A328616 Number of digits in primorial base expansion of n that are maximal possible in their positions.

0, 1, 0, 1, 1, 2, 0, 1, 0, 1, 1, 2, 0, 1, 0, 1, 1, 2, 0, 1, 0, 1, 1, 2, 1, 2, 1, 2, 2, 3, 0, 1, 0, 1, 1, 2, 0, 1, 0, 1, 1, 2, 0, 1, 0, 1, 1, 2, 0, 1, 0, 1, 1, 2, 1, 2, 1, 2, 2, 3, 0, 1, 0, 1, 1, 2, 0, 1, 0, 1, 1, 2, 0, 1, 0, 1, 1, 2, 0, 1, 0, 1, 1, 2, 1, 2, 1, 2, 2, 3, 0, 1, 0, 1, 1, 2, 0, 1, 0, 1, 1, 2, 0, 1, 0, 1

Offset: 0

Author

Antti Karttunen, Oct 22 2019

Keywords

• nonn
• base

Examples

			In primorial base (see A049345), the maximum digit value that can occur in the k-th position from the right (with k=1 standing for the rightmost, i.e., the least significant digit position) is A000040(k)-1, and it is for the terms of A057588 (primorial numbers minus one) where all significant digits are maximal allowed for their positions, e.g. 209 is written as "6421" because 209 = 6*30 + 4*6 + 2*2 + 1*1, thus a(209) = 4.
87 is written as "2411" because 87 = 2*A002110(3) + 4*A002110(2) + 1*A002110(1) + 1*A002110(0) = 2*30 + 4*6 + 1*2 + 1*1. Only the digit positions 1 and 3 are occupied with maximum digits allowed in those positions (that are 1 and 4, being one less than the corresponding primes, 2 and 5), thus a(87) = 2.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 0..32768
• Index entries for sequences related to primorial base.

Crossrefs

Cf. A002110, A049345, A276086, A328114, A328614, A328615.

Cf. A057588 (positions of records, and the first occurrence of each n > 0).

Cf. also A260736.

Programs

Mathematica

a[n_] := Module[{k = n, p = 2, s = {}, r}, While[{k, r} = QuotientRemainder[k, p]; k != 0 || r != 0, AppendTo[s, r]; p = NextPrime[p]]; Count[Prime[Range[1, Length[s]]] - s, 1]]; a[0] = 0; Array[a, 100, 0] (* Amiram Eldar, Mar 13 2024 *)

PARI

A328616(n) = { my(s=0, p=2); while(n, s += ((p-1)==(n%p)); n = n\p; p = nextprime(1+p)); (s); };

Formula

For all n >= 1, a(A057588(n)) = n.