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A371900 Numbers k such that k+1 is composite and A371641(k) != p^2 where p = A020639(k+1) is the smallest prime factor of k+1.

%I A371900 #15 Apr 13 2024 23:17:03
%S A371900 406,766,988,1036,1072,1138,1246,1396,1402,1456,1500,1642,1738,1762,
%T A371900 1768,1816,1918,1926,1942,2076,2116,2158,2182,2278,2506,2716,2746,
%U A371900 2812,2866,2920,2992,3076,3148,3172,3286,3316,3382,3496,3568,3682,3706,3712,3742,3762
%N A371900 Numbers k such that k+1 is composite and A371641(k) != p^2 where p = A020639(k+1) is the smallest prime factor of k+1.
%C A371900 If k+1 is composite, then A371641(k) <= A020639(k+1)^2. This sequence lists numbers k where the inequality is strict.
%H A371900 Chai Wah Wu, <a href="/A371900/b371900.txt">Table of n, a(n) for n = 1..10000</a>
%o A371900 (Python)
%o A371900 from itertools import count, islice
%o A371900 from sympy import isprime, primefactors, factorint, integer_log
%o A371900 def A371900_gen(startvalue=2): # generator of terms >= startvalue
%o A371900     for n in count(max(startvalue,2)):
%o A371900         if not isprime(n+1):
%o A371900             q = min(primefactors(n+1))
%o A371900             for m in range(4,q**2):
%o A371900                 f = factorint(m)
%o A371900                 if sum(f.values()) > 1:
%o A371900                     c = 0
%o A371900                     for p in sorted(f):
%o A371900                         a = pow(n,integer_log(p,n)[0]+1,m)
%o A371900                         for _ in range(f[p]):
%o A371900                             c = (c*a+p)%m
%o A371900                     if not c:
%o A371900                         yield n
%o A371900                         break
%o A371900 A371900_list = list(islice(A371900_gen(),30))
%Y A371900 Cf. A020639, A027746, A259047, A322843, A248915, A371641.
%K A371900 nonn,base
%O A371900 1,1
%A A371900 _Chai Wah Wu_, Apr 11 2024