%I #67 Apr 27 2023 22:08:06
%S 3,7,21,73,273,1057,4161,16513,65793,262657,1049601,4196353,16781313,
%T 67117057,268451841,1073774593,4295032833,17180000257,68719738881,
%U 274878431233,1099512676353,4398048608257,17592190238721,70368752566273,281474993487873,1125899940397057
%N a(n) = 1^n + 2^n + 4^n.
%C Equals A135576, except for the first term. - _Omar E. Pol_, Nov 18 2008
%C 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
%C The previous conjecture was proved by Golomb in 1978. See A051154. - _T. D. Noe_, Aug 15 2010
%C Another more elementary proof can be found in Liu link. - _Bernard Schott_, Mar 08 2019
%C 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
%H T. D. Noe, <a href="/A001576/b001576.txt">Table of n, a(n) for n = 0..200</a>
%H John Elias, <a href="/A001576/a001576.png">Illustration of Initial Terms: 1/4 Sierpinski Square Curve</a>
%H Andy Liu, <a href="https://cms.math.ca/crux/backfile/Crux_v12n05_May.pdf">West German Mathematical Olympiad 1982 - Second round, Problem 4</a>, Crux Mathematicorum, p. 105, Vol. 12, May. 86.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (7,-14,8).
%F a(n) = 6*a(n-1) - 8*a(n-2) + 3.
%F 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
%F E.g.f.: e^x + e^(2*x) + e^(4*x). - _Mohammad K. Azarian_, Dec 26 2008
%F a(n) = A024088(n)/A000225(n). - _Reinhard Zumkeller_, Feb 15 2009
%F 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
%t Table[1^n + 2^n + 4^n, {n, 0, 24}]
%o (Sage) [sigma(4,n)for n in range(0,23)] # _Zerinvary Lajos_, Jun 04 2009
%o (PARI) a(n)=1+2^n+4^n \\ _Charles R Greathouse IV_, Jun 10 2011
%Y Subsequence of A002061.
%Y Cf. A001550, A034513, A001579, A074501-A074580, A135576, A135577.
%Y See also comments in A051154.
%Y Cf. A006095, A022166.
%K nonn,easy
%O 0,1
%A _N. J. A. Sloane_