A185379 Product of exactly three distinct primes congruent to 1 mod 8 (A007519).

50881, 62033, 67609, 78761, 95489, 110449, 120377, 134521, 140233, 146761, 162401, 167977, 170017, 170969, 179129, 186337, 195857, 207281, 218161, 225913, 234889, 239513, 246041, 263177, 266377, 279497, 285073, 289153, 290321, 292009, 299081, 301801, 312953

Offset: 1

Author

Jonathan Vos Post, Feb 20 2011

Keywords

• nonn
• easy

Comments

Subset of numbers that are divisible by exactly 3 primes (counted with multiplicity), also known as triprimes or 3-almost primes, A014612. Subset of {d = p_1 * p_2 * ... * p_m where p_i == 1 (mod 8), 1 <= i <= m are distinct primes} as occurs in Wei, p.2.

Examples

			a(12) = 170017 = 17 * 73 * 137 = A007519(1) * A007519(3) * A007519(7).
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• Dasheng Wei, On the equation x^2-Dy^2=n, Feb 18, 2011.

Crossrefs

Cf. A007519, A014612, A017077, A185377.

Programs

Mathematica

p = Select[Prime[Range[100]], Mod[#, 8] == 1 &]; Sort[Reap[Do[n = p[[i]] p[[j]] p[[k]]; If[n <= p[[1]] p[[2]] p[[-1]], Sow[n]], {i, 2, Length[p]}, {j, i - 1}, {k, j - 1}]][[2, 1]]]

PARI

list(lim)=my(v=List(),u=v,t); forprime(p=2,lim\697, if(p%8==1, listput(u,p))); for(i=1,#u-2, for(j=i+1, #u-1, if(u[i]*u[j]*u[j+1]>lim, break); for(k=j+1,#u, t=u[i]*u[j]*u[k]; if(t>lim, break); listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Jan 31 2017

Formula

{A007519(i) * A007519(j) * A007519(k) for i < j < k}. {A000040(i) * A000040(j) * A000040(k) for i < j < k, and A000040(i) in A017077 and A000040(j) in A017077 and A000040(k) in A017077}.