A360944 Numbers m such that phi(m) is a triangular number, where phi is the Euler totient function (A000010).

1, 2, 7, 9, 11, 14, 18, 22, 29, 37, 57, 58, 63, 67, 74, 76, 79, 108, 114, 126, 134, 137, 143, 155, 158, 175, 183, 191, 211, 225, 231, 244, 248, 274, 277, 286, 308, 310, 329, 341, 350, 366, 372, 379, 382, 396, 417, 422, 423, 450, 453, 462, 554, 556, 604, 623, 631, 658, 682

Offset: 1

Author

Bernard Schott, Feb 26 2023

Keywords

• nonn

Comments

Subsequence of primes is A055469 because in this case phi(k(k+1)/2+1) = k(k+1)/2.

Subsequence of triangular numbers is A287472.

Examples

			phi(57) = 36 = 8*9/2, a triangular number; so 57 is a term of the sequence.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000010, A000217, A055469, A287472.

Similar, but with phi(m) is: A039770 (square), A078164 (biquadrate), A096503 (repdigit), A117296 (palindrome), A236386 (oblong).

Programs

Maple

filter := m ->  issqr(1 + 8*numtheory:-phi(m)) : select(filter, [$(1 .. 700)]);

Mathematica

Select[Range[700], IntegerQ[Sqrt[8 * EulerPhi[#] + 1]] &] (* Amiram Eldar, Feb 27 2023 *)

PARI

isok(m) = ispolygonal(eulerphi(m), 3); \\ Michel Marcus, Feb 27 2023

Python

from itertools import islice, count
from sympy.ntheory.primetest import is_square
from sympy import totient
def A360944_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:is_square((totient(n)<<3)+1), count(max(1,startvalue)))
A360944_list = list(islice(A360944_gen(),20)) # Chai Wah Wu, Feb 28 2023