cp's OEIS Frontend

A261074 Semiprimes whose prime factors are of equal binary length and which differ from each other in exactly two bit positions.

%I A261074 #20 Feb 22 2025 20:05:25
%S A261074 143,391,493,589,667,1517,1739,1927,2257,2419,2501,2773,2867,3599,
%T A261074 4891,5293,5767,5893,6499,6901,7081,7169,7171,7387,7811,7957,8137,
%U A261074 8453,8611,9379,9991,10033,10057,10379,10573,11021,11227,11413,11663,13081,13589,13843,17947,19781,21509,21877,22657,23449,23701,23707,25217,25283,26069,26441,27029
%N A261074 Semiprimes whose prime factors are of equal binary length and which differ from each other in exactly two bit positions.
%H A261074 Antti Karttunen, <a href="/A261074/b261074.txt">Table of n, a(n) for n = 1..10000</a>
%H A261074 N. S. Dattani & N.  Bryans, <a href="http://arxiv.org/abs/1411.6758">Quantum factorization of 56153 with only 4 qubits</a>, arXiv:1411.6758 [quant-ph], 2014.
%e A261074 143 = 11*13 is included because 11 ("1011" in binary) and 13 ("1101" in binary) differ from each other in exactly two bit-positions.
%e A261074 56153 = 233 * 241 is included (as term a(119)) because 233 ("11101001" in binary) and 241 ("11110001" in binary) differ from each other in exactly two bit-positions.
%t A261074 Select[Range[10^5], And[Length@ # == 2, IntegerLength[#1, 2] == IntegerLength[#2, 2] & @@ #, Total@ BitXor[IntegerDigits[#1, 2], IntegerDigits[#2, 2]] == 2 & @@ #] &@ Flatten@ Map[ConstantArray[#1, #2] & @@ # &, FactorInteger@ #] &] (* _Michael De Vlieger_, Oct 08 2016 *)
%o A261074 (PARI)
%o A261074 A000523 = n -> logint(n, 2);
%o A261074 A020639(n) = if(1==n,n,vecmin(factor(n)[, 1]));
%o A261074 isA261074(n) = { my(a,b); if(bigomega(n)!=2, 0, a = A020639(n); b = (n/a); ((A000523(a) == A000523(b)) && (2 == norml2(binary(bitxor(a,b)))))); };
%o A261074 i=0; n=0; while(i < 10000, n++; if(isA261074(n), i++; write("b261074.txt", i, " ", n)));
%o A261074 (Scheme)
%o A261074 ;; With _Antti Karttunen_'s IntSeq-library.
%o A261074 (define A261074 (MATCHING-POS 1 1 (lambda (n) (and (= 2 (A001222 n)) (= (A000523 (A020639 n)) (A000523 (A006530 n))) (= 2 (A101080bi (A020639 n) (A006530 n)))))))
%Y A261074 Cf. A000523, A001222, A006530, A020639, A101080.
%Y A261074 Cf. also A261073, A261075.
%Y A261074 Subsequence of A085721.
%K A261074 nonn,base
%O A261074 1,1
%A A261074 _Antti Karttunen_, Sep 22 2015