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A337081 Primitive complement of A337037: terms of A337080 that are not multiples of previous terms.

%I A337081 #26 Sep 14 2020 00:22:28
%S A337081 4,90,546,675,850,918,945,1026,1050,1134,1242,1365,1386,1575,1650,
%T A337081 1750,1782,1950,2205,2295,2310,2450,2475,2646,2793,2850,3250,3366,
%U A337081 3465,3626,3654,3762,3850,3969,3990,4218,4290,4374,4455,4510,4550,4650,4875,4998,5022,5166,5382,5390,5610
%N A337081 Primitive complement of A337037: terms of A337080 that are not multiples of previous terms.
%C A337081 The only semiprime in the sequence is a(1) = 4, and there are no terms with exactly 3 prime factors.
%C A337081 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.
%H A337081 David A. Corneth, <a href="/A337081/b337081.txt">Table of n, a(n) for n = 1..10000</a>
%H A337081 Math StackExchange, <a href="https://math.stackexchange.com/q/3794895/318073">Smallest power of a prime whose factorizations don't have distinct sums of factors</a>, 2020.
%e A337081 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.
%o A337081 (PARI)
%o A337081 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]]);}
%o A337081 factorz(n) = factz(n, 2);
%o A337081 isok(n) = my(vs = apply(x->vecsum(x), factorz(n))); #vs != #Set(vs);
%o A337081 isprimitive(n, va) = {for (k=1, #va, if ((n % va[k]) == 0, return (0));); return (1);}
%o A337081 lista(nn) = {my(va = []); for (n=1, nn, if (isok(n) && isprimitive(n, va), va = concat(va, n));); va;} \\ _Michel Marcus_, Aug 15 2020
%Y A337081 Cf. A337037, A337080, A337112 (smallest term with n factors).
%Y A337081 Cf. A001055 (number of unordered factorizations of n), A074206 (number of ordered factorizations of n).
%Y A337081 Cf. A056472 (all factorizations of n), A069016 (number of distinct sums).
%K A337081 nonn,easy
%O A337081 1,1
%A A337081 _Matej Veselovac_, Aug 14 2020