A110226 1 + sum of first n 4-almost primes.

1, 17, 41, 77, 117, 171, 227, 287, 368, 452, 540, 630, 730, 834, 960, 1092, 1227, 1363, 1503, 1653, 1805, 1961, 2145, 2334, 2530, 2728, 2932, 3142, 3362, 3587, 3815, 4047, 4281, 4529, 4779, 5039, 5315, 5609, 5905, 6202, 6508, 6816, 7131, 7459, 7789, 8129

Offset: 0

Author

Jonathan Vos Post, Sep 06 2005

Keywords

• easy
• nonn

Comments

First differences are the sequence of 4-almost primes (A014613). Hence a(n) is the least positive sequence whose first differences are the sequence of 4-almost primes. Primes in this sequence include: a(1) = 17, a(2) = 41, a(6) = 227, a(35) = 5039, a(43) = 7459, a(44) = 7789. Semiprimes in this sequence include: a(3) = 77 = 7 * 11, a(7) = 287 = 7 * 41, a(16) = 1227 = 3 * 409, a(17) = 1363 = 29 * 47, a(21) = 1961 = 37 * 53, a(27) = 3142 = 2 * 1571, a(29) = 3587 = 17 * 211, a(32) = 4281 = 3 * 1427, a(33) = 4529 = 7 * 647, a(36) = 5315 = 5 * 1063, a(37) = 5609 = 71 * 79, a(38) = 5905 = 5 * 1181, a(42) = 7131 = 3 * 2733a(45) = 8129 = 11 * 739. 3-almost primes in this sequence include: a(4) = 117 = 3^2 * 13, a(5) = 171 = 3^2 * 19, a(9) = 452 = 2^2 * 113, a(12) = 730 = 2 * 5 * 73, a(13) = 833 = 2 * 3 * 139, a(18) = 3^2 * 167, a(19) = 1653 = 3 * 19 * 29, a(20) = 1805 = 5 * 19^2, a(23) = 2534 = 2 * 3 * 389, a(26) = 2932 = 2^2 * 733, a(28) = 3362 = 2 * 41^2, a(30) = 3815 = 5 * 7 * 109, a(31) = 4047 = 3 * 19 * 71, a(39) = 2 * 7 * 443, a(40) = 6508 = 2^2 * 1627. 4-almost primes in this sequence include: a(22) = 2145 = 3 * 5 * 11 * 13, a(24) = 2530 = 2 * 5 * 11 * 23.

Crossrefs

Cf. A014613, A086046, A110208, A110209.

Programs

Mathematica

Accumulate[Join[{1},Select[Range[500],PrimeOmega[#]==4&]]] (* Harvey P. Dale, Dec 13 2018 *)

Formula

a(0) = 1; for n>0, a(n) = 1 + A086046(n).