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

143, 391, 493, 589, 667, 1517, 1739, 1927, 2257, 2419, 2501, 2773, 2867, 3599, 4891, 5293, 5767, 5893, 6499, 6901, 7081, 7169, 7171, 7387, 7811, 7957, 8137, 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

Offset: 1

Author

Antti Karttunen, Sep 22 2015

Keywords

• nonn
• base

Examples

			143 = 11*13 is included because 11 ("1011" in binary) and 13 ("1101" in binary) differ from each other in exactly two bit-positions.
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.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000
• N. S. Dattani & N. Bryans, Quantum factorization of 56153 with only 4 qubits, arXiv:1411.6758 [quant-ph], 2014.

Crossrefs

Cf. A000523, A001222, A006530, A020639, A101080.

Cf. also A261073, A261075.

Subsequence of A085721.

Programs

Mathematica

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 *)

PARI

A000523 = n -> logint(n, 2);
A020639(n) = if(1==n,n,vecmin(factor(n)[, 1]));
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)))))); };
i=0; n=0; while(i < 10000, n++; if(isA261074(n), i++; write("b261074.txt", i, " ", n)));

Scheme

;; With Antti Karttunen's IntSeq-library.
(define A261074 (MATCHING-POS 1 1 (lambda (n) (and (= 2 (A001222 n)) (= (A000523 (A020639 n)) (A000523 (A006530 n))) (= 2 (A101080bi (A020639 n) (A006530 n)))))))