A076680 -id:A076680 - cp's OEIS frontend

A180590 Numbers k such that k! is the sum of two triangular numbers.

0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 13, 15, 16, 17, 21, 24, 27, 28, 29, 32, 33, 34, 42, 49, 54, 59, 66, 68, 72, 79, 80, 81, 85, 86, 95, 96, 99, 102, 109, 118

Offset: 1

Robert G. Wilson v, Sep 10 2010

Numbers k such that there are nonnegative numbers x and y such that x*(x+1)/2 + y*(y+1)/2 = k!. Equivalently, (2x+1)^2 + (2y+1)^2 = 8k! + 2. A necessary and sufficient condition for this is that all the prime factors of 4k!+1 that are congruent to 3 (mod 4) occur to even powers (cf. A001481).
Based on an email from R. K. Guy to the Sequence Fans Mailing List, Sep 10 2010.
See A152089 for further links.

			0! = 1! = T(0) + T(1);
2! = T(1) + T(1);
3! = T(0) + T(3) = T(2) + T(2);
4! = T(2) + T(6);
5! = T(0) + T(15) = T(5) + T(14);
7! = T(45) + T(89);
8! = T(89) + T(269);
9! = T(210) + T(825);
10! = T(665) + T(2610) = T(1770) + T(2030);
13! = T(71504) + T(85680);
15! = T(213384) + T(1603064) = T(299894) + T(1589154);
16! = T(3631929) + T(5353005);
17! = T(12851994) + T(23370945) = T(17925060) + T(19750115);
etc.

Factor Database, Factors of the numbers 4z!+1

Complement of A152089. A171099 gives the number of solutions.

Cf. A000142, A000217, A051533, A076680.

triQ[n_] := IntegerQ@ Sqrt[8 n + 1]; fQ[n_] := Block[{k = 1, lmt = Floor@Sqrt[2*n! ], nf = n!}, While[k < lmt && ! triQ[nf - k (k + 1)/2], k++ ]; r = (Sqrt[8*(nf - k (k + 1)/2) + 1] - 1)/2; Print[{k, r, n}]; If[IntegerQ@r, True, False]]; k = 1; lst = {}; While[k < 69, If[ fQ@ k, AppendTo[lst, k]]; k++ ]; lst

from math import factorial
from itertools import count, islice
from sympy import factorint
def A180590_gen(): # generator of terms
    return filter(lambda n:all(p & 3 != 3 or e & 1 == 0 for p, e in factorint(4*factorial(n)+1).items()),count(0))
A180590_list = list(islice(A180590_gen(),15)) # Chai Wah Wu, Jun 27 2022

Edited by N. J. A. Sloane, Sep 24 2010
69 eliminated (see A152089) by N. J. A. Sloane, Sep 24 2010
Extended by Georgi Guninski and D. S. McNeil, Sep 24 2010
a(35)-a(38) from Georgi Guninski, Oct 12 2010
a(39)-a(40) from Tyler Busby, Apr 24 2025

A062538 Primes of the form 4*k! + 1.

Original entry on oeis.org

5, 97, 20161, 161281, 1451521, 24908083201, 83691159552001, 1219553378446855442006016000001, 923374789356965521888370970732110324333114258287231764529152000000000001

Offset: 1

Jason Earls, Jul 10 2001

nonn

			4*4! + 1 = 97, a prime.

Cf. A076680, A173322.

lst={};Do[p=4*n!+1;If[PrimeQ[p],AppendTo[lst,p]],{n,0,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 28 2009 *)
Select[(4Range[60]!+1),PrimeQ] (* Harvey P. Dale, May 08 2024 *)

for(n=1,55, if(isprime(4*n!+1),print(4*n!+1)))

a(n) = A173322(A076680(n+1)). - Elmo R. Oliveira, Apr 17 2025

cp's OEIS Frontend

A180590 Numbers k such that k! is the sum of two triangular numbers.

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