A276085 Primorial base log-function: a(1) = 0, a(n) = (e1*A002110(i1-1) + ... + ez*A002110(iz-1)) for n = prime(i1)^e1 * ... * prime(iz)^ez, where prime(k) is the k-th prime, A000040(k) and A002110(k) (the k-th primorial) is the product of first k primes.

0, 1, 2, 2, 6, 3, 30, 3, 4, 7, 210, 4, 2310, 31, 8, 4, 30030, 5, 510510, 8, 32, 211, 9699690, 5, 12, 2311, 6, 32, 223092870, 9, 6469693230, 5, 212, 30031, 36, 6, 200560490130, 510511, 2312, 9, 7420738134810, 33, 304250263527210, 212, 10, 9699691, 13082761331670030, 6, 60, 13, 30032, 2312, 614889782588491410, 7, 216, 33, 510512, 223092871, 32589158477190044730, 10

Offset

1,3

Comments

Additive with a(p^e) = e * A002110(A000720(p)-1).

This is a left inverse of A276086 ("primorial base exp-function"), hence the new name. When the domain is restricted to the terms of A048103, this works also as a right inverse, as A276086(a(A048103(n))) = A048103(n) for all n >= 1. - Antti Karttunen, Apr 24 2022

Links

Antti Karttunen, Table of n, a(n) for n = 1..2310

Index entries for sequences related to primorial base

Index entries for sequences related to primorial numbers

Formula

a(1) = 0; for n > 1, a(n) = a(A028234(n)) + (A067029(n) * A002110(A055396(n)-1)).

Other identities.

For all n >= 0:

a(A276086(n)) = n.

a(A000040(1+n)) = A002110(n).

a(A002110(1+n)) = A143293(n).

From Antti Karttunen, Apr 24 & Apr 29 2022: (Start)

a(A283477(n)) = A283985(n).

a(A108951(n)) = A346105(n). [The latter has a similar additive formula as this sequence, but instead of primorials, uses their partial sums]

When applied to sequences where a certain subset of the divisors of n has been multiplicatively encoded with the help of A276086, this yields a corresponding number-theoretical sequence, i.e. completes their computation:

a(A319708(n)) = A001065(n) and a(A353564(n)) = A051953(n).

a(A329350(n)) = A069359(n) and a(A329380(n)) = A323599(n).

In the following group, the sum of the rhs-sequences is n [on each row, as say, A328841(n)+A328842(n)=n], because the pointwise product of the corresponding lhs-sequences is A276086:

a(A053669(n)) = A053589(n) and a(A324895(n)) = A276151(n).

a(A328571(n)) = A328841(n) and a(A328572(n)) = A328842(n).

a(A351231(n)) = A351233(n) and a(A327858(n)) = A351234(n).

a(A351251(n)) = A351253(n) and a(A324198(n)) = A351254(n).

The sum or difference of the rhs-sequences is A108951:

a(A344592(n)) = A346092(n) and a(A346091(n)) = A346093(n).

a(A346106(n)) = A346108(n) and a(A346107(n)) = A346109(n).

Here the two sequences are inverse permutations of each other:

a(A328624(n)) = A328625(n) and a(A328627(n)) = A328626(n).

a(A346102(n)) = A328622(n) and a(A346233(n)) = A328623(n).

a(A346101(n)) = A289234(n). [Self-inverse]

Other correspondences:

a(A324350(x,y)) = A324351(x,y).

a(A003961(A276086(n))) = A276154(n). [The primorial base left shift]

a(A276076(n)) = A351576(n). [Sequence reinterpreting factorial base representation as primorial base representation]

(End)

Mathematica

nn = 60; b = MixedRadix[Reverse@ Prime@ Range@ PrimePi[nn + 1]]; Table[FromDigits[#, b] &@ Reverse@ If[n == 1, {0}, Function[k, ReplacePart[Table[0, {PrimePi[k[[-1, 1]]]}], #] &@ Map[PrimePi@ First@ # -> Last@ # &, k]]@ FactorInteger@ n], {n, nn}] (* Version 10.2, or *)
f[w_List] := Total[Times @@@ Transpose@ {Map[Times @@ # &, Prime@ Range@ Range[0, Length@ w - 1]], Reverse@ w}]; Table[f@ Reverse@ If[n == 1, {0}, Function[k, ReplacePart[Table[0, {PrimePi[k[[-1, 1]]]}], #] &@ Map[PrimePi@ First@ # -> Last@ # &, k]]@ FactorInteger@ n], {n, 60}] (* Michael De Vlieger, Aug 30 2016 *)

Prog

(Scheme, with memoization-macro definec)
(definec (A276085 n) (cond ((= 1 n) (- n 1)) (else (+ (* (A067029 n) (A002110 (+ -1 (A055396 n)))) (A276085 (A028234 n))))))
(PARI)
A002110(n) = prod(i=1, n, prime(i));
A276085(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*A002110(primepi(f[k, 1])-1)); }; \\ Antti Karttunen, Mar 15 2021
(Python)
from sympy import primorial, primepi, factorint
def a002110(n):
    return 1 if n<1 else primorial(n)
def a(n):
    f=factorint(n)
    return sum(f[i]*a002110(primepi(i) - 1) for i in f)
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 22 2017

Crossrefs

A left inverse of A276086.

Cf. A000040, A000720, A002110, A028234, A048103, A049345, A055396, A067029, A108951, A143293, A276154, A328316, A328624, A328625, A328768, A328832, A346105, A351576.

Cf. also A276075 for factorial base and A054841 for base-10 analog.

Sequence in context: A276075 A321908 A130728 * A334870 A324122 A324349

Adjacent sequences: A276082 A276083 A276084 * A276086 A276087 A276088

Keyword

nonn

Author

Antti Karttunen, Aug 21 2016

Extensions

Name amended by Antti Karttunen, Apr 24 2022

Status

approved