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A036349
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Numbers whose sum of prime factors (taken with multiplicity) is even.
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17
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1, 2, 4, 8, 9, 15, 16, 18, 21, 25, 30, 32, 33, 35, 36, 39, 42, 49, 50, 51, 55, 57, 60, 64, 65, 66, 69, 70, 72, 77, 78, 81, 84, 85, 87, 91, 93, 95, 98, 100, 102, 110, 111, 114, 115, 119, 120, 121, 123, 128, 129, 130, 132, 133, 135, 138, 140, 141, 143, 144, 145, 154, 155
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OFFSET
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1,2
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COMMENTS
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A multiplicative semigroup; if m and n are in the sequence then so is m*n. - David James Sycamore, Jul 17 2018
Also closed under the commutative binary operation A059897(.,.), forming a subgroup of the positive integers under A059897.
A number is listed if and only if it has an even number of odd prime factors, counting repetitions; equivalently, if and only if it is the product of a term of A046337 and a power of 2 (term of A000079).
(End)
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LINKS
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FORMULA
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Sum_{n>=1} 1/a(n)^s = (zeta(s) + ((2^s + 1)/(2^s - 1))*zeta(2*s)/zeta(s))/2 for Re(s)>1. - Amiram Eldar, Nov 02 2020
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EXAMPLE
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141 = 3 * 47 is a term since the sum 3 + 47 = 50 is even.
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MAPLE
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filter:= proc(n) local t; add(t[1]*t[2], t=ifactors(n)[2])::even end proc:
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MATHEMATICA
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Select[Range[160], EvenQ[Total[Times@@@FactorInteger[#]]]&] (* Harvey P. Dale, Sep 21 2011 *)
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PROG
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(PARI) isok(n) = my(f=factor(n)); (sum(k=1, #f~, f[k, 1]*f[k, 2]) % 2) == 0; \\ Michel Marcus, Jul 19 2018
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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