A226526 Slowest-growing sequence of semiprimes where 1/(sp+1) sums to 1 without actually reaching it.

4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, 34, 35, 38, 39, 46, 69, 1497, 259465, 4852747709, 3429487924785490781, 305153651313989042415043589313598477, 21932475414742921908206321699222250910796483151080020353252738457741771

Offset: 1

Author

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

Keywords

• nonn
• hard

Comments

The semiprime analogous to A181503.

Because the semiprimes are sparser than the primes in the beginning, the sequence contains more of the lesser semiprimes than the analogous sequence of primes. In fact, one has to get to the seventeenth semiprime before it, 49,is not present, whereas in A181503, one only has to get to the sixth prime before it, 13, is not present.

If you change 1/(a(n)+1) to simply 1/a(n) the sequence becomes: 4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, 34, 355, 16627, 76723511, 17218740226618333, 374886275842473712491638217368219, 9036922116709843444667289331349853231276337589593114741410804131,....

Examples

			1/(4+1) + 1/(6+1) + 1/(9+1) + … 1/(46+1) + 1/(69+1) is still less than 1. Instead of 1/69, if one were to use any semiprime between 46 and 69, {} the sum would then exceed 1.
		

Crossrefs

Cf. A181503, A226527.

Programs

Mathematica

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