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

0,6

Links

Antti Karttunen, Table of n, a(n) for n = 0..32768

Index entries for sequences related to primorial base.

Formula

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

Example

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.

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 *)

Prog

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

Crossrefs

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

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

Cf. also A260736.

Sequence in context: A321396 A141747 A239706 * A260736 A342656 A293896

Adjacent sequences: A328613 A328614 A328615 * A328617 A328618 A328619

Keyword

nonn,base

Author

Antti Karttunen, Oct 22 2019

Status

approved