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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
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: A208643 A375345 A123252 * A066168 A058024 A215464
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Mar 12 2022
EXTENSIONS
a(24) and beyond from Michael S. Branicky, Mar 12 2022.
STATUS
approved