A351070 -id:A351070 - cp's OEIS frontend

A351038 Numbers whose maximal digit in their primorial base expansion is not larger than the maximal exponent in their prime factorization.

2, 3, 4, 6, 7, 8, 9, 12, 16, 30, 31, 32, 33, 36, 37, 38, 39, 40, 44, 45, 48, 60, 63, 64, 68, 72, 75, 76, 80, 81, 96, 104, 108, 112, 128, 144, 160, 192, 210, 211, 212, 213, 216, 217, 218, 219, 220, 224, 225, 232, 240, 241, 242, 243, 244, 245, 246, 247, 248, 249, 250, 252, 256, 270, 272, 275, 276, 279, 280, 284, 288, 304

Offset: 1

Antti Karttunen, Feb 01 2022

Numbers k for which A328114(k) <= A051903(k).

Numbers such that when the map x -> A276086(x) is applied to them, the maximal exponent in the prime factorization (A051903) doesn't increase.

Antti Karttunen, Table of n, a(n) for n = 1..28472; terms <= 9699690
Index entries for sequences related to primorial base

Cf. A051903, A276086, A328114, A351039 (characteristic function), A351069 and A351070 (counts).
Positions of nonpositive terms in A350074.
Subsequences: A350075, A351075, also A369645 (after its initial 1).
Cf. also A341518.

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); };
isA351038(n) = (A051903(A276086(n)) <= A051903(n));

A351069 Number of integers in range A002110(n) .. A002110(1+n)-1 such that the maximal digit in their primorial base expansion is not larger than the maximal exponent in their prime factorization, where A002110(n) gives the n-th primorial.

Original entry on oeis.org

3, 6, 29, 122, 633, 3587, 24091, 184924

Offset: 1

Antti Karttunen, Feb 02 2022

a(n) is the number of terms of A351038 (numbers k satisfying A328114(k) <= A051903(k)) in range A002110(n) .. A002110(1+n)-1.

			There are six numbers in range A002110(2) .. A002110(3)-1 [in 6 .. (30-1)] that satisfy the condition: 6, 7, 8, 9, 12, 16, therefore a(2) = 6.

Cf. A002110, A051903, A276086, A328114, A351038, A351039, A351067, A351070 (partial sums).

Cf. also A327969.

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); };
A351039(n) = (A051903(A276086(n)) <= A051903(n));
A351069(n) = sum(k=A002110(n),A002110(1+n)-1,A351039(k));

a(n) = Sum_{k=A002110(n) .. A002110(1+n)-1} A351039(k).

For all n, a(n) > A351067(n).

A351068 Number of integers less than 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.

Original entry on oeis.org

0, 3, 14, 66, 357, 2038, 13544, 102891

Offset: 1

Antti Karttunen, Feb 02 2022

a(n) is the number of terms of A350075 smaller than A002110(1+n).

Index entries for sequences related to primorial base

Partial sums of A351067.

Cf. A002110, A051903, A276086, A328114, A350075, A351070.

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

For all n, a(n) < A351070(n).

cp's OEIS Frontend

A351038 Numbers whose maximal digit in their primorial base expansion is not larger than the maximal exponent in their prime factorization.

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A351069 Number of integers in range A002110(n) .. A002110(1+n)-1 such that the maximal digit in their primorial base expansion is not larger than the maximal exponent in their prime factorization, where A002110(n) gives the n-th primorial.

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A351068 Number of integers less than 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.

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