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%I #57 May 01 2022 23:04:48
%S 0,1,1,5,6,21,1,7,8,31,39,123,10,45,55,185,240,705,75,275,350,1075,
%T 1425,3975,500,1625,2125,6125,8250,22125,1,9,10,41,51,165,12,59,71,
%U 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 Arithmetic derivative of the primorial base exp-function: a(n) = A003415(A276086(n)).
%C Are there any other fixed points after 0, 1, 7, 8 and 2556? (A328110, see also A351087 and A351088).
%C Out of the 30030 initial terms, 19220 are multiples of 5. (See A327865).
%C 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 Antti Karttunen, <a href="/A327860/b327860.txt">Table of n, a(n) for n = 0..2310</a>
%H Antti Karttunen, <a href="/A327860/a327860.txt">Data supplement: n, a(n) computed for n = 0..30030</a>
%H <a href="/index/Pri#primorialbase">Index entries for sequences related to primorial base</a>
%F a(n) = A003415(A276086(n)).
%F a(A002110(n)) = 1 for all n >= 0.
%F From _Antti Karttunen_, Nov 03 2019: (Start)
%F 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 a(n) = a(A328841(n)+A328842(n)) = A329031(n)*A328572(n) + A329032(n)*A328571(n).
%F A051903(a(n)) = A328391(n).
%F A328114(a(n)) = A328392(n).
%F (End)
%F From _Antti Karttunen_, May 01 2022: (Start)
%F a(n) = A328572(n) * A342002(n).
%F For all n >= 0, A000035(a(n)) = A166486(n). [See comments]
%F (End)
%e 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 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 (PARI)
%o 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 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(n) = A003415(A276086(n));
%o (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 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 Cf. A345000, A351074, A351075, A351076, A351077, A351080, A351083, A351084, A351087 (numbers k such that a(k) is a multiple of k), A351088.
%Y Coincides with A329029 on positions given by A276156.
%Y Cf. A166486 (a(n) mod 2), A353630 (a(n) mod 4).
%Y Cf. A267263, A276150, A324650, A324653, A324655 for omega, bigomega, phi, sigma and tau applied to A276086(n).
%Y Cf. also A351950 (analogous sequence).
%K nonn,base,easy,look
%O 0,4
%A _Antti Karttunen_, Sep 30 2019
%E Verbal description added to the definition by _Antti Karttunen_, May 01 2022
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