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 A342002 a(n) = A327860(n) / A328572(n) = A003415(A276086(n)) / A003557(A276086(n)). 28
 0, 1, 1, 5, 2, 7, 1, 7, 8, 31, 13, 41, 2, 9, 11, 37, 16, 47, 3, 11, 14, 43, 19, 53, 4, 13, 17, 49, 22, 59, 1, 9, 10, 41, 17, 55, 12, 59, 71, 247, 106, 317, 19, 73, 92, 289, 127, 359, 26, 87, 113, 331, 148, 401, 33, 101, 134, 373, 169, 443, 2, 11, 13, 47, 20, 61, 17, 69, 86, 277, 121, 347, 24, 83, 107, 319, 142, 389, 31 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS The scatter plot shows an interesting structure. The terms are essentially the "wild part" of the arithmetic derivative (A003415) of those natural numbers (A048103) that are not immediately beyond all hope of reaching zero by iteration (as the terms of A100716 are), ordered by the primorial base expansion of n as in A276086. Sequence A342018 shows the positions of the terms here that have just moved to the "no hope" region, while A342019 shows how many prime powers in any term have breached the p^p limit. Note that the results are same as for A327860(n), as the division by "regular part", A328572(n) does not affect the "wild part" of the arithmetic derivative of A276086(n). - Antti Karttunen, Mar 12 2021 LINKS Antti Karttunen, Table of n, a(n) for n = 0..11550 Victor Ufnarovski and Bo Åhlander, How to Differentiate a Number, J. Integer Seqs., Vol. 6, 2003, #03.3.4. FORMULA a(n) = A327860(n) / A328572(n) = A003415(A276086(n)) / A003557(A276086(n)). a(n) = A342001(A276086(n)) = A083345(A276086(n)). A342007(a(n)) = A342017(n), A129251(a(n)) = A342019(n). - Antti Karttunen, Mar 11 2021 From Antti Karttunen, Jul 18 2021: (Start) There are several permutations of this sequence. The following formulas show the relations: a(n) = A344760(A289234(n)). a(n) = A346252(A328623(n)) = A346253(A328622(n)). a(n) = A344761(A328626(n)) = A344762(A328625(n)). (End) PROG (PARI) A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); }; A342002(n) = A342001(A276086(n)); \\ Uses also code from A342001. (PARI) A342002(n) = { my(s=0, m=1, p=2, e); while(n, e = (n%p); m *= p^(e>0); s += (e/p); n = n\p; p = nextprime(1+p)); (s*m); }; \\ Antti Karttunen, Mar 12 2021 (PARI) A342002(n) = { my(s=0, p=2, e); while(n, e = (n%p); s += (e/p); n = n\p; p = nextprime(1+p)); numerator(s); }; \\ (Taking denominator instead would give A328571) - Antti Karttunen, Mar 12 2021 CROSSREFS Cf. A003415, A003557, A083345, A085731, A276086, A289234, A327860, A328571, A328572, A342001 ("wild part" of the arithmetic derivative of n), A342005, A342006, A342016, A342017, A342019, A342022 (rgs-transform), A342417 (Dirichlet inverse, from the term a(1)=1 onward), A342419, A342463 [= a(A329886(n))], A342920 [= a(A108951(n))], A342921 [= a(A276156(n))]. Cf. A344760, A344761, A344762, A346252, A346253 for permuted variants. Cf. also A345930. Sequence in context: A108399 A094772 A263832 * A344760 A343422 A214969 Adjacent sequences:  A341999 A342000 A342001 * A342003 A342004 A342005 KEYWORD nonn,base,look AUTHOR Antti Karttunen, Feb 28 2021 STATUS approved

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Last modified November 30 08:16 EST 2021. Contains 349419 sequences. (Running on oeis4.)