Search: a353574 -id:a353574
|
| |
|
|
A342002
|
|
Čiurlionis sequence: Arithmetic derivative without its inherited divisor applied to the primorial base exp-function: a(n) = A342001(A276086(n)).
|
|
+10
48
|
|
|
|
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" 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
|
|
|
|
FORMULA
|
There are several permutations of this sequence. The following formulas show the relations:
(End)
|
|
|
PROG
|
(PARI)
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
(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. A002110 (positions of 1's), A003415, A003557, A083345, A085731, A276086, A289234, A327860, A328571, A328572, A342001, A342005, A342006, A342016, A342022 (rgs-transform), A342417, A342419.
Cf. A351952 (similar definition, but using factorial base, with quite a different look).
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
EXTENSIONS
|
Sequence renamed as "Čiurlionis sequence" to honor Lithuanian artist Mikalojus Čiurlionis - Antti Karttunen, Apr 30 2022
|
|
|
STATUS
|
approved
|
| |
|
|
| |
| |
|
|
|
0, 1, 1, 8, 2, 11, 1, 10, 12, 71, 19, 92, 2, 13, 17, 86, 24, 107, 3, 16, 22, 101, 29, 122, 4, 19, 27, 116, 34, 137, 1, 14, 16, 103, 27, 136, 18, 131, 167, 886, 244, 1117, 29, 164, 222, 1051, 299, 1282, 40, 197, 277, 1216, 354, 1447, 51, 230, 332, 1381, 409, 1612, 2, 17, 21, 118, 32, 151, 25, 152, 202, 991, 279
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,4
|
|
|
LINKS
|
|
|
|
FORMULA
|
For all n >= 0, a(n) >= A342002(n).
|
|
|
PROG
|
(PARI)
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003557(n) = (n/factorback(factorint(n)[, 1]));
A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn,base
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
| |
|
|
A353575
|
|
Primepi based arithmetic derivative applied to the prime shadow of the primorial base exp-function: a(n) = A258851(A181819(A276086(n))).
|
|
+10
4
|
|
|
|
0, 1, 1, 4, 2, 7, 1, 4, 4, 12, 7, 20, 2, 7, 7, 20, 12, 33, 3, 11, 11, 32, 19, 53, 4, 15, 15, 44, 26, 73, 1, 4, 4, 12, 7, 20, 4, 12, 12, 32, 20, 52, 7, 20, 20, 52, 33, 84, 11, 32, 32, 84, 53, 136, 15, 44, 44, 116, 73, 188, 2, 7, 7, 20, 12, 33, 7, 20, 20, 52, 33, 84, 12, 33, 33, 84, 54, 135, 19, 53, 53, 136, 87, 219
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,4
|
|
|
LINKS
|
|
|
|
FORMULA
|
|
|
|
PROG
|
(PARI)
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A181819(n) = factorback(apply(e->prime(e), (factor(n)[, 2])));
A258851(n) = (n*sum(i=1, #n=factor(n)~, n[2, i]*primepi(n[1, i])/n[1, i])); \\ From A258851
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
| |
| |
|
|
|
0, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 8, 1, 3, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 12, 1, 1, 1, 2, 1, 1, 1, 8, 1, 1, 1, 2, 1, 1, 1, 4, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,5
|
|
|
LINKS
|
|
|
|
FORMULA
|
|
|
|
PROG
|
(PARI)
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003557(n) = (n/factorback(factorint(n)[, 1]));
A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
Search completed in 0.007 seconds
|
