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A260258 T(n,k) is the array read by rows, n>0 and k=1..q (with q = number of prime distinct divisors of n^2+1) giving the number of occurrences of the k-th prime divisor of n^2+1 counted from the prime divisors of m^2+1 for m=1..n.

%I A260258 #22 Oct 29 2021 06:21:35
%S A260258 1,1,2,2,1,3,1,1,4,3,4,2,5,1,1,6,1,5,1,7,6,2,1,8,1,1,9,7,2,8,3,10,1,1,
%T A260258 11,4,3,9,1,12,10,1,1,13,1,1,14,11,1,12,1,15,1,4,2,16,5,2,13,2,17,14,
%U A260258 1,6,1,18,1,1,19,15,1,16,5,20,1,1,21,3,17,1
%N A260258 T(n,k) is the array read by rows, n>0 and k=1..q (with q = number of prime distinct divisors of n^2+1) giving the number of occurrences of the k-th prime divisor of n^2+1 counted from the prime divisors of m^2+1 for m=1..n.
%C A260258 A002313(n) are the numbers such that T(n,k)>1 for all k=1..q.
%C A260258 T(2n-1,1)=n and T(m,1)=1 if m =1, 2, 4, 6, 10, 14, ... = A005574(n)(numbers n such that n^2 + 1 is prime). The length of row n is A128428(n).
%H A260258 Michel Lagneau, <a href="/A260258/b260258.txt">Table of n, a(n) for n = 1..5000</a>
%e A260258 T(13,k) = [7,6,2] for k = 1,2,3 because 13^2+1 = 2*5*17 =>
%e A260258 The number of occurrences of the prime divisor 2 is 7: 1^2+1=2, 3^2+1=2*5, 5^2+1=2*13, 7^2+1=2*5^2, 9^2+1=2*41, 11^2+1=2*61 and 13^2+1=2*5*17;
%e A260258 The number of occurrences of the prime divisor 5 is 6: 2^2+1=5, 3^2+1=2*5, 7^2+1=2*5^2, 8^2+1=5*13, 12^2+1=5*29;
%e A260258 The number of occurrences of the prime divisor 17 is 2: 4^2+1=17 and 13^2+1=2*5*17.
%e A260258 The array begins:
%e A260258   [1]
%e A260258   [1]
%e A260258   [2,2]
%e A260258   [1]
%e A260258   [3,1]
%e A260258   [1]
%e A260258   [4,3]
%e A260258   [4,2]
%e A260258   [5,2]
%e A260258   [1]
%e A260258   ...
%p A260258 with(numtheory):lst:={2}:nn:=1000:T:=array(1..270,[0$270]):
%p A260258 for j from 1 to nn do:
%p A260258    p:=4*j+1:
%p A260258    if isprime(p)
%p A260258    then
%p A260258    lst:=lst union {p}:
%p A260258    fi:
%p A260258 od:
%p A260258    nn0:=nops(lst):
%p A260258    for n from 1 to 60 do:
%p A260258      q:=factorset(n^2+1):n0:=nops(q):
%p A260258      for k from 1 to n0 do:
%p A260258       for m from 1 to 270 do:
%p A260258       if q[k]=lst[m] then T[m]:=T[m]+1:printf(`%d, `, T[m]):
%p A260258       fi:
%p A260258      od:
%p A260258     od:
%p A260258 od:
%Y A260258 Cf. A005574, A002312, A002313, A128428.
%K A260258 nonn,tabf
%O A260258 1,3
%A A260258 _Michel Lagneau_, Jul 21 2015