A073490 Number of prime gaps in factorization of n.

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 2, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0

Offset: 1

Author

Reinhard Zumkeller, Aug 03 2002

Keywords

• nonn
• nice

Comments

A137723(n) is the smallest number of the first occurring set of exactly n consecutive numbers with at least one prime gap in their factorization: a(A137723(n)+k)>0 for 0<=kA137723(n)-1)=a(A137723(n)+n)=0. - Reinhard Zumkeller, Feb 09 2008

Examples

			84 = 2*2*3*7 with one gap between 3 and 7, therefore a(84) = 1;
110 = 2*5*11 with two gaps: between 2 and 5 and between 5 and 11, therefore a(110) = 2.
		

Links

• R. Zumkeller, Table of n, a(n) for n = 1..50000

Crossrefs

Cf. A073491, A073492, A073493, A073494, A073495.

Cf. A137721, A137722, A027748, A049084.

Programs

Haskell

a073490 1 = 0
a073490 n = length $ filter (> 1) $ zipWith (-) (tail ips) ips
   where ips = map a049084 $ a027748_row n
-- Reinhard Zumkeller, Jul 04 2012

Maple

A073490 := proc(n)
    local a,plist ;
    plist := sort(convert(numtheory[factorset](n),list)) ;
    a := 0 ;
    for i from 2 to nops(plist) do
        if op(i,plist) <> nextprime(op(i-1,plist)) then
            a := a+1 ;
        end if;
    end do:
    a;
end proc:
seq(A073490(n),n=1..110) ; # R. J. Mathar, Oct 27 2019

Mathematica

gaps[n_Integer/;n>0]:=If[n===1, 0, Complement[Prime[PrimePi[Rest[ # ]]-1], # ]&[First/@FactorInteger[n]]]; Table[Length[gaps[n]], {n, 1, 120}] (* Wouter Meeussen, Oct 30 2004 *)
pa[n_, k_] := If[k == NextPrime[n], 0, 1]; Table[Total[pa @@@ Partition[First /@ FactorInteger[n], 2, 1]], {n, 120}] (* Jayanta Basu, Jul 01 2013 *)

Python

from sympy import primefactors, nextprime
def a(n):
    pf = primefactors(n)
    return sum(p2 != nextprime(p1) for p1, p2 in zip(pf[:-1], pf[1:]))
print([a(n) for n in range(1, 121)]) # Michael S. Branicky, Oct 14 2021

Formula

a(n) = A073484(A007947(n)).

a(A000040(n))=0; a(A000961(n))=0; a(A006094(n))=0; a(A002110(n))=0; a(A073485(n))=0.

a(A073486(n))>0; a(A073487(n)) = 1; a(A073488(n))=2; a(A073489(n))=3.

a(n)=0 iff A073483(n) = 1.

a(A097889(n)) = 0. - Reinhard Zumkeller, Nov 20 2004

0 <= a(m*n) <= a(m) + a(n) + 1. A137794(n) = 0^a(n). - Reinhard Zumkeller, Feb 11 2008

Extensions

More terms from Franklin T. Adams-Watters, May 19 2006