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%I #17 Aug 16 2015 12:17:55
%S 21,33,42,57,66,69,77,84,93,105,114,129,132,133,138,141,154,161,165,
%T 168,177,186,189,201,209,210,213,217,228,231,237,249,253,258,264,266,
%U 273,276,282,285,297,301,308,309,321,322,329,330,336,341,345,354,357,372,378,381,385,393,399,402,413,417,418,420,426,429,434,437,441,453,456,462,465,469,473,474,483,489,497,498,501,506,513,516,517,525,528,532,537,546,552
%N Numbers n for which A065339(n) > A260728(n).
%C Numbers n such that when the exponents in the prime factorization of A097706(n) are added in base-2 they produce at least one carry-bit. In other words, in that set of exponents {e1, e2, ..., en} there is at least one pair e_i, e_j that their binary representations have at least one 1-bit in the same position. (Here i and j are distinct as e_i and e_j are exponents of different primes, although e_i could be equal to e_j. See the examples.)
%C This differs from A119973 for the first time at n=30 where a(30)=231, term which is not present in A119973. Note that n=231 is the first position where the difference A065339(n) - A260728(n) > 1 as 231 = 3*7*11, a product of three distinct 4k+3 primes, thus A065339(231) = 3, while A260728(231) = 1.
%H Antti Karttunen, <a href="/A260730/b260730.txt">Table of n, a(n) for n = 1..10000</a>
%e 21 = 3^1 * 7^1 is present, because in its prime factors of the form 4k+3 (which are 3 and 7) the exponents 1 and 1 have at least one 1-bit in the same position, thus producing a carry-bit when summed in base-2.
%e 63 = 3^2 * 7^1 is NOT present, because in its prime factors of the form 4k+3 the exponents 2 and 1 ("10" and "1" in binary) do NOT produce a carry-bit when summed in base-2, as those binary representations do not have any 1's in a common position.
%e 189 = 3^3 * 7^1 is present, because in its prime factors of the form 4k+3 the exponents 3 and 1 ("11" and "1" in binary) have at least one 1-bit in the same position, thus producing a carry-bit when summed in base-2.
%o (Scheme, with _Antti Karttunen_'s IntSeq-library)
%o (define A260730 (NONZERO-POS 1 1 (lambda (n) (- (A065339 n) (A260728 n)))))
%Y Cf. A065339, A260728, A097706, A119973.
%K nonn
%O 1,1
%A _Antti Karttunen_, Aug 12 2015
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