A293424 Hamming distance between two consecutive semiprimes.

1, 4, 2, 1, 1, 3, 2, 4, 2, 5, 2, 1, 2, 1, 2, 5, 1, 1, 3, 2, 1, 7, 1, 4, 3, 5, 3, 2, 1, 2, 2, 2, 1, 4, 2, 3, 2, 1, 3, 2, 1, 6, 1, 2, 3, 2, 1, 4, 2, 2, 2, 1, 5, 3, 4, 2, 2, 2, 3, 1, 5, 3, 2, 1, 2, 2, 5, 1, 2, 1, 3, 2, 1, 2, 6, 2, 2, 3, 3, 1, 2, 8, 2, 4, 1, 3, 1, 2, 5, 1, 1, 3, 1, 2, 2, 1, 4, 1, 4, 2, 6, 1, 2, 1, 3

Offset

1,2

Comments

The least semiprime whose Hamming distance between it and its successor semiprime is k: 4, 9, 15, 6, 26, 123, 62, 254, 511, 3071, 2047, 8189, 32765, 16382, 98303, 65531, 393215, 262142, 1572863, 2621438, 1048574, 16777207, 8388607, 50331647, 33554429, 134217721, 268435451, etc.

Not surprisingly, the above are often the largest semiprime < 2^j.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Example

a(1) = 1 because the semiprimes 4 & 6, 100_2 & 110_2 have a Hamming distance of 1;
a(2) = 4 because the semiprimes 6 & 9, 110_2 & 1001_2 have a Hamming distance of 4;
a(3) = 2 because the semiprimes 9 & 10, 1001_2 & 1010_2 have a Hamming distance of 2; etc.

Maple

semiprimes:= select(t -> numtheory:-bigomega(t)=2, [$4..1023]):
L:=map(t -> convert(t+1024, base, 2), semiprimes):
map(t -> 11 - numboccur(0, t), L[2..-1]-L[1..-2]); # Robert Israel, Oct 08 2017
# alternative
read("transforms") :
A293424 := proc(n)
    local s1, s2 ;
    s1 := A001358(n) ;
    s2 := A001358(n+1) ;
    XORnos(s1, s2) ;
    wt(%) ;
end proc: # R. J. Mathar, Jan 06 2018

Mathematica

Count[ IntegerDigits[ BitXor[ #[[1]], #[[2]]], 2], 1] & /@ Partition[ Select[ Range@330, PrimeOmega@# == 2 &], 2, 1]

Prog

(PARI) lista(nn) = my(v = select(x->bigomega(x)==2, vector(nn, k, k))); vector(#v-1, k, norml2(binary(bitxor(v[k], v[k+1])))); \\ Michel Marcus, Oct 11 2017

Crossrefs

Cf. A001358 (semiprimes), A205510 (between consecutive primes).

Sequence in context: A144260 A097526 A051149 * A152145 A288251 A024553

Adjacent sequences: A293421 A293422 A293423 * A293425 A293426 A293427

Keyword

base,nonn

Author

Robert G. Wilson v, Oct 08 2017

Status

approved