935 a(n) = A000005(n) - A345936(n).
%F A345935 a(n) <= A345934(n).
%e A345935 36 has 9 divisors: 1, 2, 3, 4, 6, 9, 12, 18, 36. When A002034 is applied to them, one obtains values [1, 2, 3, 4, 3, 6, 4, 6, 6], thus there are three divisors that obtain the maximal value 6 obtained at 36 itself, therefore a(36) = 3.
%t A345935 a[n_]:=(m=1;While[Mod[m!,n]!=0,m++];m);Table[Length@Select[Divisors@k,a@#==a@k&],{k,100}] (* _Giorgos Kalogeropoulos_, Jul 03 2021 *)
%o A345935 (PARI)
%o A345935 A002034(n) = if(1==n,n,my(s=factor(n)[, 1], k=s[#s], f=Mod(k!, n)); while(f, f*=k++); (k)); \\ After code in A002034.
%o A345935 A345935(n) = { my(x=A002034(n)); sumdiv(n,d,A002034(d)==x); };
%Y A345935 Cf. A000005, A002034, A345934, A345936, A345944 (positions of 1's), A345945 (of terms > 1), A345950.
%Y A345935 Cf. also A344590.
%K A345935 nonn
%O A345935 1,6
%A A345935 _Antti Karttunen_, Jul 02 2021