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A337081
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Primitive complement of A337037: terms of A337080 that are not multiples of previous terms.
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5
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4, 90, 546, 675, 850, 918, 945, 1026, 1050, 1134, 1242, 1365, 1386, 1575, 1650, 1750, 1782, 1950, 2205, 2295, 2310, 2450, 2475, 2646, 2793, 2850, 3250, 3366, 3465, 3626, 3654, 3762, 3850, 3969, 3990, 4218, 4290, 4374, 4455, 4510, 4550, 4650, 4875, 4998, 5022, 5166, 5382, 5390, 5610
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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.
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PROG
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(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
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CROSSREFS
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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).
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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