A351567 -id:A351567 - cp's OEIS frontend

A351566 Radix of the second least significant nonzero digit in the primorial base expansion of n, or 1 if there is no such digit.

1, 1, 1, 3, 1, 3, 1, 5, 5, 3, 5, 3, 1, 5, 5, 3, 5, 3, 1, 5, 5, 3, 5, 3, 1, 5, 5, 3, 5, 3, 1, 7, 7, 3, 7, 3, 7, 5, 5, 3, 5, 3, 7, 5, 5, 3, 5, 3, 7, 5, 5, 3, 5, 3, 7, 5, 5, 3, 5, 3, 1, 7, 7, 3, 7, 3, 7, 5, 5, 3, 5, 3, 7, 5, 5, 3, 5, 3, 7, 5, 5, 3, 5, 3, 7, 5, 5, 3, 5, 3, 1, 7, 7, 3, 7, 3, 7, 5, 5, 3, 5, 3, 7, 5, 5, 3

Offset: 0

Antti Karttunen, Apr 01 2022

The terms larger than one are given by the k-th prime (A000040), where k is the position of the second least significant nonzero digit in the primorial base expansion of n, counted from the right. See the example.

			For n = 13, its primorial base representation (see A049345) is "201" as 13 = 2*A002110(2) + 1*A002110(0). The one-based index of the second least significant nonzero digit ("2"), when counted from the right, is 3, therefore a(13) = A000040(3) = 5.

Antti Karttunen, Table of n, a(n) for n = 0..60060
Index entries for sequences related to primorial base.

Cf. A060735 (gives the positions of ones after the initial one at a(0)=1).

Cf. A000040, A002110, A049345, A053669, A119288, A276086, A351567.

a[n_] := Module[{k = n, p = 2, s = {}, r, i}, While[{k, r} = QuotientRemainder[k, p]; k != 0 || r != 0, AppendTo[s, r]; p = NextPrime[p]]; i = Position[s, ?(# > 0 &)] // Flatten; If[Length[i] < 2, 1, Prime[i[[2]]]]]; Array[a, 100, 0] (* _Amiram Eldar, Mar 13 2024 *)

A119288(n) = if(1>=omega(n), 1, (factor(n))[2,1]);
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A351566(n) = A119288(A276086(n));

a(n) = A119288(A276086(n)).
For all n, a(n) > A351567(n).
If a(n) > 1, then a(n) > A053669(n).

A351563 a(n) is the exponent of the second smallest prime factor of n, or 0 if n is a power of a prime.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Apr 01 2022

nonn

			For n = 4 = 2^2 there is no second smallest prime factor as 4 is a power of prime, therefore a(4) = 0.
For n = 18 = 2^1 * 3^2, the exponent of the second smallest prime factor (3) is 2, therefore a(18) = 2.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000961 (positions of zeros), A001221, A028234, A067029.

Cf. also A119288, A351567.

Array[If[Length[#] < 2, 0, #[[2, -1]]] &@ FactorInteger[#] &, 108] (* Michael De Vlieger, Apr 01 2022 *)

A351563(n) = if(1>=omega(n), 0, (factor(n))[2,2]);

a(n) = A067029(A028234(n)).

cp's OEIS Frontend

A351566 Radix of the second least significant nonzero digit in the primorial base expansion of n, or 1 if there is no such digit.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula