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A383257 Let p = prime(n), then a(n) is the non-p-smooth part of (p-1)!+1.

%I A383257 #53 May 16 2025 03:49:55
%S A383257 1,1,1,103,329891,2834329,1230752346353,336967037143579,
%T A383257 48869596859895986087,10513391193507374500051862069,
%U A383257 8556543864909388988268015483871,10053873697024357228864849950022572972973,19900372762143847179161250477954046201756097561,32674560877973951128910293168477013254334511627907
%N A383257 Let p = prime(n), then a(n) is the non-p-smooth part of (p-1)!+1.
%C A383257 If x is an integer > 1 and p is a prime divisor of x, then a tower of x subordinate to p is an integer t such that there exists a prime divisor q of x such that q <= p, and t is the highest power of q that is a divisor of x.
%C A383257 If (p-1)!+1 = Product_{k} q_k^(e_k), then a(n) = Product_{k>n} q_k^(e_k). - _Sean A. Irvine_, May 05 2025
%C A383257 Let p = prime(n) and k = (p-1)!+1. If m<p, then k == 1 mod m, so a(n) = k/(p-adic valuation of k). By Wilson's theorem, a(n)<=k/p. Conjecture: a(n) = k/p^2 if n = 3, 6, or 103 and a(n) = k/p otherwise. - _Chai Wah Wu_, May 11 2025
%e A383257 a(6) = 2834329 because ((13 - 1)! + 1)/w = (12! + 1)/w = (13^2*2834329)/w = 2834329, where w is the product of the towers of ((13 - 1)! + 1) subordinate to 13, w equaling 13^2.
%o A383257 (PARI) a(n) = my(p=prime(n), f=factor((p-1)! + 1, nextprime(p+1))); for (i=1, #f~, if (f[i,1] <= p, f[1,1] = 1)); factorback(f); \\ _Michel Marcus_, Apr 30 2025
%o A383257 (Python)
%o A383257 from sympy import prime, factorial
%o A383257 def A383257(n):
%o A383257     p = prime(n)
%o A383257     f = factorial(p-1)+1
%o A383257     a, b = divmod(f,p)
%o A383257     while not b:
%o A383257         f = a
%o A383257         a, b = divmod(f,p)
%o A383257     return f # _Chai Wah Wu_, May 12 2025
%Y A383257 Cf. A060371, A383578.
%K A383257 nonn
%O A383257 1,4
%A A383257 _Mike Jones_, Apr 29 2025
%E A383257 More terms from _Michel Marcus_, Apr 30 2025