The OEIS is supported by the many generous donors to the OEIS Foundation.
%I #44 Mar 21 2022 09:11:16
%S 3,5,7,11,11,13,19,19,19,31,31,31,31,31,31,31,31,47,47,61,61,61,61,
%T 127,127,127,127,127,139,139,139,139,139,193,193,229,229,229,229,283,
%U 283,283,283,283,283,337,337,337,337,337,409,409,409,409,409,409,409,461
%N a(n) = smallest integer m for which there is an integer k, with 0 < k < m and gcd(k,m)=1, such that the arithmetic progression m+k, 2*m+k, ..., n*m+k contains only composite numbers.
%C Richard Guy reports that the sequence originated with Victor Pambuccian, who asks, among other things, if a(n) is always prime. [The answer is no - read on. - _N. J. A. Sloane_, Mar 13 2022]
%C For the corresponding values of k see A352186.
%C a(135) = 8207 is the first nonprime term and there are many other counterexamples to the conjecture: a(150..173) = 12311, a(193..195) = 40247, a(196..202) = 40951, ... . - _Michael S. Branicky_, Mar 13 2022
%H Michael S. Branicky, <a href="/A352185/b352185.txt">Table of n, a(n) for n = 1..232</a>
%H Richard K. Guy, <a href="http://www.jstor.org/stable/2322320">What are the smallest arithmetic progressions of composite numbers?</a>, Amer. Math. Monthly, Vol. 93, No. 8 (1986), p. 627.
%F a(n+1) >= a(n) by construction. - _Michael S. Branicky_, Mar 12 2022
%e For n=1, m=3, k=1, the AP is [4].
%e For n=4, m=11, k=5, the AP is [16, 27, 38, 49].
%o (Python)
%o from math import gcd
%o from sympy import isprime
%o from itertools import count, islice, takewhile
%o def comp(n): return not isprime(n)
%o def agen(): # generator of terms
%o n = 1
%o for m in count(2):
%o for k in range(1, m):
%o if gcd(k, m) != 1:
%o continue
%o ap = len(list(takewhile(comp, (i*m+k for i in count(1)))))
%o if ap >= n:
%o for i in range(n, ap+1):
%o yield m
%o n = ap + 1
%o print(list(islice(agen(), 58))) # _Michael S. Branicky_, Mar 12 2022
%Y Cf. A352186.
%K nonn
%O 1,1
%A _N. J. A. Sloane_, Mar 12 2022
%E a(24) and beyond from _Michael S. Branicky_, Mar 12 2022.