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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.

%I A136106 #34 Jan 09 2023 13:25:32
%S A136106 2,5,103,1867,491851,17681491,35565206671,43194825904693
%N 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.
%F A136106 a(n) >= A086560(n). - _R. J. Mathar_, Feb 05 2008
%F A136106 A001221(a(n)+k) = k+1 for 0 <= k <= n-1. - _Pontus von Brömssen_, Jan 09 2023
%e A136106 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.
%t A136106 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 *)
%o A136106 (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
%Y A136106 Cf. A001221, A072875, A086560.
%K A136106 more,nonn
%O A136106 1,1
%A A136106 _Enoch Haga_, Dec 14 2007
%E A136106 Edited by _N. J. A. Sloane_, Dec 23 2007
%E A136106 a(5)-a(6) from _Fred Schneider_, Dec 18 2007
%E A136106 a(7) from _Donovan Johnson_, Sep 19 2009
%E A136106 a(8) from _Donovan Johnson_, Jul 19 2011
%E A136106 Name clarified by _Pontus von Brömssen_, Jan 09 2023