A242109 First of two consecutive (primes of the form n^2+1) with no semiprime of the same form between them.

2, 2917, 13457, 15377, 15877, 21317, 78401, 147457, 190097, 215297, 217157, 287297, 401957, 414737, 577601, 1299601, 1308737, 1313317, 1378277, 1547537, 1623077, 1664101, 1731857, 1742401, 1822501, 1887877, 1976837, 2044901, 2390117, 2421137, 2446097, 2483777

Offset

1,1

Links

Harvey P. Dale, Table of n, a(n) for n = 1..104 (* All terms up to and including the 2 millionth prime. *)

Example

2 is in the sequence because there is no semiprime between the two primes 1^2 + 1 = 2 and 2^2 + 1 = 5 of the form k^2 + 1.
2917 is in the sequence because there is no semiprime between the two primes 54^2 + 1 = 2917 and 56^2 + 1 = 3127 : 55^2 + 1 = 3026 = 2*17*89 is not a semiprime.

Maple

with(numtheory):nn:=2000: lst:={}:
for n from 1 to nn do:
  if type(n^2+1, prime)=true
    then
    lst:=lst union {n}:
    else
  fi:
od:
n1:=nops(lst):
  for m from 1 to n1-1 do:
    i1:=lst[m]:i2:=lst[m+1]:ii:=0:
     for k from i1+1 to i2-1 do:
       x:=k^2+1:y:=factorset(x):
         if bigomega(x)=2 and nops(y)=2
         then
         ii:=ii+1:
         else
       fi:
    od:
    if ii=0
    then
    printf(`%d, `, i1^2+1):
    else
    fi:
od:

Mathematica

nspQ[{a_, b_}]:=Select[Range[a+1, b-1], IntegerQ[Sqrt[#-1]]&&PrimeOmega[#]==2&]=={}; With[{cp=Partition[Select[Prime[Range[200000]], IntegerQ[Sqrt[#-1  ]]&], 2, 1]}, Select[cp, nspQ]][[;; , 1]] (* Harvey P. Dale, Oct 23 2025 *)

Prog

(PARI)
for(n=1, 10^4, if(isprime(n^2+1), k=1; while(!isprime((n+k)^2+1), k++); c=0; for(i=1, k-1, d=factor((n+i)^2+1); s=sum(j=1, #d[, 1], d[j, 2]); if(s==2, c++; break)); if(c==0, print1(n^2+1, ", ")))) \\ Derek Orr, Aug 15 2014

Crossrefs

Cf. A002496, A005574.

Sequence in context: A109119 A002495 A353126 * A078457 A128148 A308575

Adjacent sequences: A242106 A242107 A242108 * A242110 A242111 A242112

Keyword

nonn

Author

Michel Lagneau, Aug 15 2014

Status

approved