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

1,2

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.

Links

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

Example

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.

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 *)

Prog

(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
(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

Crossrefs

Cf. A010051.

Sequence in context: A349081 A192045 A084021 * A072835 A367035 A096229

Adjacent sequences: A138663 A138664 A138665 * A138667 A138668 A138669

Keyword

nice,nonn

Author

T. D. Noe, Mar 26 2008

Status

approved