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

Author

Jonathan Vos Post, Dec 26 2007

Keywords

• easy
• nonn

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).

Examples

			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.
		

Crossrefs

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

Programs

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 *)