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A327860 a(n) = A003415(A276086(n)). 21
0, 1, 1, 5, 6, 21, 1, 7, 8, 31, 39, 123, 10, 45, 55, 185, 240, 705, 75, 275, 350, 1075, 1425, 3975, 500, 1625, 2125, 6125, 8250, 22125, 1, 9, 10, 41, 51, 165, 12, 59, 71, 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Are there any other fixed points after 0, 1, 7, 8 and 2556? (A328110).

Out of the 30030 initial terms, 19220 are multiples of 5. (See A327865).

LINKS

Antti Karttunen, Table of n, a(n) for n = 0..2310

Antti Karttunen, Data supplement: n, a(n) computed for n = 0..30030

Index entries for sequences related to primorial base

FORMULA

a(n) = A003415(A276086(n)).

a(A002110(n)) = 1 for all n >= 0.

From Antti Karttunen, Nov 03 2019: (Start)

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:

a(n) = a(A328841(n)+A328842(n)) = A329031(n)*A328572(n) + A329032(n)*A328571(n).

A051903(a(n)) = A328391(n).

A328114(a(n)) = A328392(n).

(End)

EXAMPLE

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.

PROG

(PARI)

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

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; };

A327860(n) = A003415(A276086(n));

(PARI) A327860(n) = { my(m=1, i=0, s=0, pr=1, nextpr); while((n>0), i=i+1; nextpr = prime(i)*pr; if((n%nextpr), my(e=((n%nextpr)/pr)); m *= (prime(i)^e); s += (e / prime(i)); n-=(n%nextpr)); pr=nextpr); (s*m); }; \\ (Stand-alone implementation)

(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); }; \\ (This is even better) - Antti Karttunen, Nov 07 2019

CROSSREFS

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.

Coincides with A329029 on positions given by A276156.

Cf. A267263, A276150, A324650, A324653, A324655 for omega, bigomega, phi, sigma and tau applied to A276086(n).

Sequence in context: A231182 A231181 A259775 * A057520 A060423 A037951

Adjacent sequences:  A327857 A327858 A327859 * A327861 A327862 A327863

KEYWORD

nonn,look

AUTHOR

Antti Karttunen, Sep 30 2019

STATUS

approved

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Last modified February 18 03:33 EST 2020. Contains 332006 sequences. (Running on oeis4.)