A226527 Slowest-growing sequence of 3-almost primes (trientprimes) where 1/(tp+1) sums to 1 without actually reaching it.

8, 12, 18, 20, 27, 28, 30, 42, 44, 45, 50, 52, 63, 66, 68, 70, 75, 76, 78, 92, 98, 99, 102, 105, 110, 114, 116, 117, 124, 125, 130, 138, 147, 148, 153, 154, 164, 165, 170, 171, 172, 174, 175, 182, 186, 188, 190, 195, 207, 212, 222, 230, 231, 236, 238, 242, 244, 245, 246, 255, 258, 261, 266, 268, 273, 275, 279, 282, 284, 285, 286, 290, 292, 310, 316, 318, 322, 325, 332, 333, 338, 343, 345, 354, 356, 357, 363, 366, 369, 370, 374, 385, 387, 388, 399, 402, 404, 406, 410, 412, 418, 423, 425, 426, 428, 429, 430, 434, 435, 436, 8662, 44335708, 1251938572491943, 1505273212784203338150808798466, 680617602541158152398258079780439819108542271775727566330763

Offset: 1

Author

Aaron Meyerowitz, Jonathan Vos Post, and Robert G. Wilson v, Jun 09 2013

Keywords

• nonn
• hard

Comments

See comments in A226526.

Deviates from A014612 after the 110th term.

Crossrefs

Cf. A181503, A226526.

Programs

Mathematica

kAlmostPrimeQ[n_, k_: 2] := Plus @@ Last /@ FactorInteger@ n == k (* For those who have Mmca v or later, you could use PrimeOmega@ n == k *) NextkAlmostPrime[n_, k_: 2, m_: 1] := Block[{c = 0, sgn = Sign[m]}, kap = n + sgn; While[c < Abs[m], While[ PrimeOmega[kap] != k, If[sgn < 0, kap--, kap++]]; If[ sgn < 0, kap--, kap++]; c++]; kap + If[sgn < 0, 1, -1]]; a[n_] := a[n] = Block[{sm = Sum[1/(a[i] + 1), {i, n - 1}]}, NextkAlmostPrime[ Max[a[n - 1], Floor[1/(1 - sm)]]]]; a[0] = 1; Do[ Print[{n, a[n] // Timing}], {n, 25}]