The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

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