A342017 -id:A342017 - cp's OEIS frontend

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

Antti Karttunen, Feb 28 2021

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

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

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).

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.

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

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

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)

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

A342018 Numbers k such that the arithmetic derivative of A276086(k) is divisible by at least one prime power divisor of the form p^p, where A276086 gives the prime product form of primorial base expansion of its argument.

Original entry on oeis.org

8, 16, 24, 36, 44, 52, 64, 72, 80, 88, 92, 100, 108, 116, 120, 126, 128, 136, 144, 156, 164, 172, 184, 192, 200, 208, 216, 222, 224, 232, 244, 252, 260, 268, 271, 272, 280, 288, 296, 300, 308, 316, 324, 336, 344, 348, 352, 364, 372, 380, 388, 392, 397, 400, 408, 416, 424, 432, 440, 444, 448, 452, 460, 468, 476, 480, 488, 493, 496

Offset: 1

Antti Karttunen, Mar 04 2021

nonn

Numbers k for which A342019(k) > 0, or equally, A342007(A327860(k)) = A342017(k) is larger than one, or equally A342007(A342002(k)) > 1, that is, k for which A342023(A342002(k)) = 1.

The first odd term is a(35) = 271.

			8 is present as A276086(8) = 15, A003415(15) = 8 = 2^3, which is thus divisible by p^p (with p=2 in this case).
271 is present as A276086(271) = 1078, A003415(1078) = 945 = 3^3 * 5 * 7, which is thus divisible by p^p (with p=3 in this case).

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences related to primorial base

Positions of terms larger than one in A342017, of nonzero terms in A342019.
Cf. A003415, A276086, A327860, A342002, A342007, A342017, A342019, A342023.
Not a subsequence of A342006.

Block[{b = MixedRadix[Reverse@ Prime@ Range@ 12]}, Position[#, ?(# > 1 &)][[All, 1]] &@ Array[Function[k, Times @@ Map[#1^Floor[#2/#1] & @@ # &, FactorInteger[#]] &@ If[# < 2, 0, # Total[#2/#1 & @@@ FactorInteger[#]] ] &@ Abs[Times @@ Power @@@ # &@ Transpose@ {Prime@ Range@ Length@ k, Reverse@ k}]]@ IntegerDigits[#, b] &, 500]] (* _Michael De Vlieger, Mar 12 2021 *)

PARI
```
isA342018(n) = (A342017(n)>1);
```

Name changed by Antti Karttunen, Mar 12 2021

A342019 Number of prime power divisors of the form p^p in the arithmetic derivative of A276086(n), the prime product form of primorial base expansion of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1

Offset: 1

Antti Karttunen, Mar 11 2021

nonn

When considering the iterations of the arithmetic derivative (i.e., of map x -> A003415(x)) and where such process will eventually end, it is known that any divisor of the form p^p (with p prime) in x guarantees that instead of ever reaching zero, it will either get stuck at a fixed point (of the form p^p) or diverge forever towards infinity (see, e.g., Ufnarovski and Åhlander paper). Because such (new) "doom divisors" are born only in the "wild part" of the arithmetic derivative (i.e., what is left after A003557(n) has been divided out of the derivative of n), it follows that when the derivative is applied to terms of A276086 (that do not contain any doom divisors), the count of such divisors in the result must be equal to that for A342002(n).

Positions of the records (and the first occurrence of each n) start as: 1, 8, 1164, 18675300, ...

			For n=108, A342002(108) = 36 = 2^2 * 3^2. Only the first prime power divisor is of the form p^p, thus a(108) = 1. Note that A276086(108) = A003415(42875) = 42875 = 5^3 * 7^3, and A327860(108) = 44100 = 2^2 * 3^2 * 5^2 * 7^2. The same "doom divisors" are always found both in A327860(n) and in A342002(n).
For n=1164, A342002(1164) = 648 = 2^3 * 3^4. In both prime power divisors the exponent attains its base prime (3 >= 2 and 4 >= 3), thus a(1164) = 2. Note that A276086(1164) = 34525308125 = 5^4 * 7^3 * 11^5, and A327860(1164) = 58110129000 = 2^3 * 3^4 * 5^3 * 7^2 * 11^4.
For n=18675300, A342002(18675300) = 3037500 = 2^2 * 3^5 * 5^5. Here all three prime power divisors are "doom divisors" because they reach the p^p limit, thus a(18675300) = 3.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Victor Ufnarovski and Bo Åhlander, How to Differentiate a Number, J. Integer Seqs., Vol. 6, 2003, #03.3.4.
Index entries for sequences related to primorial base

Cf. A001221, A003415, A003557, A129251, A276086, A327860, A342002, A342005, A342017, A342018 (positions of nonzero terms), A342026.

A129251(n) = { my(f = factor(n)); sum(k=1, #f~, (f[k, 2]>=f[k, 1])); };
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); };
A342019(n) = A129251(A327860(n));

a(n) = A129251(A327860(n)) = A129251(A342002(n)).

a(n) = A001221(A342017(n)).

A342007 Multiplicative with a(p^e) = p^floor(e/p).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 1, 8, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 4, 3, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 2, 1

Offset: 1

Antti Karttunen, Mar 04 2021

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A327936, A327938, A327939, A341997, A342017 [= a(A276086(n))].

Array[Times @@ Map[#1^Floor[#2/#1] & @@ # &, FactorInteger[#]] &, 105] (* Michael De Vlieger, Mar 12 2021 *)

A342007(n) = { my(f = factor(n)); for(k=1, #f~, f[k, 2] = floor(f[k, 2]/f[k, 1])); factorback(f); };

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + (p-1)/(p^p-p)) = 1.6270951877598772517... . - Amiram Eldar, Nov 07 2022

A342022 Lexicographically earliest infinite sequence such that a(i) = a(j) => A342002(i) = A342002(j) for all i, j >= 0.

Original entry on oeis.org

1, 2, 2, 3, 4, 5, 2, 5, 6, 7, 8, 9, 4, 10, 11, 12, 13, 14, 15, 11, 16, 17, 18, 19, 20, 8, 21, 22, 23, 24, 2, 10, 25, 9, 21, 26, 27, 24, 28, 29, 30, 31, 18, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 4, 11, 8, 14, 49, 50, 21, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 7, 62, 63, 64, 65, 66

Offset: 0

Antti Karttunen, Mar 12 2021

Restricted growth sequence transform of A342002.
For all i, j >= 1:
  a(i) = a(j) => A342017(i) = A342017(j) => A342019(i) = A342019(j).

Antti Karttunen, Table of n, a(n) for n = 0..65537
Index entries for sequences related to primorial base

Cf. A003415, A276086, A327860, A328572, A342002, A342017, A342019.

Cf. also A286626, A328388.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
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); };
v342022 = rgs_transform(vector(1+up_to,n,A342002(n-1)));
A342022(n) = v342022[1+n];

cp's OEIS Frontend

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

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A342018 Numbers k such that the arithmetic derivative of A276086(k) is divisible by at least one prime power divisor of the form p^p, where A276086 gives the prime product form of primorial base expansion of its argument.

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A342019 Number of prime power divisors of the form p^p in the arithmetic derivative of A276086(n), the prime product form of primorial base expansion of n.

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A342007 Multiplicative with a(p^e) = p^floor(e/p).

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A342022 Lexicographically earliest infinite sequence such that a(i) = a(j) => A342002(i) = A342002(j) for all i, j >= 0.

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