A123652 - OEIS

%I #13 Sep 08 2022 08:45:28

%S 14,2339155617965,36923966682271786990,4854597644377050732053585,

%T 45547499507677574921923909526,80266855145143309588022024772829,

%U 44586202603279528645530450127574150

%N a(n) = 1 + n^2 + n^3 + n^5 + n^7 + n^11 + n^13 + n^17 + n^19 + n^23 + n^29 + n^31 + n^37 + n^41.

%C 13th row, A(13,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 sequence A(13,n) = a(n) can never be prime because of the polynomial factorization. It can be semiprime, as with a(1) = 14 and a(2) = 2339155617965 = 5 * 467831123593 and a(6) and 100010000010100000100010100010100010101101 = 101 * 990198019901980199010000990199010001001. We similarly have polynomial factorization for the 7th row, A123651 = A(7,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).

%H G. C. Greubel, <a href="/A123652/b123652.txt">Table of n, a(n) for n = 1..1000</a>

%F a(n) = 1+n^2+n^3+n^5+n^7+n^11+n^13+n^17+n^19+n^23+n^29+n^31+n^37+n^41 = 100010000010100000100010100010100010101101(base n) = +/-(n^2+1)*(n^39-n^37+2n^35-2n^33+2n^31-n^29+2n^27-2n^25+2n^23-n^21+n^19+n^15-n^13+2n^11-n^9+n^7+n^3+1).

%t Table[1+Total[n^Prime[Range[PrimePi[41]]]],{n,8}] (* _Harvey P. Dale_, Dec 20 2010 *)

%o (PARI) for(n=1,25, print1(1 + n^2 + n^3 + n^5 + n^7 + n^11 + n^13 + n^17 + n^19 + n^23 + n^29 + n^31 + n^37 + n^41, ", ")) \\ _G. C. Greubel_, Oct 17 2017

%o (Magma) [1 + n^2 + n^3 + n^5 + n^7 + n^11 + n^13 + n^17 + n^19 + n^23 + n^29 + n^31 + n^37 + n^41: n in [1..25]]; // _G. C. Greubel_, Oct 17 2017

%Y Cf. A002522, A098547, A123111, A123113, A123650, A123651.

%K easy,nonn

%O 1,1

%A _Jonathan Vos Post_, Oct 04 2006