1,1

Semiprime analogue of A123113 Main diagonal of prime power sum array. a(n) is prime for n = 1, 4; what is the next prime value in this sequence?

a(n) = 1 + n^4 + n^4 + n^6 + ... + n^semiprime(n) = 1 + SUM[i=1..n]n^semiprime(i) = Main diagonal A(n,n), of the infinite array A(k,n) = 1 + SUM[i=1..k]n^semiprime(i) = 1 + SUM[i=1..k]n^A001358(i). If we deem semiprime(0) = 1, the array is A(k,n) = SUM[i=0..k]n^A001358(i).

a(1) = 1 + 1^semiprime(1) = 1 + 1^4 = 2.

a(2) = 1 + 2^semiprime(1) + 2^semiprime(2) = 1 + 2^4 + 2^6 = 81.

a(3) = 1 + 3^semiprime(1) + 3^semiprime(2) + 3^semiprime(3) = 1 + 3^4 + 3^6 + 3^9 = 20494.

a(4) = 1 + 4^semiprime(1) + 4^semiprime(2) + 4^semiprime(3) + 4^semiprime(4) = 1 + 4^4 + 4^6 + 4^9 + 4^10 = 1315073 (which is prime).

a(5) = 1 + 5^semiprime(1) + 5^semiprime(2) + 5^semiprime(3) + 5^semiprime(4) + 5^semiprime(5) = 1 + 5^4 + 5^6 + 5^9 + 5^10 + 5^14 = 6115250626.

a(6) = 1 + 6^semiprime(1) + 6^semiprime(2) + 6^semiprime(3) + 6^semiprime(4) + 6^semiprime(5) + 6^semiprime(6) = 1 + 6^4 + 6^6 + 6^9 + 6^10 + 6^14 + 6^15 = 548619740497.

a(7) = 1 + 7^4 + 7^6 + 7^9 + 7^10 + 7^14 + 7^15 + 7^21 = 558551290190815706.

a(8) = 1 + 8^4 + 8^6 + 8^9 + 8^10 + 8^14 + 8^15 + 8^21 + 8^22 = 83010387915319808001.

a(9) = 1 + 9^4 + 9^6 + 9^9 + 9^10 + 9^14 + 9^15 + 9^21 + 9^22 + 9^25 = 718992177811939511654842.

a(10) = 1 + 10^4 + 10^6 + 10^9 + 10^10 + 10^14 + 10^15 + 10^21 + 10^22 + 10^25 + 10^26 = 110011000001100011001010001.

a(11) = 1 + 11^4 + 11^6 + 11^9 + 11^10 + 11^14 + 11^15 + 11^21 + 11^22 + 11^25 + 11^26 + 11^33 = 23225155720141324351494556519644062.

a(12) = 1 + 12^4 + 12^6 + 12^9 + 12^10 + 12^14 + 12^15 + 12^21 + 12^22 + 12^25 + 12^26 + 12^33 + 12^34 = 5332421525600135159678023844770734337.

a(13) = 1 + 13^4 + 13^6 + 13^9 + 13^10 + 13^14 + 13^15 + 13^21 + 13^22 + 13^25 + 13^26 + 13^33 + 13^34 + 13^35 = 1053371868623897220377558669169756037622.

Cf. A001358, A123113.

Sequence in context: A051391 A041799 A072408 this_sequence A090603 A020955 A120826

Adjacent sequences: A123174 A123175 A123176 this_sequence A123178 A123179 A123180

easy,nonn

Jonathan Vos Post (jvospost3(AT)gmail.com), Oct 03 2006