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Numbers n such that n^2 + 1 has three distinct prime divisors less than n.
2

%I #17 Nov 26 2019 04:17:53

%S 21,43,57,72,99,111,117,119,128,132,142,172,174,185,192,193,200,211,

%T 212,216,251,268,294,305,322,336,338,342,351,360,378,394,408,418,431,

%U 443,448,450,460,485,498,509,515,524,552,560,562,568,580,601,606,612,616

%N Numbers n such that n^2 + 1 has three distinct prime divisors less than n.

%C Subsequence of A256011.

%C The "triprimes n^2+1 numbers" are the numbers that are the product of exactly three (not necessarily distinct) primes less than n.

%C If the three prime divisors are distinct, the corresponding subsequence is 21, 72, 111, 119, 128, 142, 172, 174, 185, 192, 200, 211, 212, 216, 294, 305, 322, 336, 338, 342, 351, 360, 394, 431, 448, 450, 460, 485, 498, 509, 524, 552, 560, 562, 580, ...

%C The corresponding sequence of the number of prime divisors with multiplicity is 3, 4, 5, 3, 4, 3, 4, 3, 3, 4, 3, 3, 3, 3, 3, 5, 3, 3, 3, 3, 4, 5, 3, 3, 3, 3, 3, 3, 3, 3, 4, 3, 4, 4, 3, 6, 3, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 5, 3, 3, 4, 3, 4, 3, 3, 3, 3, 4, 3, 4, ...

%H Amiram Eldar, <a href="/A263877/b263877.txt">Table of n, a(n) for n = 1..10000</a>

%e 72 is in the sequence because 72^2 + 1 = 5*17*61 and 5, 17 and 61 are less than 72.

%t Select[Range[800], PrimeNu[#^2+1] == 3&&FactorInteger[#^2+1][[1,1]]<#&& FactorInteger[#^2+1][[2,1]]<#&&FactorInteger[#^2+1][[3,1]]<#&]

%o (PARI) for(n=1, 1e3, t=n^2+1; if ((omega(t) == 3) && (factor(t)[, 1][3] < n), print1(n, ", "))); \\ _Altug Alkan_, Oct 28 2015

%Y Cf. A014612, A256011, A263876.

%K nonn

%O 1,1

%A _Michel Lagneau_, Oct 28 2015