A352185 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.

3, 5, 7, 11, 11, 13, 19, 19, 19, 31, 31, 31, 31, 31, 31, 31, 31, 47, 47, 61, 61, 61, 61, 127, 127, 127, 127, 127, 139, 139, 139, 139, 139, 193, 193, 229, 229, 229, 229, 283, 283, 283, 283, 283, 283, 337, 337, 337, 337, 337, 409, 409, 409, 409, 409, 409, 409, 461

Offset

1,1

Comments

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]

For the corresponding values of k see A352186.

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

Links

Michael S. Branicky, Table of n, a(n) for n = 1..232

Richard K. Guy, What are the smallest arithmetic progressions of composite numbers?, Amer. Math. Monthly, Vol. 93, No. 8 (1986), p. 627.

Formula

a(n+1) >= a(n) by construction. - Michael S. Branicky, Mar 12 2022

Example

For n=1, m=3, k=1, the AP is [4].
For n=4, m=11, k=5, the AP is [16, 27, 38, 49].

Prog

(Python)
from math import gcd
from sympy import isprime
from itertools import count, islice, takewhile
def comp(n): return not isprime(n)
def agen(): # generator of terms
    n = 1
    for m in count(2):
        for k in range(1, m):
            if gcd(k, m) != 1:
                continue
            ap = len(list(takewhile(comp, (i*m+k for i in count(1)))))
            if ap >= n:
                for i in range(n, ap+1):
                    yield m
                n = ap + 1
print(list(islice(agen(), 58)))  # Michael S. Branicky, Mar 12 2022

Crossrefs

Cf. A352186.

Sequence in context: A112070 A208643 A123252 * A066168 A058024 A215464

Adjacent sequences: A352182 A352183 A352184 * A352186 A352187 A352188

Keyword

nonn

Author

N. J. A. Sloane, Mar 12 2022

Extensions

a(24) and beyond from Michael S. Branicky, Mar 12 2022.

Status

approved