A178313 Absolute difference between prime factors of n-th semiprime mod n.

0, 1, 0, 3, 0, 2, 4, 1, 0, 1, 8, 3, 2, 3, 10, 5, 0, 14, 6, 16, 6, 7, 8, 20, 10, 4, 12, 12, 12, 26, 6, 28, 12, 14, 16, 34, 18, 19, 10, 0, 18, 38, 40, 12, 20, 44, 22, 2, 24, 21, 26, 25, 50, 16, 26, 0, 56, 29, 58, 32, 6, 33, 1, 35, 22, 36, 34, 8, 68, 35, 38, 24, 34, 70, 4, 35, 42, 76, 6, 0

Offset

1,4

Links

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Example

a(2)=1 because semiprime(2) = 6 = 3*2 and (3-2) mod 2 = 1.

Mathematica

semiPrimeQ[n_] := Plus @@ Last /@ FactorInteger@ n == 2; f[n_] := Subtract @@ Reverse@ Flatten[ Table[ #[[1]], {#[[2]]}] & /@ FactorInteger@ n]; t = Select[ Range@ 215, semiPrimeQ]; Table[ Mod[ f[ t[[n]]], n], {n, 80}]
f[{a_, b_}]:=Module[{c=FactorInteger[b][[;; , 1]]}, If[Length[c]==1, 0, Mod[Differences[c][[1]], a]]]; Module[{nn=300, spr}, spr=Select[Range[nn], PrimeOmega[#]==2&]; f/@Thread[{Range[ Length[ spr]], spr}]] (* Harvey P. Dale, May 29 2024 *)

Prog

(Python)
from math import isqrt
from sympy import primepi, primerange, primefactors
from sympy.ntheory.primetest import is_square
def A178313(n):
    def bisection(f, kmin=0, kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    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)))
    return 0 if is_square(m:=bisection(f, n, n)) else ((p:=primefactors(m))[1]-p[0])%n # Chai Wah Wu, Apr 03 2025

Crossrefs

Cf. A001358, A178102, A109313.

Sequence in context: A112974 A113069 A136163 * A190013 A171088 A058624

Adjacent sequences: A178310 A178311 A178312 * A178314 A178315 A178316

Keyword

nonn

Author

Juri-Stepan Gerasimov, Dec 20 2010

Status

approved