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 A269818 Numbers coprime to the number of their even divisors. 2
 1, 2, 8, 32, 50, 98, 128, 162, 200, 242, 338, 392, 512, 578, 722, 968, 1058, 1352, 1458, 1568, 1682, 1922, 2048, 2312, 2450, 2592, 2738, 2888, 3200, 3362, 3698, 3872, 4232, 4418, 4802, 5408, 5618, 6050, 6728, 6962, 7442, 7688, 8192, 8450, 8978, 9248, 9800, 10082, 10368, 10658, 10952, 11552, 11858, 12482, 12800 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS This sequence is characterized by the following property (theorem). Theorem. If n is coprime to the number of its even divisors, then n is 1 or of the form 2m^2, m>0. Proof. If n is odd, its number of even divisors is 0 and since gcd(n,0)=|n| (for any n), n must be 1 to be coprime to 0. If n is even, then it is of the form 2^k*p^a*q*^b*...*r^c, where p, q, r are odd primes and k, a, b, c are positive integers, and its sum of even divisors is k*(1+a)*(1+b)*...*(1+c). The latter number can be coprime to an even number only if it is odd, implying that k must be odd and a, b, ..., c must be even; thus n is twice a square. LINKS Amiram Eldar, Table of n, a(n) for n = 1..10000 EXAMPLE For n = 3, a(3) = 8 is a member for the number of even divisors of 8, (2,4,8), is 3, which is coprime with 8. MATHEMATICA Select[Range@13000, CoprimeQ[#, Length@Select[Divisors[#], EvenQ]]&] PROG (PARI) for(n=1, 13000, gcd(n, if(n%2, 0, numdiv(n/2)))==1&&print1(n, ", ")) CROSSREFS Cf. A183063 (number of even divisors), A046642 (numbers coprime to the number of their divisors), A269870 (counterpart for the number of odd divisors), A268066 (related sequence). Sequence in context: A238984 A071389 A026369 * A331823 A006552 A134753 Adjacent sequences: A269815 A269816 A269817 * A269819 A269820 A269821 KEYWORD nonn AUTHOR Waldemar Puszkarz, Mar 05 2016 STATUS approved

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Last modified July 23 19:18 EDT 2024. Contains 374553 sequences. (Running on oeis4.)