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A276085 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. 17
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

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

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..210

Index entries for sequences related to primorial base

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).

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))))))

(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 xrange(1, 101)] # Indranil Ghosh, Jun 22 2017

CROSSREFS

Cf. A000040, A000720, A002110, A028234, A049345, A055396, A067029, A143293.

Left inverse of A276086.

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

Sequence in context: A276075 A321908 A130728 * A324122 A324349 A092384

Adjacent sequences:  A276082 A276083 A276084 * A276086 A276087 A276088

KEYWORD

nonn

AUTHOR

Antti Karttunen, Aug 21 2016

STATUS

approved

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Last modified March 25 12:11 EDT 2019. Contains 321470 sequences. (Running on oeis4.)