A271720 a(1)=1; for n>1, define a sequence {b(m), m >= 1} by b(1)=n b(2)=13, and b(m) = A020639(b(m-2)) + A006530(b(m-1)); then a(n) is the number of terms in that sequence before the first of the infinite string of 4s.

8, 13, 15, 13, 4, 13, 12, 13, 15, 13, 4, 13, 16, 13, 15, 13, 12, 13, 15, 13, 15, 13, 4, 13, 4, 13, 15, 13, 8, 13, 12, 13, 15, 13, 4, 13, 12, 13, 15, 13, 4, 13, 8, 13, 15, 13, 12, 13, 12, 13, 15, 13, 12, 13, 4, 13, 15, 13, 4, 13, 8, 13, 15, 13, 4, 13, 12, 13, 15, 13, 8, 13, 11, 13, 15, 13, 12, 13

Offset

1,1

Comments

Note that the majority of the terms (every other term, initially) are equal to b(2), which is 13. This happens with several other values of b(2) less than 20. Many other values for b(2) have been tested, and it seems that for all b(2) < 100000000, a(n) < 20.

Records 8, 13, 15, 16, 19, 20, 24, ... occur at 1, 2, 3, 13, 349, 3919, 55633, ...

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Example

n = 6; the sequence is:
6, 13, 15, 18, 6, 5, 7, 12, 10, 7, 9, 10, 8, 4, 4, 4, ...
There are 13 terms before the first of the infinite 4s; a(6) = 13.
For n = 55633 the sequence is: 55633, 13, 55646, 27836, 6961, 6963, 7172, 166, 85, 19, 24, 22, 13, 15, 18, 6, 5, 7, 12, 10, 7, 9, 10, 8, 4, 4, 4, ... . As the first 4 comes as the 25th term, a(55633) = 24. - Antti Karttunen, Oct 01 2018

Mathematica

Table[
Clear[h];
h[1]=x;
h[2]=13;
h[n_]:=FactorInteger[h[n-1]][[-1, 1]]+FactorInteger[h[n-2]][[1, 1]];
Position[Array[h, 100], 4][[1, 1]]-1,
{x, 1, 100}] (*This only works for x≠4*)

Prog

(PARI)
A006530(n) = if(n>1, vecmax(factor(n)[, 1]), 1); \\ From A006530
A020639(n) = if(1==n, n, factor(n)[1, 1]);
A271720(n) = { my(up=1001, bvec = vector(up), m=1); bvec[1] = n; bvec[2] = 13; for(n=3, oo, bvec[n] = A020639(bvec[n-2])+A006530(bvec[n-1]); if(4==bvec[n], return(n-1))); }; \\ Antti Karttunen, Oct 01 2018

Crossrefs

Sequence in context: A253775 A168137 A252458 * A243436 A380549 A214412

Adjacent sequences: A271717 A271718 A271719 * A271721 A271722 A271723

Keyword

nonn

Author

Cody M. Haderlie, Apr 12 2016

Status

approved