A261075 -id:A261075 - cp's OEIS frontend

A085721 Semiprimes whose prime factors have an equal number of digits in binary representation.

4, 6, 9, 25, 35, 49, 121, 143, 169, 289, 323, 361, 391, 437, 493, 527, 529, 551, 589, 667, 713, 841, 899, 961, 1369, 1517, 1591, 1681, 1739, 1763, 1849, 1927, 1961, 2021, 2173, 2183, 2209, 2257, 2279, 2419, 2491, 2501, 2537, 2623, 2773, 2809

Offset: 1

Reinhard Zumkeller, Jul 20 2003

A138510(A174956(a(n))) <= 2. - Reinhard Zumkeller, Dec 19 2014

			A078972(35) = 527 = 17*31 -> 10001*11111, therefore 527 is a term;
A078972(37) = 533 = 13*41 -> 1101*101001, therefore 533 is not a term;
A001358(1920) = 7169 = 67*107 -> 1000011*1101011: therefore 7169 a term, but not of A078972.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Dario A. Alpern, Brilliant Numbers.

Cf. A078972, A007088, A070939, A055642.
Cf. A084126, A084127, A070939.
Cf. A138510, A174956.
Cf. A261073, A261074, A261075 (subsequences).
Intersection of A001358 and A266346.

a085721 n = a085721_list !! (n-1)
a085721_list = [p*q | (p,q) <- zip a084126_list a084127_list,
                      a070939 p == a070939 q]
-- Reinhard Zumkeller, Nov 10 2013

fQ[n_] := Block[{fi = FactorInteger@ n}, Plus @@ Last /@ fi == 2 && IntegerLength[ fi[[1, 1]], 2] == IntegerLength[ fi[[-1, 1]], 2]]; Select[ Range@ 2866, fQ] (* Robert G. Wilson v, Oct 29 2011 *)
Select[Range@ 3000, And[Length@ # == 2, IntegerLength[#1, 2] == IntegerLength[#2, 2] & @@ #] &@ Flatten@ Map[ConstantArray[#1, #2] & @@ # &, FactorInteger@ #] &] (* Michael De Vlieger, Oct 08 2016 *)

is(n)=bigomega(n)==2&&#binary(factor(n)[1,1])==#binary(n/factor(n)[1,1]) \\ Charles R Greathouse IV, Nov 08 2011

Edited by Charles R Greathouse IV, Aug 02 2010

A261073 Semiprimes whose prime factors are of equal binary length and which differ from each other in one bit position only.

Original entry on oeis.org

6, 35, 323, 437, 713, 899, 1763, 1961, 2021, 2537, 3233, 4757, 5561, 5609, 6497, 7313, 9797, 10403, 10961, 11009, 18209, 19043, 21353, 22499, 23393, 26969, 27221, 29177, 37001, 38021, 39203, 45113, 71273, 72899, 79523, 87953, 95477, 98201, 99221, 106793, 114857, 114929, 123353

Offset: 1

Antti Karttunen, Sep 22 2015

			6 = 2*3 is present, as 2 in binary is "10" and 3 in binary is "11", so both have two (significant) bits and they differ only in one bit-position from each other.
35 = 5*7 is present, as 5 in binary is "101" and 7 in binary is "111", which both have three bits, differing only in the middle position from each other.

Antti Karttunen, Table of n, a(n) for n = 1..5000

Cf. A000523, A001222, A006530, A020639, A101080.
Cf. also A261074, A261075.
Cf. A071697 (a subsequence).
Intersection of A085721 and A261077.

Select[Range[10^6], And[Length@ # == 2, IntegerLength[#1, 2] == IntegerLength[#2, 2] & @@ #, Total@ BitXor[IntegerDigits[#1, 2], IntegerDigits[#2, 2]] == 1 & @@ #] &@ Flatten@ Map[ConstantArray[#1, #2] & @@ # &, FactorInteger@ #] &] (* Michael De Vlieger, Oct 08 2016 *)

A000523 = n -> logint(n, 2);
A020639(n) = if(1==n,n,vecmin(factor(n)[, 1]));
isA261073(n) = { my(a,b); if(bigomega(n)!=2, 0, a=A020639(n); b = (n/a); ((A000523(a) == A000523(b)) && (1 == norml2(binary(bitxor(a,b)))))); };
i=0; n=0; while(i < 5000, n++; if(isA261073(n), i++; write("b261073.txt", i, " ", n)));

;; With Antti Karttunen's IntSeq-library.
(define A261073 (MATCHING-POS 1 1 (lambda (n) (and (= 2 (A001222 n)) (= (A000523 (A020639 n)) (A000523 (A006530 n))) (= 1 (A101080bi (A020639 n) (A006530 n)))))))

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

Original entry on oeis.org

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

Antti Karttunen, Sep 22 2015

			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.

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.

Cf. A000523, A001222, A006530, A020639, A101080.
Cf. also A261073, A261075.
Subsequence of A085721.

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 *)

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)));

;; 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)))))))

A266346 Numbers that can be represented as a product of two numbers with an equal number of significant digits (bits) in binary system.

Original entry on oeis.org

0, 1, 4, 6, 9, 16, 20, 24, 25, 28, 30, 35, 36, 42, 49, 64, 72, 80, 81, 88, 90, 96, 99, 100, 104, 108, 110, 112, 117, 120, 121, 126, 130, 132, 135, 140, 143, 144, 150, 154, 156, 165, 168, 169, 180, 182, 195, 196, 210, 225, 256, 272, 288, 289, 304, 306, 320, 323, 324, 336, 340, 342, 352, 357, 360, 361, 368, 374, 378, 380, 384, 391

Offset: 0

Antti Karttunen, Dec 28 2015

Indexing starts from zero as a(0) = 0 is a special case in this sequence.

			1 can be represented as 1*1 (1 being "1" also in base-2 system), thus it is included.
4 can be represented as 2*2, and like any square, is included.
6 can be represented as 2*3, and both "10" and "11" require two bits in binary system, thus 6 is included.

Antti Karttunen, Table of n, a(n) for n = 0..9616

Positions of nonzeros in A266342.
Cf. A266347 (complement).
Cf. A000290, A085721, A261073, A261074, A261075 (subsequences).
Cf. also A266342.

{0}~Join~Flatten[Position[#, k_ /; k > 0] &@ Table[Length@ DeleteCases[Flatten@ Map[Differences@ IntegerLength[#, 2] &, Transpose@ {#, n/#}] &@ TakeWhile[Divisors@ n, # <= Sqrt@ n &], k_ /; k > 0], {n, 400}]] (* Michael De Vlieger, Dec 30 2015 *)

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A085721 Semiprimes whose prime factors have an equal number of digits in binary representation.

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A261073 Semiprimes whose prime factors are of equal binary length and which differ from each other in one bit position only.

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

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A266346 Numbers that can be represented as a product of two numbers with an equal number of significant digits (bits) in binary system.

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