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

%I A276085 #77 Nov 13 2024 13:53:22
%S A276085 0,1,2,2,6,3,30,3,4,7,210,4,2310,31,8,4,30030,5,510510,8,32,211,
%T A276085 9699690,5,12,2311,6,32,223092870,9,6469693230,5,212,30031,36,6,
%U A276085 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 A276085 Primorial base log-function: fully additive with a(p) = p#/p, where p# = A034386(p).
%C A276085 Completely additive with a(p^e) = e * A002110(A000720(p)-1).
%C A276085 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
%C A276085 On average, every third term is a multiple of 4. See A369001. - _Antti Karttunen_, May 26 2024
%H A276085 Antti Karttunen, <a href="/A276085/b276085.txt">Table of n, a(n) for n = 1..2310</a>
%H A276085 <a href="/index/Pri#primorialbase">Index entries for sequences related to primorial base</a>
%H A276085 <a href="/index/Pri#primorial_numbers">Index entries for sequences related to primorial numbers</a>
%F A276085 a(1) = 0; for n > 1, a(n) = a(A028234(n)) + (A067029(n) * A002110(A055396(n)-1)).
%F A276085 a(1) = 0, a(n) = (e1*A002110(i1-1) + ... + ez*A002110(iz-1)) when n = prime(i1)^e1 * ... * prime(iz)^ez.
%F A276085 Other identities.
%F A276085 For all n >= 0:
%F A276085 a(A276086(n)) = n.
%F A276085 a(A000040(1+n)) = A002110(n).
%F A276085 a(A002110(1+n)) = A143293(n).
%F A276085 From _Antti Karttunen_, Apr 24 & Apr 29 2022: (Start)
%F A276085 a(A283477(n)) = A283985(n).
%F A276085 a(A108951(n)) = A346105(n). [The latter has a similar additive formula as this sequence, but instead of primorials, uses their partial sums]
%F A276085 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:
%F A276085 a(A319708(n)) = A001065(n) and a(A353564(n)) = A051953(n).
%F A276085 a(A329350(n)) = A069359(n) and a(A329380(n)) = A323599(n).
%F A276085 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:
%F A276085 a(A053669(n)) = A053589(n) and a(A324895(n)) = A276151(n).
%F A276085 a(A328571(n)) = A328841(n) and a(A328572(n)) = A328842(n).
%F A276085 a(A351231(n)) = A351233(n) and a(A327858(n)) = A351234(n).
%F A276085 a(A351251(n)) = A351253(n) and a(A324198(n)) = A351254(n).
%F A276085 The sum or difference of the rhs-sequences is A108951:
%F A276085 a(A344592(n)) = A346092(n) and a(A346091(n)) = A346093(n).
%F A276085 a(A346106(n)) = A346108(n) and a(A346107(n)) = A346109(n).
%F A276085 Here the two sequences are inverse permutations of each other:
%F A276085 a(A328624(n)) = A328625(n) and a(A328627(n)) = A328626(n).
%F A276085 a(A346102(n)) = A328622(n) and a(A346233(n)) = A328623(n).
%F A276085 a(A346101(n)) = A289234(n). [Self-inverse]
%F A276085 Other correspondences:
%F A276085 a(A324350(x,y)) = A324351(x,y).
%F A276085 a(A003961(A276086(n))) = A276154(n). [The primorial base left shift]
%F A276085 a(A276076(n)) = A351576(n). [Sequence reinterpreting factorial base representation as a primorial base representation]
%F A276085 (End)
%t A276085 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 A276085 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 A276085 (Scheme, with memoization-macro definec)
%o A276085 (definec (A276085 n) (cond ((= 1 n) (- n 1)) (else (+ (* (A067029 n) (A002110 (+ -1 (A055396 n)))) (A276085 (A028234 n))))))
%o A276085 (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
%o A276085 (Python)
%o A276085 from sympy import primorial, primepi, factorint
%o A276085 def a002110(n):
%o A276085     return 1 if n<1 else primorial(n)
%o A276085 def a(n):
%o A276085     f=factorint(n)
%o A276085     return sum(f[i]*a002110(primepi(i) - 1) for i in f)
%o A276085 print([a(n) for n in range(1, 101)]) # _Indranil Ghosh_, Jun 22 2017
%Y A276085 A left inverse of A276086.
%Y A276085 Cf. A000040, A000720, A002110, A028234, A034386, A048103, A049345, A055396, A067029, A108951, A143293, A276154, A328316, A328624, A328625, A328768, A328832, A346105, A351576, A376398 (partial sums).
%Y A276085 Positions of multiples of k in this sequence, for k=2, 3, 4, 5, 8, 27, 3125: A003159, A339746, A369002, A373140, A373138, A377872, A377878.
%Y A276085 Cf. A036554 (positions of odd terms), A035263, A096268 (parity of terms).
%Y A276085 Cf. A372575 (rgs-transform), A372576 [a(n) mod 360], A373842 [= A003415(a(n))].
%Y A276085 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)].
%Y A276085 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#).
%Y A276085 Cf. also A276075 for factorial base and A048675, A054841 for base-2 and base-10 analogs.
%K A276085 nonn
%O A276085 1,3
%A A276085 _Antti Karttunen_, Aug 21 2016
%E A276085 Name amended by _Antti Karttunen_, Apr 24 2022
%E A276085 Name simplified, the old name moved to the comments - _Antti Karttunen_, Jun 23 2024