A342002 Čiurlionis sequence: Arithmetic derivative without its inherited divisor applied to the primorial base exp-function: a(n) = A342001(A276086(n)).

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

Offset: 0

Author

Antti Karttunen, Feb 28 2021

Keywords

• nonn
• base
• easy
• look

Comments

The scatter plot shows an interesting structure.

The terms are essentially the "wild" or "unherited" 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

I decided to name this sequence in honor of Lithuanian artist Mikalojus Čiurlionis, 1875 - 1911, as the scatter plot of this sequence reminds me thematically of his work "Pyramid sonata", with similar elements: fractal repetition in different scales and high tension present, discharging as lightning. Like Čiurlionis's paintings, this sequence has many variations, see the Formula and Crossrefs sections. - Antti Karttunen, Apr 30 2022

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.
• Wikimedia, Čiurlionis: Piramidziu sonata, Allegro (a medium resolution scan of the painting "Pyramid Sonata, Allegro")
• Wikipedia, Mikalojus Konstantinas Čiurlionis
• Index entries for sequences related to primorial base

Crossrefs

Cf. A002110 (positions of 1's), A003415, A003557, A083345, A085731, A276086, A289234, A327860, A328571, A328572, A342001, A342005, A342006, A342016, A342022 (rgs-transform), A342417, A342419.

Cf. A342463 [= a(A329886(n))], A342920 [= a(A108951(n))], A342921 [= a(A276156(n))], A342017 [= A342007(a(n))], A342019 [= A129251(a(n))].

Cf. A166486 (a(n) mod 2, parity of terms, see comment in A327860), A353640 (a(n) mod 4).

Cf. A344760, A344761, A344762, A346252, A346253 and A345930, A353572, A353574 for permuted and other variants.

Cf. A351952 (similar definition, but using factorial base, with quite a different look).

Programs

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

Formula

a(n) = A342001(A276086(n)) = A083345(A276086(n)).

a(n) = A327860(n) / A328572(n) = A003415(A276086(n)) / A003557(A276086(n)).

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)

Extensions

Sequence renamed as "Čiurlionis sequence" to honor Lithuanian artist Mikalojus Čiurlionis - Antti Karttunen, Apr 30 2022