

A005381


Numbers k such that k and k1 are composite.
(Formerly M4598)


45



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
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OFFSET

1,1


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

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].


FORMULA

Conjecture: pi(n)=Sum_{k=1..n} k mod a(m) mod a(m1) ... mod a(1) mod 2, for all values 1<n<=a(m), where the mod are evaluated from left to right. Verified for first 10000 a(n).  Benedict 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


MAPLE

isA005381 := proc(n)
not isprime(n) and not isprime(n1) ;
end proc:
local a;
option remember;
if n = 1 then
9;
else
for a from procname(n1)+1 do
if isA005381(a) then
return a;
end if;
end do:
end if;
# second Maple program:
q:= n> ormap(isprime, [n, n1]):


MATHEMATICA

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


PROG

(PARI) is(n)=!isprime(n)&&!isprime(n1) \\ M. F. Hasler, Jan 07 2019
(Python)
from sympy import isprime
def ok(n): return n > 3 and not isprime(n) and not isprime(n1)


CROSSREFS

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


KEYWORD

nonn,easy


AUTHOR



STATUS

approved



