A337037 Numbers whose every unordered factorization has a distinct sum of factors.

1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 50, 51, 53, 54, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 81, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 98, 99, 101

Offset

1,2

Comments

The number 1 is in the sequence by convention.

All primes p are trivially in the sequence.

All semiprimes greater than 4 are in the sequence because they have only two unordered factorizations pq = p*q whose sums are distinct. They are distinct because the only solution to p*q = p+q is p=q=2.

If a number m is not in the sequence, then all multiples of m are not in the sequence. For example, multiples of 4 are not 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.

The complement is in A337080.

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

Links

Table of n, a(n) for n=1..75.

Eric Weisstein's World of Mathematics, Unordered Factorization.

Example

All unordered factorization of 30 are 30 = 2*15 = 3*10 = 5*6 = 2*3*5. Corresponding sums of factors are distinct: 30, 17 = 15+2, 13 = 10+3, 11 = 6+5, 10 = 2+3+5. Therefore 30 is in the sequence.
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 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 13 2020

Crossrefs

Cf. A337080 (complement), A337081 (primitive complement).

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: A059557 A195291 A042968 * A048103 A276078 A193303

Adjacent sequences: A337034 A337035 A337036 * A337038 A337039 A337040

Keyword

nonn,easy

Author

Matej Veselovac, Aug 12 2020

Status

approved