A337080 Complement of A337037.

4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 90, 92, 96, 100, 104, 108, 112, 116, 120, 124, 128, 132, 136, 140, 144, 148, 152, 156, 160, 164, 168, 172, 176, 180, 184, 188, 192, 196, 200, 204, 208, 212, 216, 220, 224, 228, 232, 236, 240, 244, 248

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.

Numbers m such that A069016(m) < A001055(m). - Michel Marcus, Aug 15 2020

Links

Table of n, a(n) for n=1..63.

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

Cf. A334911 (census-taker numbers).

Cf. A337037 (complement), A337081.

Cf. A001055 (number of unordered factorizations of n), A074206 (number of ordered factorizations of n).

Cf. A056472 (all factorizations of n), A069016 (number of distinct sums).

Sequence in context: A191677 A076310 A161352 * A295774 A008586 A059558

Adjacent sequences: A337077 A337078 A337079 * A337081 A337082 A337083

Keyword

nonn,easy

Author

Matej Veselovac, Aug 14 2020

Extensions

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

Status

approved