A337903 a(0) = 0, a(1) = 1; for n>1, if a(n-1) is composite then a(n) = a(n-1)/A107286(n-1), where A107286(n) = the smallest prime factor of n, otherwise a(n) = a(n-1) + n.

0, 1, 3, 6, 3, 8, 4, 2, 10, 5, 15, 5, 17, 30, 15, 5, 21, 7, 25, 5, 25, 5, 27, 9, 3, 28, 14, 7, 35, 7, 37, 68, 34, 17, 51, 17, 53, 90, 45, 15, 5, 46, 23, 66, 33, 11, 57, 19, 67, 116, 58, 29, 81, 27, 9, 3, 59, 116, 58, 29, 89, 150, 75, 25, 5, 70, 35, 7, 75, 25, 5, 76, 38, 19, 93, 31, 107, 184, 92

Offset

0,3

Comments

The sequence can only increase for two consecutive terms at most as if a(n) is even then a(n+1) will be a(n)/2, while if a(n) is odd and a(n+1) is prime then a(n+2) will be even and thus a(n+3) = a(n+2)/2.

For the first 100 million terms the lowest number not to have appeared is 888. It is likely all numbers eventually appear although this is unknown.

Links

Table of n, a(n) for n=0..78.

Scott R. Shannon, Graph of the first 10 million terms.

Example

a(3) = 6 as a(2) = 3 which is prime thus a(3) = 3 + 3 = 6.
a(4) = 3 as a(3) = 6 which has the smallest divisor 2 thus a(4) = 6/2 = 3.
a(11) = 5 as a(10) = 15 which has the smallest divisor 3 thus a(11) = 15/3 = 5.

Mathematica

a[0] = 0;  a[1] = 1; a[n_] := a[n] = If[CompositeQ[a[n - 1]], a[n - 1]/FactorInteger[a[n - 1]][[1, 1]], a[n - 1] + n]; Array[a, 100, 0] (* Amiram Eldar, Sep 30 2020 *)

Crossrefs

Cf. A107286, A064413, A098550, A336957, A330252.

Sequence in context: A135003 A350877 A090895 * A358787 A264767 A065231

Adjacent sequences: A337900 A337901 A337902 * A337904 A337905 A337906

Keyword

nonn

Author

Scott R. Shannon, Sep 29 2020

Status

approved