1,1

The only semiprime in the sequence is a(1) = 4, and there are no terms with exactly 3 prime factors.

Numbers of form p^k where p >= 5 is a prime number are terms of the sequence if and only if k = 4p+6. The only terms of the form 2^k or 3^k have k = 2, 12 respectively.

David A. Corneth, Table of n, a(n) for n = 1..10000

Math StackExchange, Smallest power of a prime whose factorizations don't have distinct sums of factors, 2020.

Numbers of the form m = 2*p*q*((p-1)*q-(p-2)) where p, q and (p-1)*q-(p-2) are odd prime numbers are even terms of the sequence. First, notice that m is a term of A337080 because the factorizations m = (2*((p-1)*q-(p-2)))*(p)*(q) = (2)*(((p-1)*q-(p-2)))*(p*q) have equal sums of factors. Second, m is not a multiple of any of the previous terms of the sequence because m has exactly 4 prime factors and the only term with less than 4 prime factors is 4, but 4 does not divide m.

(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);

isprimitive(n, va) = {for (k=1, #va, if ((n % va[k]) == 0, return (0)); ); return (1); }

lista(nn) = {my(va = []); for (n=1, nn, if (isok(n) && isprimitive(n, va), va = concat(va, n)); ); va; } \\ Michel Marcus, Aug 15 2020

Cf. A337037, A337080, A337112 (smallest term with n factors).

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: A205295 A376299 A292304 * A218385 A189882 A067767

Adjacent sequences: A337078 A337079 A337080 * A337082 A337083 A337084

nonn,easy

Matej Veselovac, Aug 14 2020

approved