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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. 18

%I

%S 0,1,2,2,6,3,30,3,4,7,210,4,2310,31,8,4,30030,5,510510,8,32,211,

%T 9699690,5,12,2311,6,32,223092870,9,6469693230,5,212,30031,36,6,

%U 200560490130,510511,2312,9,7420738134810,33,304250263527210,212,10,9699691,13082761331670030,6,60,13,30032,2312,614889782588491410,7,216,33,510512,223092871,32589158477190044730,10

%N 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.

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

%H Antti Karttunen, <a href="/A276085/b276085.txt">Table of n, a(n) for n = 1..210</a>

%H <a href="/index/Pri#primorialbase">Index entries for sequences related to primorial base</a>

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

%F Other identities.

%F For all n >= 0:

%F a(A276086(n)) = n.

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

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

%t 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 *)

%t 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 *)

%o (Scheme, with memoization-macro definec)

%o (definec (A276085 n) (cond ((= 1 n) (- n 1)) (else (+ (* (A067029 n) (A002110 (+ -1 (A055396 n)))) (A276085 (A028234 n))))))

%o (Python)

%o from sympy import primorial, primepi, factorint

%o def a002110(n): return 1 if n<1 else primorial(n)

%o def a(n):

%o f=factorint(n)

%o return sum([f[i]*a002110(primepi(i) - 1) for i in f])

%o print [a(n) for n in xrange(1, 101)] # _Indranil Ghosh_, Jun 22 2017

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

%Y Left inverse of A276086.

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

%K nonn

%O 1,3

%A _Antti Karttunen_, Aug 21 2016

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Last modified April 25 13:31 EDT 2019. Contains 322461 sequences. (Running on oeis4.)