A374846 Numbers appearing exactly once in a Pythagorean triple.

3, 4, 6, 7, 11, 14, 19, 22, 23, 31, 38, 43, 46, 47, 59, 62, 67, 71, 79, 83, 86, 94, 103, 107, 118, 127, 131, 134, 139, 142, 151, 158, 163, 166, 167, 179, 191, 199, 206, 211, 214, 223, 227, 239, 251, 254, 262, 263, 271, 278, 283, 302, 307, 311, 326, 331, 334, 347, 358, 359, 367, 379, 382, 383, 398

Offset: 1

Author

Manfred Boergens, Jul 22 2024

Keywords

• nonn

Comments

Positions of the ones in A046081.

With the exception a(2) = 4, the terms are given by A374845, thus providing a simple formula for the sequence.

Links

• A. Tripathi, On Pythagorean triples containing a fixed integer, Fib. Q., 46/47 (2008/2009), 331-340. See Theorem 8.

Crossrefs

Cf. A374845, A046081.

Programs

Mathematica

t={}; Do[If[(PrimeQ[n] && Mod[n, 4] == 3) || (PrimeQ[n/2] && Mod [n/2, 4] == 3), t = Join[t, {n}]], {n, 445}]; t = Insert[t, 4, 2]
(* Positions of the ones in  A046081; based on program by Jean-François Alcover *)
a[1] = 0; a[n_] := Module[{f}, f = Select[FactorInteger[n], Mod[#[[1]], 4] == 1 &][[All, 2]]; (DivisorSigma[0, If[OddQ[n], n, n/2]^2] - 1)/2 + (Times @@ (2*f + 1) - 1)/2]; arr = Array[a, 445]; fl = Flatten[Position[arr, 1]]

Python

from itertools import count, islice
from sympy import isprime
def A374846_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:n==4 or (isprime(n) and n&3==3) or (isprime(n>>1) and n&7==6), count(max(startvalue,1)))
A374846_list = list(islice(A374846_gen(),20)) # Chai Wah Wu, Jul 31 2024

Formula

p or 2p with p prime and p = 3 mod 4, with 4 added to the sequence, in ascending order.