A020505 -id:A020505

Search: a020505 -id:a020505

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A138924

Indices k such that A020505(k)=Phi[k](-6) is prime, where Phi is a cyclotomic polynomial.

+20
2

3, 4, 6, 8, 9, 11, 14, 15, 21, 24, 25, 31, 42, 43, 45, 47, 58, 59, 77, 107, 124, 142, 144, 177, 192, 254, 279, 360, 407, 437, 480, 525, 542, 551, 579, 764, 811, 822, 891, 917, 1018, 1028, 1150, 1326, 1376, 1464, 1468, 1650, 1719, 1924, 2096, 2098, 2176, 2226

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Robert Price, Table of n, a(n) for n = 1..104

Yves Gallot, Cyclotomic polynomials and prime numbers

MATHEMATICA

Select[ Range[ 2000], PrimeQ[ Cyclotomic[#, -6]] &] (* Robert G. Wilson v, Mar 25 2012 *)

PROG

(PARI) for( i=1, 999, ispseudoprime( polcyclo(i, -6)) && print1( i", ")) /* use ...subst(polcyclo(i), x, -6)... in PARI < 2.4.2 */

CROSSREFS

Cf. A020505, A138920-A138940.

KEYWORD

nonn

AUTHOR

M. F. Hasler, Apr 03 2008

EXTENSIONS

a(51)-a(54) from Robert Price, Apr 02 2012

STATUS

approved

A179897

a(n) = (n^(2*n+1) + 1) / (n+1).

+10
3

1, 1, 11, 547, 52429, 8138021, 1865813431, 593445188743, 250199979298361, 135085171767299209, 90909090909090909091, 74619186937936447687211, 73381705110822317661638341, 85180949465178001182799643437, 115244915978498073437814463065839, 179766618030828831251710653305053711

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

a(n) is the arithmetic mean of the multiset consisting of n lots of 1/n and one lot of n^(2*n+1). This multiset also has an integer valued geometric mean which is equal to n for n > 0.

According to search at OEIS for particular sequence members, a(n) is also: (1+2*n)-th q-integer for q=-n, (2*(n+1))-th cyclotomic polynomial at q=-n, Gaussian binomial coefficient [2*n+1, 2*n] for q=-n, number of walks of length 1+2*n between any two distinct vertices of the complete graph K_(n+1).

LINKS

Andrew Howroyd, Table of n, a(n) for n = 0..100

Google Groups, Integer-valued arithmetic and geometric means of sequences with non-integer numbers

FORMULA

a(n) = Sum_{i=0..2*n} (-n)^i.

EXAMPLE

For n = 2, a(2) = 11 which is the arithmetic mean of {1/2, 1/2, 2^5} = 33 / 3 = 11. The geometric mean is 8^(1/3) = 2, i.e. both are integral.

PROG

(Python) [(n**(2*n+1)+1)//(n+1) for n in range(1, 11)]

(PARI) a(n) = (n^(2*n + 1) + 1)/(n + 1) \\ Andrew Howroyd, May 03 2023

CROSSREFS

Main diagonal of A362783.

Values for n = 5, 6 via other ways. Q-integers: A014986, A014987, K_n paths: A015531, A015540, Cyclotomic polynomials: A020504, A020505, Gaussian binomial coefficients: A015391, A015429.

KEYWORD

easy,nonn

AUTHOR

Martin Saturka (martin(AT)saturka.net), Jul 31 2010

EXTENSIONS

Edited, a(0)=1 prepended and more terms from Andrew Howroyd, May 03 2023

STATUS

approved

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