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A327860 Arithmetic derivative of the primorial base exp-function: a(n) = A003415(A276086(n)).

%I A327860 #57 May 01 2022 23:04:48
%S A327860 0,1,1,5,6,21,1,7,8,31,39,123,10,45,55,185,240,705,75,275,350,1075,
%T A327860 1425,3975,500,1625,2125,6125,8250,22125,1,9,10,41,51,165,12,59,71,
%U A327860 247,318,951,95,365,460,1445,1905,5385,650,2175,2825,8275,11100,30075,4125,12625,16750,46625,63375,166125,14,77,91,329,420
%N A327860 Arithmetic derivative of the primorial base exp-function: a(n) = A003415(A276086(n)).
%C A327860 Are there any other fixed points after 0, 1, 7, 8 and 2556? (A328110, see also A351087 and A351088).
%C A327860 Out of the 30030 initial terms, 19220 are multiples of 5. (See A327865).
%C A327860 Proof that a(n) is even if and only if n is a multiple of 4: Consider _Charlie Neder_'s Feb 25 2019 comment in A235992. As A276086 is never a multiple of 4, and as it toggles the parity, we only need to know when A001222(A276086(n)) = A276150(n) is even. The condition for that is given in the latter sequence by _David A. Corneth_'s Feb 27 2019 comment. From this it also follows that A166486 gives similarly the parity of terms of A342002, A351083 and A345000. See also comment in A327858. - _Antti Karttunen_, May 01 2022
%H A327860 Antti Karttunen, <a href="/A327860/b327860.txt">Table of n, a(n) for n = 0..2310</a>
%H A327860 Antti Karttunen, <a href="/A327860/a327860.txt">Data supplement: n, a(n) computed for n = 0..30030</a>
%H A327860 <a href="/index/Pri#primorialbase">Index entries for sequences related to primorial base</a>
%F A327860 a(n) = A003415(A276086(n)).
%F A327860 a(A002110(n)) = 1 for all n >= 0.
%F A327860 From _Antti Karttunen_, Nov 03 2019: (Start)
%F A327860 Whenever A329041(x,y) = 1, a(x + y) = A003415(A276086(x)*A276086(y)) = a(x)*A276086(y) + a(y)*A276086(x). For example, we have:
%F A327860 a(n) = a(A328841(n)+A328842(n)) = A329031(n)*A328572(n) + A329032(n)*A328571(n).
%F A327860 A051903(a(n)) = A328391(n).
%F A327860 A328114(a(n)) = A328392(n).
%F A327860 (End)
%F A327860 From _Antti Karttunen_, May 01 2022: (Start)
%F A327860 a(n) = A328572(n) * A342002(n).
%F A327860 For all n >= 0, A000035(a(n)) = A166486(n). [See comments]
%F A327860 (End)
%e A327860 2556 has primorial base expansion [1,1,1,1,0,0] as 1*A002110(5) + 1*A002110(4) + 1*A002110(3) + 1*A002110(2) = 2310 + 210 + 30 + 6 = 2556. That in turn is converted by A276086 to 13^1 * 11^1 * 7^1 * 5^1 = 5005, whose arithmetic derivative is 5' * 1001 + 1001' * 5 = 1*1001 + 311*5 = 2556, thus 2556 is one of the rare fixed points (A328110) of this sequence.
%t A327860 Block[{b = MixedRadix[Reverse@ Prime@ Range@ 12]}, Array[Function[k, If[# < 2, 0, # Total[#2/#1 & @@@ FactorInteger[#]] ] &@ Abs[Times @@ Power @@@ # &@ Transpose@{Prime@ Range@ Length@ k, Reverse@ k}]]@ IntegerDigits[#, b] &, 65, 0]] (* _Michael De Vlieger_, Mar 12 2021 *)
%o A327860 (PARI)
%o A327860 A003415(n) = {my(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1]))}; \\ From A003415
%o A327860 A276086(n) = { my(i=0,m=1,pr=1,nextpr); while((n>0),i=i+1; nextpr = prime(i)*pr; if((n%nextpr),m*=(prime(i)^((n%nextpr)/pr));n-=(n%nextpr));pr=nextpr); m; };
%o A327860 A327860(n) = A003415(A276086(n));
%o A327860 (PARI) A327860(n) = { my(s=0, m=1, p=2, e); while(n, e = (n%p); m *= (p^e); s += (e/p); n = n\p; p = nextprime(1+p)); (s*m); }; \\ (Standalone version) - _Antti Karttunen_, Nov 07 2019
%Y A327860 Cf. A002110 (positions of 1's), A003415, A048103, A276086, A327858, A327859, A327865, A328110 (fixed points), A328233 (positions of primes), A328242 (positions of squarefree terms), A328388, A328392, A328571, A328572, A329031, A329032, A329041, A342002.
%Y A327860 Cf. A345000, A351074, A351075, A351076, A351077, A351080, A351083, A351084, A351087 (numbers k such that a(k) is a multiple of k), A351088.
%Y A327860 Coincides with A329029 on positions given by A276156.
%Y A327860 Cf. A166486 (a(n) mod 2), A353630 (a(n) mod 4).
%Y A327860 Cf. A267263, A276150, A324650, A324653, A324655 for omega, bigomega, phi, sigma and tau applied to A276086(n).
%Y A327860 Cf. also A351950 (analogous sequence).
%K A327860 nonn,base,easy,look
%O A327860 0,4
%A A327860 _Antti Karttunen_, Sep 30 2019
%E A327860 Verbal description added to the definition by _Antti Karttunen_, May 01 2022