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 A001576 a(n) = 1^n + 2^n + 4^n. 99
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS Equals A135576, except for the first member. - Omar E. Pol, Nov 18 2008 Conjecture: Let n>1, if a(n)= 1^n+2^n+4^n is a prime number then n is the form 3^h. Example, for h=1, n=3, a(n)= 1^3+2^3+4^3=73 (prime); 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 Crux Mathematicorum Link. - Bernard Schott, Mar 08 2019 LINKS T. D. Noe, Table of n, a(n) for n = 0..200 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). 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 MATHEMATICA Table[1^n + 2^n + 4^n, {n, 0, 24}] PROG (Sage) [sigma(4, n)for n in range(0, 23)] # Zerinvary Lajos, Jun 04 2009 (PARI) a(n)=1+2^n+4^n \\ Charles R Greathouse IV, Jun 10 2011 CROSSREFS Cf. A001550, A034513, A001579, A074501-A074580, A135576, A135577. See also comments in A051154. Cf. A006095, A022166. Sequence in context: A148678 A148679 A148680 * A169587 A075211 A075212 Adjacent sequences:  A001573 A001574 A001575 * A001577 A001578 A001579 KEYWORD nonn,easy AUTHOR STATUS approved

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Last modified May 30 05:35 EDT 2020. Contains 334712 sequences. (Running on oeis4.)