A133576 Numbers which are sums of consecutive composites.

4, 6, 8, 9, 10, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 74, 75, 76, 77, 78, 79, 80, 81

Offset

1,1

Comments

This is to composites A002808 as A034707 is to primes A000040. The complement of this sequence, numbers which are not sums of consecutive composites, begins 1, 2, 3, 5, 7, ... (A140464).

Links

Table of n, a(n) for n=1..71.

Example

Every composite is in this sequence as one consecutive composite. We account for primes thus:
a(10) = 17 = 8 + 9.
a(12) = 19 = 9 + 10.
a(16) = 23 = 6 + 8 + 9.
a(22) = 29 = 14 + 15.
a(24) = 31 = 9 + 10 + 12.
a(30) = 37 = 4 + 6 + 8 + 9 + 10.
a(34) = 41 = 20 + 21 = 12 + 14 + 15.
a(36) = 43 = 21 + 22.
Not included = 47.
a(45) = 53 = 26 + 27 = 8 + 9 + 10 + 12 + 14.
a(51) = 59 = 18 + 20 + 21 = 6 + 8 + 9 + 10 + 12 + 14.
Not included = 61.
a(58) = 67 = 33 + 34 = 21 + 22 + 24 = 10 + 12 + 14 + 15 + 16.
a(62) = 71 = 35 + 36 = 22 + 24 + 25 = 4 + 6 + 8 + 9 + 10 + 12 + 14.
Not included = 73.
a(69) = 79 = 39 + 40.
a(73) = 83 = 14 + 15 + 16 + 18 + 20.
a(79) = 89 = 44 + 45.
a(87) = 97 = 48 + 49 = 22 + 24 + 25 + 26.
a(91) = 101 = 50 + 51.
a(93) = 103 = 51 + 52.

Maple

isA133576 := proc(n)
    local i, j ;
    for i from 1 do
        if A002808(i) > n then
            return false;
        end if;
        for j from i do
            s := add( A002808(l), l=i..j) ;
            if s > n then
                break;
            elif s = n then
                return true;
            end if;
        end do:
    end do:
end proc:
A133576 := proc(n)
    local a;
    if n = 1 then
        return A002808(1) ;
    else
        for a from procname(n-1)+1 do
            if isA133576(a) then
                return a;
            end if;
        end do:
    end if ;
end proc:
seq(A133576(n), n=1..71) ; # R. J. Mathar, Feb 14 2015

Mathematica

okQ[n_] := If[CompositeQ[n], True, MemberQ[IntegerPartitions[n, All, Select[Range[n], CompositeQ]], p_List /; Length[p] == Length[Union[p]] && AllTrue[Complement[Range[p[[-1]], p[[1]]], p], PrimeQ]]];
Select[Range[150], okQ] (* Jean-François Alcover, Oct 27 2023 *)

Crossrefs

Cf. A002808, A034707, A037174, A140464 (complement).

Sequence in context: A137353 A336371 A167376 * A192607 A088224 A002808

Adjacent sequences: A133573 A133574 A133575 * A133577 A133578 A133579

Keyword

easy,nonn

Author

Jonathan Vos Post, Dec 26 2007

Status

approved