A384996 Numbers m such that phi(m) is a hexagonal number.

1, 2, 7, 9, 14, 18, 29, 58, 67, 134, 143, 155, 175, 183, 191, 225, 231, 244, 248, 277, 286, 308, 310, 329, 350, 366, 372, 379, 382, 396, 417, 423, 450, 462, 554, 556, 631, 658, 758, 834, 846, 869, 917, 947, 1129, 1179, 1262, 1327, 1415, 1633, 1738, 1834, 1894, 2017, 2117, 2147, 2159, 2258, 2264, 2358, 2359

Offset

1,2

Links

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

Example

Since phi(67) = 66 = 6*(2*6-1), a hexagonal number, 67 is a term of this sequence.

Maple

ishex:= proc(y) issqr(1+8*y) and sqrt(1+8*y) mod 4 = 3 end proc:
select(t -> ishex(NumberTheory:-phi(t)), [$1..2000]); # Robert Israel, Dec 23 2025

Mathematica

Select[Range[2500], IntegerQ[(1 + Sqrt[1 + 8 EulerPhi[#]])/4] &]

Prog

(PARI) isok(m) = ispolygonal(eulerphi(m), 6); \\ Michel Marcus, Sep 09 2025
(Python)
from math import isqrt
from sympy import totient as phi
ok = lambda n: (d:=1+8*phi(n)) == (s:=isqrt(d))**2 and (1+s)%4==0
print([m for m in range(1, 2500) if ok(m)]) # Aidan Chen, Sep 19 2025

Crossrefs

Cf. A000010, A000384, A384995, A384997.

Sequence in context: A205559 A047352 A184400 * A074244 A190565 A341631

Adjacent sequences: A384993 A384994 A384995 * A384997 A384998 A384999

Keyword

nonn,easy

Author

Aidan Chen, Sep 07 2025

Status

approved