OFFSET
1,1
COMMENTS
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.
LINKS
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.
EXAMPLE
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.
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);
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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Matej Veselovac, Aug 14 2020
STATUS
approved