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).
62, 149, 257, 281, 286, 365, 403, 418, 526, 534, 573, 577, 579, 712, 744, 825, 849, 877, 973, 992, 1016, 1106, 1191, 1243, 1251, 1257, 1286, 1341, 1388, 1440, 1487, 1526, 1636, 1656, 1841, 1844, 1846, 1953, 1966, 2028, 2108, 2120, 2142, 2225, 2272, 2392, 2409
Offset: 1
Keywords
Examples
62 is in the sequence:
436^2 + 1 = 190097 is the 62nd prime of the form m^2 + 1;
437^2 + 1 = 190970 = 2 * 5 * 13^2 * 113;
438^2 + 1 = 191845 = 5 * 17 * 37 * 61;
439^2 + 1 = 192722 = 2 * 173 * 557;
440^2 + 1 = 193601 is the 63rd prime of the form m^2 + 1;
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:
1^2 + 1 = 2,
3^2 + 1 = 2 * 5,
4^2 + 1 = 17,
5^2 + 1 = 2 * 13,
6^2 + 1 = 37,
11^2 + 1 = 2 * 61,
15^2 + 1 = 2 * 113,
80^2 + 1 = 37 * 173,
118^2 + 1 = 5^2 * 557.
Programs
-
Maple
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:
Extensions
Edited by Jon E. Schoenfield, Sep 09 2017
Comments