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

1,110

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<=k<n and a(A137723(n)-1)=a(A137723(n)+n)=0. - Reinhard Zumkeller, Feb 09 2008

Links

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

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

Example

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.

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, 105}] (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 *)

Prog

(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
(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

Crossrefs

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

Cf. A137721, A137722, A027748, A049084.

Sequence in context: A265892 A324966 A005090 * A279907 A225654 A236747

Adjacent sequences: A073487 A073488 A073489 * A073491 A073492 A073493

Keyword

nonn,nice

Author

Reinhard Zumkeller, Aug 03 2002

Extensions

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

Status

approved