A334099 The least k for which A329697(k) = n; Position of first occurrence of n (and also records) in A329697.

1, 3, 7, 19, 43, 127, 283, 659, 1319, 3957, 9227, 21599, 50123, 129263, 258527, 775581, 1551163, 4340087, 9750239, 27353747, 65148847, 156067127, 340997113, 955523423

Offset: 0

Author

Antti Karttunen, Apr 14 2020

Keywords

• nonn
• more

Comments

Note that although most of the terms after 1 are primes, we also have a few composites: a(9) = a(1)*a(8) = 3*1319 = 3957, a(15) = a(1)*a(14) = 3*258527 = 775581, a(22) = a(8)*a(14) = 340997113.

a(n) <= 3^n and in particular, a(n+1) <= 3*a(n), n > 0 and more generally a(n + m) <= a(n) * a(m) where m, n >= 0. - David A. Corneth, Apr 15 2020

The above follows because A329697 is totally additive.

Crossrefs

The leftmost column of A334100.

Cf. A329697 (a left inverse).

Cf. A067513.

Cf. A007755, A105017, and also A329662 (analogous sequence when using the map k -> k + k/p).

Programs

Mathematica

With[{s = Array[Length@ NestWhileList[# - #/FactorInteger[#][[-1, 1]] &, #, # != 2^IntegerExponent[#, 2] &] - 1 &, 10^6]}, {1}~Join~Array[FirstPosition[s, #][[1]] &, Max@ s]] (* Michael De Vlieger, Apr 30 2020 *)

PARI

A329697(n) = if(!bitand(n,n-1),0,1+A329697(n-(n/vecmax(factor(n)[, 1]))));
m=-1; k=0; for(n=1,2^32, t=A329697(n); if(t>m, m=t; write("b334099.txt", k, " ", n); k++));

Formula

For all n >= 0, A329697(a(n)) = n.