A138666 Numbers n such that every sum of consecutive positive numbers ending in n is not prime.

1, 8, 14, 18, 20, 25, 26, 28, 32, 33, 35, 38, 39, 44, 46, 48, 50, 56, 58, 60, 62, 63, 65, 68, 72, 74, 77, 78, 80, 81, 85, 86, 88, 92, 93, 94, 95, 98, 102, 104, 105, 108, 110, 111, 116, 118, 119, 122, 123, 124, 125, 128, 130, 133, 134, 138, 140, 143, 144, 145, 146, 148

Offset: 1

Author

T. D. Noe, Mar 26 2008

Keywords

• nice
• nonn

Comments

Also numbers n such that all terms in row n of A087401 are not prime. Also the index of the triangular numbers in A076768. See A087572 for the least prime, if it exists. David Wasserman points out (in A087572) that n is in this sequence if and only if n and 2n-1 are both not prime. This sequence is infinite because 2k^2 is a term for all k>1.

Examples

			8 is in this sequence because 8, 15=7+8, 21=6+7+8, 26=5+6+7+8, 30=4+5+6+7+8, 33=3+4+5+6+7+8, 35=2+3+4+5+6+7+8 and 36=1+2+3+4+5+6+7+8 are all composite.
		

Links

• T. D. Noe, Table of n, a(n) for n=1..1000

Crossrefs

Cf. A010051.

Programs

Haskell

a138666 n = a138666_list !! (n-1)
a138666_list = map (head . tail) $
   filter (all (== 0) . map a010051 . tail) $ drop 2 a087401_tabl
-- Reinhard Zumkeller, Oct 03 2012

Mathematica

Select[Range[200], !PrimeQ[ # ] && !PrimeQ[2#-1] &]
Select[Range[150],AllTrue[Accumulate[Reverse[Range[#]]],!PrimeQ[#]&]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Mar 18 2017 *)

Python

from sympy import isprime
from itertools import accumulate
def ok(n): return all(not isprime(s) for s in accumulate(range(n, 0, -1)))
def aupto(nn): return [m for m in range(1, nn+1) if ok(m)]
print(aupto(148)) # Michael S. Branicky, Jan 08 2021