A123651 a(n) = 1 + n^2 + n^3 + n^5 + n^7 + n^11 + n^13 + n^17.

8, 141485, 130914100, 17251189841, 764209065776, 16940083223773, 232729381165100, 2252358161564225, 16679754951397336, 100010100010101101, 505481836542757988, 2218718842990269265, 8650720586711446400

Offset

1,1

Comments

7th row, A(7,n), of the infinite array A(k,n) = 1 + Sum_{i=1..k} n^prime(i). If we deem prime(0) = 1, the array is A(k,n) = Sum_{i=0..k} n^prime(i). The first row is A002522 = 1 + n^2. The second row is A098547 = 1 + n^2 + n^3. The 3rd row, A(3,n), is A123650. The 4th row, A(4,n), is A123111 1 + n^2 + n^3 + n^5 + n^7. 10101101 (base n). A(n,n) is A123113 Main diagonal of prime power sum array. The current sequence, A(7,n), can never be prime, because of the polynomial factorization a(n) = 1 + n^2 + n^3 + n^5 + n^7 + n^11 + n^13 + n^17 = +/- (n^2+1)*(n^15 -n^13 +2n^11 -n^9 +n^7 +n^3 +1). It can be semiprime, as with a(2) and with a(10) = 100010100010101101 = 101 * 990199010001001 and a(14). We similarly have polynomial factorization for A123652 = A(13,n) = 1 +n^2 +n^3 +n^5 +...+ n^41.

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Formula

a(n) = 1 + n^2 + n^3 + n^5 + n^7 + n^11 + n^13 + n^17 = 100010100010101101 (base n) = +/- (n^2+1)*(n^15-n^13+2n^11-n^9+n^7+n^3+1).

Mathematica

Table[Total[n^Prime[Range[7]]]+1, {n, 20}] (* Harvey P. Dale, Aug 22 2012 *)

Prog

(PARI) for(n=1, 25, print1(1 + n^2 + n^3 + n^5 + n^7 + n^11 + n^13 + n^17, ", ")) \\ G. C. Greubel, Oct 17 2017
(Magma) [1 + n^2 + n^3 + n^5 + n^7 + n^11 + n^13 + n^17: n in [1..25]]; // G. C. Greubel, Oct 17 2017

Crossrefs

Cf. A002522, A098547, A123111, A123113, A123650, A123652.

Sequence in context: A269877 A123276 A308138 * A210812 A223939 A013846

Adjacent sequences: A123648 A123649 A123650 * A123652 A123653 A123654

Keyword

easy,nonn

Author

Jonathan Vos Post, Oct 04 2006

Status

approved