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A383578 Let p = prime(n), then a(n) is the p-smooth part of (p-1)!+1.

%I A383578 #29 May 15 2025 08:24:13
%S A383578 2,3,25,7,11,169,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,
%T A383578 89,97,101,103,107,109,113,127,131,137,139,149,151,157,163,167,173,
%U A383578 179,181,191,193,197,199,211,223,227,229,233,239,241,251,257,263,269,271,277,281,283,293
%N A383578 Let p = prime(n), then a(n) is the p-smooth part of (p-1)!+1.
%C A383578 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 A383578 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 A383578 Let p = prime(n). If m<p, then (p-1)!+1 == 1 mod m, so a(n) = p-adic valuation of (p-1)!+1. By Wilson's theorem, a(n)>=p. Conjecture: a(n) = p^2 if n = 3, 6 or 103 and a(n) = p otherwise. - _Chai Wah Wu_, May 11 2025
%F A383578 a(n) = ((prime(n) - 1)! + 1) / A383257(n).
%e A383578 a(6) = 169 because the prime factorization of ((13 - 1)! + 1) is 13^2*2834329, and 13^2 = 169.
%o A383578 (PARI) a(n) = my(p=prime(n), x=(p-1)! + 1, f=factor((p-1)! + 1, nextprime(p+1))); for (i=1, #f~, if (f[i, 1] <= p, f[1, 1] = 1)); x/factorback(f); \\ _Michel Marcus_, Apr 30 2025
%o A383578 (Python)
%o A383578 from sympy import prime, factorial
%o A383578 def A383578(n):
%o A383578     p, c = prime(n), 1
%o A383578     f = factorial(p-1)+1
%o A383578     a, b = divmod(f,p)
%o A383578     while not b:
%o A383578         c *= p
%o A383578         f = a
%o A383578         a, b = divmod(f,p)
%o A383578     return c # _Chai Wah Wu_, May 12 2025
%Y A383578 Cf. A007540, A060371, A383257.
%K A383578 nonn
%O A383578 1,1
%A A383578 _Mike Jones_, Apr 30 2025
%E A383578 More terms from _Michel Marcus_, Apr 30 2025