A276085 Primorial base log-function: fully additive with a(p) = p#/p, where p# = A034386(p).

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

Author

Antti Karttunen, Aug 21 2016

Keywords

• nonn

Comments

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

This is a left inverse of A276086 ("primorial base exp-function"), hence the name "primorial base log-function". 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

On average, every third term is a multiple of 4. See A369001. - Antti Karttunen, May 26 2024

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

Crossrefs

A left inverse of A276086.

Cf. A000040, A000720, A002110, A028234, A034386, A048103, A049345, A055396, A067029, A108951, A143293, A276154, A328316, A328624, A328625, A328768, A328832, A346105, A351576, A376398 (partial sums).

Positions of multiples of k in this sequence, for k=2, 3, 4, 5, 8, 27, 3125: A003159, A339746, A369002, A373140, A373138, A377872, A377878.

Cf. A036554 (positions of odd terms), A035263, A096268 (parity of terms).

Cf. A372575 (rgs-transform), A372576 [a(n) mod 360], A373842 [= A003415(a(n))].

Cf. A373145 [= gcd(A003415(n), a(n))], A373361 [= gcd(n, a(n))], A373362 [= gcd(A001414(n), a(n))], A373485 [= gcd(A083345(n), a(n))], A373835 [= gcd(bigomega(n), a(n))], and also A373367 and A373147 [= A003415(n) mod a(n)], A373148 [= a(n) mod A003415(n)].

Other completely additive sequences with primes p mapped to a function of p include: A001222 (with a(p)=1), A001414 (with a(p)=p), A059975 (with a(p)=p-1), A341885 (with a(p)=p*(p+1)/2), A373149 (with a(p)=prevprime(p)), A373158 (with a(p)=p#).

Cf. also A276075 for factorial base and A048675, A054841 for base-2 and base-10 analogs.

Programs

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 *)

PARI

A276085(n) = { my(f = factor(n), pr=1, i=1, s=0); for(k=1, #f~, while(i <= primepi(f[k, 1])-1, pr *= prime(i); i++); s += f[k, 2]*pr); (s); }; \\ Antti Karttunen, Nov 11 2024

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

Formula

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

a(1) = 0, a(n) = (e1*A002110(i1-1) + ... + ez*A002110(iz-1)) when n = prime(i1)^e1 * ... * prime(iz)^ez.

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 a primorial base representation]

(End)

Extensions

Name amended by Antti Karttunen, Apr 24 2022

Name simplified, the old name moved to the comments - Antti Karttunen, Jun 23 2024