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)=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
KEYWORD
nonn,nice
AUTHOR
Reinhard Zumkeller, Aug 03 2002
EXTENSIONS
More terms from Franklin T. Adams-Watters, May 19 2006
STATUS
approved