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A109313 Difference between prime factors of n-th semiprime.

%I A109313 #36 Apr 22 2025 04:00:40
%S A109313 0,1,0,3,5,2,4,9,0,11,8,15,2,17,10,21,0,14,6,16,27,29,8,20,35,4,39,12,
%T A109313 41,26,6,28,45,14,51,34,18,57,10,0,59,38,40,12,65,44,69,2,24,71,26,77,
%U A109313 50,16,81,0,56,87,58,32,6,95,64,99,22,36,101,8,68,105,38,24,107,70,4
%N A109313 Difference between prime factors of n-th semiprime.
%C A109313 a(n)=0 iff sp(n) is a square of prime, sp(n) = n-th semiprime = A001358(n).
%H A109313 Alois P. Heinz, <a href="/A109313/b109313.txt">Table of n, a(n) for n = 1..20000</a> (first 1000 terms from Zak Seidov)
%e A109313 a(1)=0 because sp(1)=4=2*2 and 2-2=0; a(2)=1 because sp(2)=6=2*3 and 3-2=1; sp(n)=n-th semiprime.
%p A109313 with(numtheory): a:=proc(n) if bigomega(n)=2 and nops(factorset(n))=2 then factorset(n)[2]-factorset(n)[1] elif bigomega(n)=2 then 0 else fi end: seq(a(n),n=1..225); # _Emeric Deutsch_
%p A109313 # second Maple program:
%p A109313 b:= proc(n) option remember; local k;
%p A109313       if n=1 then 4
%p A109313     else for k from 1+b(n-1) do if not isprime(k) and
%p A109313             numtheory[bigomega](k)=2 then return k fi
%p A109313          od
%p A109313       fi
%p A109313     end:
%p A109313 a:= n-> (s-> max(s)-min(s))(numtheory[factorset](b(n))):
%p A109313 seq(a(n), n=1..100);  # _Alois P. Heinz_, Feb 05 2017
%t A109313 spQ[n_] := PrimeOmega[n] == 2; fi[n_] := FactorInteger[n];
%t A109313 f[n_] := fi[n][[-1, 1]] - fi[n][[1, 1]];
%t A109313 f[#] & /@ Select[Range@215, spQ] (* _Zak Seidov_, Oct 16 2014 *)
%o A109313 (Python)
%o A109313 from math import isqrt
%o A109313 from sympy.ntheory.primetest import is_square
%o A109313 from sympy import primepi, primerange, primefactors
%o A109313 def A109313(n):
%o A109313     def bisection(f,kmin=0,kmax=1):
%o A109313         while f(kmax) > kmax: kmax <<= 1
%o A109313         kmin = kmax >> 1
%o A109313         while kmax-kmin > 1:
%o A109313             kmid = kmax+kmin>>1
%o A109313             if f(kmid) <= kmid:
%o A109313                 kmax = kmid
%o A109313             else:
%o A109313                 kmin = kmid
%o A109313         return kmax
%o A109313     def f(x): return int(n+x+((t:=primepi(s:=isqrt(x)))*(t-1)>>1)-sum(primepi(x//p) for p in primerange(s+1)))
%o A109313     return 0 if is_square(m:=bisection(f,n,n)) else (p:=primefactors(m))[1]-p[0] # _Chai Wah Wu_, Apr 03 2025
%Y A109313 Cf. A001358, A068318, A178313.
%K A109313 nonn
%O A109313 1,4
%A A109313 _Zak Seidov_, Jun 27 2005
%E A109313 Edited by _Zak Seidov_, Oct 16 2014