A096067 Number of 3-smooth numbers between successive numbers that are powers of 2 or of 3.

0, 0, 0, 1, 0, 1, 2, 0, 3, 1, 2, 4, 0, 5, 2, 3, 6, 0, 6, 5, 2, 8, 2, 6, 8, 1, 10, 4, 6, 11, 0, 11, 8, 4, 13, 3, 10, 12, 2, 15, 6, 9, 16, 0, 17, 9, 8, 18, 2, 16, 14, 5, 20, 6, 14, 19, 2, 22, 10, 12, 23, 1, 22, 16, 8, 25, 6, 19, 22, 4, 27, 11, 16, 28, 0, 29, 16, 13, 30, 4

Offset: 1

Author

Reinhard Zumkeller, Jul 21 2004

Keywords

• nonn
• look

Comments

a(n) = {k: A006899(n) < A003586(k) < A006899(n+1)}.

Examples

			n=16: there are three 3-smooth numbers between A006899(16)=3^6=729 and A006899(17)=2^10=1024: A003586(38)=2^8*3=768, A003586(39)=2^5*3^3=864 and A003586(40)=2^2*3^5=972, therefore a(16)=3.
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A003586, A006899, A071521.

Programs

Mathematica

spi[n_] := Sum[Floor@Log[2, n/3^k] + 1, {k, 0, Floor@Log[3, n]}];
seq[n_] := Module[{a = Table[0, {n}], p = 1, s = 1}, For[i = 1, i <= Length[a], i++, p = Min[2^(1 + Floor@Log[2, p]), 3^(1 + Floor@Log[3, p])]; With[{t = spi[p]}, a[[i]] = t - s - 1; s = t]]; a];
seq[100] (* Jean-François Alcover, Dec 17 2021, after Andrew Howroyd's PARI code *)

PARI

\\ here spi(n) is A071521(n).
spi(n)={sum(k=0, logint(n, 3), logint(n\3^k, 2)+1)}
seq(n)={my(a=vector(n), p=1, s=1); for(i=1, #a, p=min(2^(1+logint(p,2)), 3^(1+logint(p,3))); my(t=spi(p)); a[i]=t-s-1; s=t); a} \\ Andrew Howroyd, Jan 07 2020

Extensions

Terms a(40) and beyond from Andrew Howroyd, Jan 06 2020