1,1

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).

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).

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

Sequence in context: A013800 A013870 A164526 this_sequence A127622 A106239 A055965

Adjacent sequences: A123649 A123650 A123651 this_sequence A123653 A123654 A123655

easy,nonn

Jonathan Vos Post (jvospost3(AT)gmail.com), Oct 04 2006