cp's OEIS Frontend

A258840 a(n) is the least integer k such that there are n values of i <= k for which gpf(i^2 + 1) = gpf(k^2 + 1), where gpf(x) is the greatest prime factor of x.

%I A258840 #13 Sep 09 2017 23:27:28
%S A258840 1,3,7,38,47,157,302,327,515,616,697,798,818,1303,2818,3141,3323,5648,
%T A258840 6962,9193,9872,13213,13747,15445,16271,17149,18263,20491,20727,24389,
%U A258840 26915,29078,31867,37848,38007,40182,41508,43328,46349,55025,62258,63133,66893
%N A258840 a(n) is the least integer k such that there are n values of i <= k for which gpf(i^2 + 1) = gpf(k^2 + 1), where gpf(x) is the greatest prime factor of x.
%C A258840 A014442(n) gives the largest prime factor of n^2 + 1.
%C A258840 The primes of the sequence are 3, 7, 47, 157, 1303, 3323, 46349, ...
%C A258840 The corresponding sequence Gpf(a(n)^2+1) is 2, 5, 5, 17, 17, 29, 37, 37, 101, 101, 101, 101, 101, 101, 101, 101, 101, 97, 97, 97, 97, 401, 349, 389, 557, 557, 557, 557, 557, 421, 421, 421, 557, ... and it is interesting to observe the frequency of repetitions for the numbers 5, 17, 37, 97, 101, 557, ...
%e A258840 a(3) = 7 because gpf(7^2 + 1) = gpf(3^2 + 1) = gpf(2^2 + 1) = 5 => 3 occurrences.
%e A258840 a(4) = 38 because gpf(38^2 + 1) = gpf(21^2 + 1) = gpf(13^2 + 1) = gpf(4^2 + 1) = 17 => 4 occurrences.
%p A258840 with(numtheory):nn:=70000:T:=array(1..nn):k:=0:kk:=1:
%p A258840 for m from 1 to nn do:
%p A258840 x:=factorset(m^2+1):n1:=nops(x):p:=x[n1]:k:=k+1:T[k]:=p:
%p A258840 od:
%p A258840 for n from 1 to 43 do:jj:=0:for k from kk to nn while(jj=0) do:
%p A258840   q:=T[k]:ii:=0:jj:=0:
%p A258840     for i from 1 to k do:
%p A258840       if T[i]=q then ii:=ii+1:
%p A258840       else
%p A258840       fi:
%p A258840     od:if ii=n then jj:=1:kk:=k:
%p A258840     printf ( "%d %d \n",n,k):else fi:
%p A258840   od:od:
%o A258840 (PARI) gpf(n) = my(f=factor(n^2+1)); f[#f~,1];
%o A258840 nboc(k) = my(gpfk = gpf(k)); sum(i=1, k, gpf(i) == gpfk);
%o A258840 a(n) = my(k = 1); while (nbo(k) != n, k++); k; \\ _Michel Marcus_, Jun 12 2015
%Y A258840 Cf. A014442, A242012.
%K A258840 nonn
%O A258840 1,2
%A A258840 _Michel Lagneau_, Jun 12 2015