A261077 -id:A261077 - cp's OEIS frontend

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

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)))))))

A260737 Sum of Hamming distances between binary representations of prime factors of n, summed over all nonordered pairs of primes present (with multiplicity) in the prime factorization of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 3, 0, 2, 0, 2, 2, 0, 0, 2, 0, 6, 1, 2, 0, 3, 0, 4, 0, 4, 0, 6, 0, 0, 1, 3, 1, 4, 0, 2, 3, 9, 0, 4, 0, 4, 4, 3, 0, 4, 0, 6, 2, 8, 0, 3, 3, 6, 1, 5, 0, 10, 0, 4, 2, 0, 1, 4, 0, 6, 2, 6, 0, 6, 0, 4, 4, 4, 2, 8, 0, 12, 0, 4, 0, 7, 2, 3, 4, 6, 0, 9, 2, 6, 3, 4, 3, 5, 0, 4, 2, 12, 0, 6, 0, 12, 4, 5, 0, 6, 0, 8, 3, 8, 0, 4, 2, 10, 6, 4, 3, 14

Offset: 1

Antti Karttunen, Sep 22 2015

			For n = 1 the prime factorization is empty, thus there is nothing to sum, so a(1) = 0.
For n = 6 = 2*3, a(6) = 1 because the Hamming distance between 2 and 3 is 1 as 2 = "10" in binary and 3 = "11" in binary.
For n = 10 = 2*5, a(10) = 3 because the Hamming distance between 2 and 5 is 3 as 2 = "10" in binary (extended with a leading zero to make it "010") and 5 = "101" in binary.
For n = 12 = 2*2*3, a(12) = 2 because the Hamming distance between 2 and 3 is 1, and the pair (2,3) occurs twice as one can pick either one of the two 2's present in the prime factorization to be a pair of a single 3. Note that the Hamming distance between 2 and 2 is 0, thus the pair (2,2) of prime divisors does not contribute to the sum.
For n = 36 = 2*2*3*3, a(36) = 4 because the Hamming distance between 2 and 3 is 1, and the prime factor pair (2,3) occurs four times in total. Note that the Hamming distance is zero between 2 and 2 as well as between 3 and 3, thus the pairs (2,2) and (3,3) do not contribute to the sum.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A101080.
Cf. A000961 (positions of the zeros), A261077 (positions of the ones).
Cf. A072594, A072595, A235488.
Cf. also A261079.

A261078 Semiprimes p*q such that q = p + 2^k for some k >= 0.

Original entry on oeis.org

6, 15, 21, 33, 35, 57, 65, 77, 143, 161, 185, 201, 209, 221, 323, 377, 393, 437, 473, 497, 713, 899, 1073, 1457, 1517, 1529, 1577, 1763, 1769, 1841, 1961, 2021, 2537, 2993, 3233, 3473, 3497, 3599, 3713, 3737, 3953, 4553, 4601, 4757, 5183, 5561, 5609, 5753, 6497, 6557, 7217, 7313, 8633, 8777, 9593, 9797, 10001, 10265, 10403, 10841, 10961, 11009, 11021

Offset: 1

Antti Karttunen, Sep 22 2015

nonn

Terms ending with digit 5 (in decimal) are very rare, because terms of A123250 are rare.

			6 = 2*3 is present as 3 = 2 + 2^0.
15 = 3*5 is present as 5 = 3 + 2^1.
35 = 5*7 is present as 7 = 5 + 2^1.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A001222, A004198, A006530, A020639, A123250.

Cf. also A261073, A261077 (subsequences).

A020639(n) = if(1==n,n,vecmin(factor(n)[, 1]));
isA261078(n) = { my(d); if(bigomega(n)!=2, return(0), d = (n/A020639(n)) - A020639(n); (d && !bitand(d,d-1))); };
i=0; n=0; while(i < 10000, n++; if(isA261078(n), i++; write("b261078.txt", i, " ", n)));

;; With Antti Karttunen's IntSeq-library.
(define A261078 (MATCHING-POS 1 1 (lambda (n) (and (= 2 (A001222 n)) (pow2? (- (A006530 n) (A020639 n)))))))
(define (pow2? n) (and (> n 0) (zero? (A004198bi n (- n 1))))) ;; A004198bi implements bitwise-AND (Cf. A004198)

A261080 Semiprimes p*q for which p and q are successive primes and their binary representations differ from each other in one bit position only.

Original entry on oeis.org

6, 35, 323, 437, 899, 1763, 2021, 4757, 9797, 10403, 19043, 22499, 27221, 38021, 39203, 72899, 79523, 95477, 99221, 131753, 145157, 154433, 164009, 205193, 210677, 213443, 250997, 272483, 324899, 381923, 412163, 416021, 455621, 549077, 557993, 594437, 656099, 675683, 736163, 741317, 777923, 783221, 826277, 870473, 881717, 974153, 1022117, 1102499, 1127843, 1238753

Offset: 1

Antti Karttunen, Sep 23 2015

nonn

Numbers n for which A260737(n) = A261079(n) = 1.

			6 is included as 6 = 2*3, 2 and 3 are successive primes, and 2 (in binary "10") and 3 (in binary "11") differ by only one bit from each other.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A205302, A205511, A260737, A261079.

Intersection of A006094 and A261077.

brdQ[{a_,b_}]:=Module[{c=IntegerDigits[a,2],d=IntegerDigits[b,2]}, Length[ c] == Length[d]&&Count[Total/@Transpose[{c,d}],1]==1]; Times@@@ Select[ Partition[Prime[Range[200]],2,1],brdQ] (* Harvey P. Dale, Jan 29 2016 *)

a(n) = A205511(n) * A205302(n).

cp's OEIS Frontend

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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A260737 Sum of Hamming distances between binary representations of prime factors of n, summed over all nonordered pairs of primes present (with multiplicity) in the prime factorization of n.

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A261078 Semiprimes p*q such that q = p + 2^k for some k >= 0.

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A261080 Semiprimes p*q for which p and q are successive primes and their binary representations differ from each other in one bit position only.

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