cp's OEIS Frontend

A238138 Let p(k) be the k-th prime of the form m^2 + 1. Sequence lists the numbers k such that each of the prime divisors of the composite numbers of the form m^2 + 1 between p(k) and p(k+1) is also a divisor of some m^2 + 1 < p(k).

%I A238138 #19 Sep 09 2017 23:24:16
%S A238138 62,149,257,281,286,365,403,418,526,534,573,577,579,712,744,825,849,
%T A238138 877,973,992,1016,1106,1191,1243,1251,1257,1286,1341,1388,1440,1487,
%U A238138 1526,1636,1656,1841,1844,1846,1953,1966,2028,2108,2120,2142,2225,2272,2392,2409
%N A238138 Let p(k) be the k-th prime of the form m^2 + 1. Sequence lists the numbers k such that each of the prime divisors of the composite numbers of the form m^2 + 1 between p(k) and p(k+1) is also a divisor of some m^2 + 1 < p(k).
%e A238138 62 is in the sequence:
%e A238138 436^2 + 1 = 190097 is the 62nd prime of the form m^2 + 1;
%e A238138 437^2 + 1 = 190970 = 2 * 5 * 13^2 * 113;
%e A238138 438^2 + 1 = 191845 = 5 * 17 * 37 * 61;
%e A238138 439^2 + 1 = 192722 = 2 * 173 * 557;
%e A238138 440^2 + 1 = 193601 is the 63rd prime of the form m^2 + 1;
%e A238138 and each of the prime divisors of 437^2 + 1, 438^2 + 1, and 439^2 + 1 is also a divisor of some m^2 + 1 < 436^2 + 1:
%e A238138     1^2 + 1 = 2,
%e A238138     3^2 + 1 = 2 * 5,
%e A238138     4^2 + 1 = 17,
%e A238138     5^2 + 1 = 2 * 13,
%e A238138     6^2 + 1 = 37,
%e A238138    11^2 + 1 = 2 * 61,
%e A238138    15^2 + 1 = 2 * 113,
%e A238138    80^2 + 1 = 37 * 173,
%e A238138   118^2 + 1 = 5^2 * 557.
%p A238138 with(numtheory):lst:={}: lst2:={}:T:=array(1..2000000):kk:=1:k:=0:for n from 2 by 2 to 200000 do: p:=n^2+1:if type(p, prime)=true then k:=k+1:T[k]:=p:else fi:od:for i from 1 to k do:lst1:={}:a:=sqrt(T[i]-1):b:=sqrt(T[i+1]-1):for j from a+1 to b-1 do:y:=factorset(j^2+1):lst1:=lst1 union y:od:lst1:=lst1 minus lst: if lst1<>{} then kk:=kk+1:lst:=lst union {lst1[1]}:else kk:=kk+1: printf(`%d, `,kk):fi:od:
%Y A238138 Cf. A002496, A002522, A234739, A238139.
%K A238138 nonn
%O A238138 1,1
%A A238138 _Michel Lagneau_, Feb 18 2014
%E A238138 Edited by _Jon E. Schoenfield_, Sep 09 2017