A001576 a(n) = 1^n + 2^n + 4^n.

3, 7, 21, 73, 273, 1057, 4161, 16513, 65793, 262657, 1049601, 4196353, 16781313, 67117057, 268451841, 1073774593, 4295032833, 17180000257, 68719738881, 274878431233, 1099512676353, 4398048608257, 17592190238721, 70368752566273, 281474993487873, 1125899940397057

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Equals A135576, except for the first term. - Omar E. Pol, Nov 18 2008

Conjecture: For n > 1, if a(n) = 1^n + 2^n + 4^n is a prime number then n is of the form 3^h. For example, for h=1, n=3, a(n) = 1^3 + 2^3 + 4^3 = 73 (prime); for h=2, n=9, a(n) = 1^9 + 2^9 + 4^9 = 262657 (prime); for h=3, n=27, a(n) is not prime. - Vincenzo Librandi, Aug 03 2010

The previous conjecture was proved by Golomb in 1978. See A051154. - T. D. Noe, Aug 15 2010

Another more elementary proof can be found in Liu link. - Bernard Schott, Mar 08 2019

Fills in one quarter section of the figurate form of the Sierpinski square curve. See illustration in links and A141725. - John Elias, Mar 29 2023

Links

• T. D. Noe, Table of n, a(n) for n = 0..200
• John Elias, Illustration of Initial Terms: 1/4 Sierpinski Square Curve
• Andy Liu, West German Mathematical Olympiad 1982 - Second round, Problem 4, Crux Mathematicorum, p. 105, Vol. 12, May. 86.
• Index entries for linear recurrences with constant coefficients, signature (7,-14,8).

Crossrefs

Subsequence of A002061.

Cf. A001550, A034513, A001579, A074501-A074580, A135576, A135577.

See also comments in A051154.

Cf. A006095, A022166.

Programs

Mathematica

Table[1^n + 2^n + 4^n, {n, 0, 24}]

PARI

a(n)=1+2^n+4^n \\ Charles R Greathouse IV, Jun 10 2011

Sage

[sigma(4,n)for n in range(0,23)] # Zerinvary Lajos, Jun 04 2009

Formula

a(n) = 6*a(n-1) - 8*a(n-2) + 3.

O.g.f.: -1/(-1+x) - 1/(-1+2*x) - 1/(-1+4*x) = ( -3+14*x-14*x^2 ) / ( (x-1)*(2*x-1)*(4*x-1) ). - R. J. Mathar, Feb 29 2008

E.g.f.: e^x + e^(2*x) + e^(4*x). - Mohammad K. Azarian, Dec 26 2008

a(n) = A024088(n)/A000225(n). - Reinhard Zumkeller, Feb 15 2009

Exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + 7*x + 35*x^2 + 155*x^3 + ... is the o.g.f. for the 2nd subdiagonal of triangle A022166, essentially A006095. - Peter Bala, Apr 07 2015