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Numbers one less than composite numbers.
15

%I #29 Aug 02 2024 12:04:38

%S 3,5,7,8,9,11,13,14,15,17,19,20,21,23,24,25,26,27,29,31,32,33,34,35,

%T 37,38,39,41,43,44,45,47,48,49,50,51,53,54,55,56,57,59,61,62,63,64,65,

%U 67,68,69,71,73,74,75,76,77,79,80,81,83,84,85,86,87,89,90,91,92,93,94,95

%N Numbers one less than composite numbers.

%C Complement of A006093 (primes minus 1).

%C Numbers which can be written as i*j+i+j, 0<i<=j: A072670(a(n))>0 for n>1.

%C a(n)! is divisible by a(n)*(a(n)+1)/2, see A060462.

%H Michael De Vlieger, <a href="/A072668/b072668.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A002808(n) - 1.

%F a(n) = 2*A002808(n) - A079696(n). - _Juri-Stepan Gerasimov_, Oct 22 2009

%F a(n) = A060462(n).

%t Select[Range[4, 96], CompositeQ] - 1 (* _Michael De Vlieger_, Dec 10 2020 *)

%o (PARI) for(n=2,100,if(!isprime(n),print1(n-1,", "))) \\ _Derek Orr_, Jun 08 2015

%o (Magma) [n-1: n in [2..120] | not IsPrime(n)]; // _Vincenzo Librandi_, Jun 09 2015

%o (Python)

%o from sympy import composite

%o def A072668(n): return composite(n)-1 # _Chai Wah Wu_, Aug 02 2024

%Y Cf. A060462, A066938, A072672.

%Y Cf. A002808, A006093, A079696.

%K nonn,easy

%O 1,1

%A _Henry Bottomley_, Apr 11 2001