A263877 Numbers n such that n^2 + 1 has three distinct prime divisors less than n.

21, 43, 57, 72, 99, 111, 117, 119, 128, 132, 142, 172, 174, 185, 192, 193, 200, 211, 212, 216, 251, 268, 294, 305, 322, 336, 338, 342, 351, 360, 378, 394, 408, 418, 431, 443, 448, 450, 460, 485, 498, 509, 515, 524, 552, 560, 562, 568, 580, 601, 606, 612, 616

Offset: 1

Author

Michel Lagneau, Oct 28 2015

Keywords

• nonn

Comments

Subsequence of A256011.

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

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, ...

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, ...

Examples

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

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A014612, A256011, A263876.

Programs

Mathematica

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

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