A005381 Numbers k such that k and k-1 are composite.

9, 10, 15, 16, 21, 22, 25, 26, 27, 28, 33, 34, 35, 36, 39, 40, 45, 46, 49, 50, 51, 52, 55, 56, 57, 58, 63, 64, 65, 66, 69, 70, 75, 76, 77, 78, 81, 82, 85, 86, 87, 88, 91, 92, 93, 94, 95, 96, 99, 100, 105, 106, 111, 112, 115, 116, 117, 118, 119, 120, 121, 122

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Position where the composites first outnumber the primes by n, among the first natural numbers. - Lekraj Beedassy, Jul 11 2006

References

• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 844.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• J. Stauduhar, Table of n, a(n) for n = 1..10000
• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
• R. P. Boas & N. J. A. Sloane, Correspondence, 1974
• B. W. J. Irwin, Recursive Modular Conjecture for pi(n).

Crossrefs

Equals A068780 + 1. Cf. A007921.

Cf. A093515 (complement, apart from 1 which is in neither sequence), A323162 (characteristic function).

Programs

Maple

isA005381 := proc(n)
    not isprime(n) and not isprime(n-1) ;
end proc:
A005381 := proc(n)
    local a;
    option remember;
    if n = 1 then
        9;
    else
        for a from procname(n-1)+1 do
            if isA005381(a) then
                return a;
            end if;
        end do:
    end if;
end proc: # R. J. Mathar, Jul 14 2015
# second Maple program:
q:= n-> ormap(isprime, [n, n-1]):
remove(q, [$2..130])[];  # Alois P. Heinz, Dec 26 2021

Mathematica

Select[Range[2, 200], ! PrimeQ[# - 1] && ! PrimeQ[#] &]

PARI

is(n)=!isprime(n)&&!isprime(n-1) \\ M. F. Hasler, Jan 07 2019

Python

from sympy import isprime
def ok(n): return n > 3 and not isprime(n) and not isprime(n-1)
print([k for k in range(122) if ok(k)]) # Michael S. Branicky, Dec 26 2021

Formula

Conjecture: pi(n)=Sum_{k=1..n} k mod a(m) mod a(m-1) ... mod a(1) mod 2, for all values 1Benedict W. J. Irwin, May 04 2016

As a check, take n=9, m=2, a(m)=10. Then we must take the numbers 1 through 9 and reduce them mod 10 then mod 9 then mod 2. The results are 1,0,1,0,1,0,1,0,0, whose sum is 4 = pi(9), as predicted. - N. J. A. Sloane, May 05 2016

For an attempt at a proof for the conjecture above, see the link. If it is true, then for n>2, isprime(n)=(n mod x) mod 2, where x is the largest a(n)<=n. - Benedict W. J. Irwin, May 06 2016