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

Author

Robert G. Wilson v, Oct 08 2017

Keywords

• base
• nonn

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.

Examples

			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.
		

Links

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

Crossrefs

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

Programs

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]

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