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a(n) = 1 if A002034(n), the smallest positive integer k such that n divides k!, is larger than A006530(n), the greatest prime factor of n, otherwise 0.
3

%I #12 Jul 04 2021 18:08:42

%S 0,0,0,1,0,0,0,1,1,0,0,1,0,0,0,1,0,1,0,0,0,0,0,1,1,0,1,0,0,0,0,1,0,0,

%T 0,1,0,0,0,0,0,0,0,0,1,0,0,1,1,1,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,

%U 0,0,0,1,0,0,1,0,0,0,0,1,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,1,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1

%N a(n) = 1 if A002034(n), the smallest positive integer k such that n divides k!, is larger than A006530(n), the greatest prime factor of n, otherwise 0.

%C a(n) = 1 if n does not divide A000142(A006530(n)), and 0 otherwise.

%C a(n) = 1 if A002034(n), the smallest positive integer k such that n divides k!, is composite, 0 otherwise.

%H Antti Karttunen, <a href="/A345951/b345951.txt">Table of n, a(n) for n = 1..65537</a>

%H <a href="/index/Ch#char_fns">Index entries for characteristic functions</a>

%H <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>

%F a(n) = 1 if A345940(n) > 0, otherwise 0.

%F a(n) <= A345952(n) for all n >= 1.

%o (PARI)

%o A006530(n) = if(1==n, n, my(f=factor(n)); f[#f~, 1]);

%o A345951(n) = !!(A006530(n)!%n);

%o (PARI)

%o 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 A345951(n) = (A006530(n) < A002034(n));

%o (PARI) A345951(n) = ((n>1)&&!isprime(A002034(n)));

%Y Characteristic function of A057109.

%Y Cf. A000142, A002034, A006530, A345940, A345952.

%K nonn

%O 1

%A _Antti Karttunen_, Jul 04 2021