A136106 a(n) is the smallest prime p such that in the sequence of n numbers p, p+1, p+2, ..., p+n-1, the i-th term has exactly i distinct prime factors, for i = 1, ..., n.
2, 5, 103, 1867, 491851, 17681491, 35565206671, 43194825904693
Offset: 1
Examples
a(4) = 1867 because it begins with the prime 1867 followed by 1868 with two distinct prime factors, 2 and 467; then 1869 with three distinct prime factors, 3, 7 and 89; then 1870 with four distinct prime factors, 2, 5, 11 and 17.
Programs
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Mathematica
Table[First[Select[Prime@Range@100000,(n=1; k=#; While[Length[First/@FactorInteger@k]==n, k++; n++]; n-1==t)&]],{t, 5}] (* Giorgos Kalogeropoulos, May 07 2019 *) -
PARI
/* a brute force program */ a136106(st,ed,ct)={ forprime(x=st,ed, if ((x%6)!=1,next); goodFlag = 1; c = 1; while(goodFlag, if (!(c%2) && isprime(x+c), goodFlag=0, v = factor(x+c); if (length(v[,2]) == c+1, c+=1; if (c > ct,print("Level = ",c," at ",x+c-1,"=",v); ct+=1), goodFlag = 0 ) ) ) ); } \\ Fred Schneider, Dec 18 2007
Formula
a(n) >= A086560(n). - R. J. Mathar, Feb 05 2008
A001221(a(n)+k) = k+1 for 0 <= k <= n-1. - Pontus von Brömssen, Jan 09 2023
Extensions
Edited by N. J. A. Sloane, Dec 23 2007
a(5)-a(6) from Fred Schneider, Dec 18 2007
a(7) from Donovan Johnson, Sep 19 2009
a(8) from Donovan Johnson, Jul 19 2011
Name clarified by Pontus von Brömssen, Jan 09 2023