

A226526


Slowestgrowing sequence of semiprimes where 1/(sp+1) sums to 1 without actually reaching it.


1



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

1,1


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,....


LINKS

Table of n, a(n) for n=1..23.


EXAMPLE

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.


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}]


CROSSREFS

Cf. A181503, A226527.
Sequence in context: A108764 A193801 A129336 * A103607 A264815 A108574
Adjacent sequences: A226523 A226524 A226525 * A226527 A226528 A226529


KEYWORD

nonn,hard


AUTHOR

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


STATUS

approved



