OFFSET

1,1

COMMENTS

Numbers with a pair of unordered factorizations whose sums of factors are the same.

All terms of the sequence are composite.

The smallest odd term of the sequence is a(174) = 675. This is a term of the sequence because 675 = 27*5*5 = 9*3*25 and 27+5+5 = 9+3+25 = 37.

Terms of the sequence are used in variations of a logic puzzle known as "Ages of Three Children Puzzle" or "Census-taker problem". For the original puzzle, see A334911.

If a number m is in the sequence, then all multiples of m are in the sequence. For example, multiples of 4 are in the sequence because there always exist at least two factorizations 4*k = 2*2*k whose factors sum to the same value 4+k = 2+2+k.

LINKS

Eric Weisstein's World of Mathematics, Unordered Factorization.

EXAMPLE

All unordered factorization of 90 are 90 = 45*2 = 30*3 = 18*5 = 15*6 = 15*3*2 = 10*9 = 9*5*2 = 10*3*3 = 6*5*3 = 5*3*3*2. Corresponding sums of factors are not all distinct: 90, 57, 33, 23, 21, 20, 19, 16, 16, 14, 13 because the sum 16 = 10+3+3 = 9+5+2 appears twice. Therefore 90 is in the sequence.

All unordered factorization of 30 are 30 = 15*2 = 10*3 = 6*5 = 5*3*2. Corresponding sums of factors are all distinct: 30 = 30, 17 = 15+2, 13 = 10+3, 11 = 6+5, 10 = 2+3+5. Therefore 30 is not in the sequence.

PROG

(PARI) factz(n, minn) = {my(v=[]); fordiv(n, d, if ((d>=minn) && (d<=sqrtint(n)), w = factz(n/d, d); for (i=1, #w, w[i] = concat([d], w[i]); ); v = concat(v, w); ); ); concat(v, [[n]]); }

factorz(n) = factz(n, 2);

isok(n) = my(vs = apply(x->vecsum(x), factorz(n))); #vs != #Set(vs); \\ Michel Marcus, Aug 14 2020

CROSSREFS

KEYWORD

nonn,easy

AUTHOR

Matej Veselovac, Aug 14 2020

EXTENSIONS

Edited by N. J. A. Sloane, Sep 14 2020

STATUS

approved