A351067 Number of integers between the n-th and the (n+1)-th primorial such that the maximal exponent in their prime factorization is larger than the maximal digit in their primorial base expansion.

0, 3, 11, 52, 291, 1681, 11506, 89347

Offset: 1

Author

Antti Karttunen, Feb 02 2022

Keywords

• nonn
• more

Comments

a(n) is the number of terms of A350075 in range A002110(n) .. A002110(1+n)-1.

The ratio a(n) / A061720(n) develops as:

n = 1: 0 / 4 = 0

2: 3 / 24 = 0.125

3: 11 / 180 = 0.061111...

4: 52 / 2100 = 0.247619...

5: 291 / 27720 = 0.010498...

6: 1681 / 480480 = 0.003499...

7: 11506 / 9189180 = 0.001252...

8: 89347 / 213393180 = 0.000419...

Examples

			Between A002110(2) = 6 and A002110(3) = 30, there are exactly three numbers that satisfy the condition: 8, 9, 16, therefore a(2) = 3.
		

Links

• Index entries for sequences related to primorial base

Crossrefs

Cf. A002110, A051903, A276086, A328114, A350075, A351068 (partial sums), A351069.

Cf. also A327969.

Programs

PARI

A002110(n) = prod(i=1,n,prime(i));
A051903(n) = if((1==n),0,vecmax(factor(n)[, 2]));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
isA350075(n) = (A051903(A276086(n)) < A051903(n));
A351067(n) = sum(k=A002110(n),A002110(1+n)-1,isA350075(k));

Formula

a(n) = Sum_{k=A002110(n) .. A002110(1+n)-1} [A328114(k) < A051903(k)], where [ ] is the Iverson bracket.

For all n, a(n) < A351069(n).