A353571 -id:A353571 - cp's OEIS frontend

A353572 Shifted variant of A342002: a(n) = A353571(A276086(n)), where A353571(x) = A003415(A003961(x)) / A003557(A003961(x)) and A276086 is the primorial base exp-function.

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

Offset: 0

Antti Karttunen, Apr 27 2022

Cf. A003415, A003557, A003961, A276154, A342002, A349905, A353571, A353573 [= gcd(a(n), A342002(n))], A353574.

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); };
A353571(n) = { my(s=A003961(n)); (A003415(s)/A003557(s)); };
A353572(n) = A353571(A276086(n));

a(n) = A353571(A276086(n)).
a(n) = A342002(A276154(n)).
For all n >= 0, a(n) >= A342002(n).

A342001 Arithmetic derivative without its inherited divisor; the arithmetic derivative of n divided by A003557(n), which is a common divisor of both n and A003415(n).

Original entry on oeis.org

0, 1, 1, 2, 1, 5, 1, 3, 2, 7, 1, 8, 1, 9, 8, 4, 1, 7, 1, 12, 10, 13, 1, 11, 2, 15, 3, 16, 1, 31, 1, 5, 14, 19, 12, 10, 1, 21, 16, 17, 1, 41, 1, 24, 13, 25, 1, 14, 2, 9, 20, 28, 1, 9, 16, 23, 22, 31, 1, 46, 1, 33, 17, 6, 18, 61, 1, 36, 26, 59, 1, 13, 1, 39, 11, 40, 18, 71, 1, 22, 4, 43, 1, 62, 22, 45, 32, 35, 1, 41, 20

Offset: 1

Antti Karttunen, Feb 28 2021

nonn

See also the scatter plot of A342002 that seems to reveal some interesting internal structure in this sequence, not fully explained by the regularity of primorial base expansion used in the latter sequence. - Antti Karttunen, May 09 2022

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A003415, A003557, A341998, A342416, A342459.
Cf. A342002 [= a(A276086(n))], A342463 [= a(A342456(n))], A351945 [= a(A181819(n))], A353571 [= a(A003961(n))].
Cf. A346485 (Möbius transform), A347395 (convolution with Liouville's lambda), A347961 (with itself), and A347234, A347235, A347954, A347959, A347963, A349396, A349612 (for convolutions with other sequences).
Cf. A007947.

Array[#1/#2 & @@ {If[# < 2, 0, # Total[#2/#1 & @@@ FactorInteger[#]]] &@ Abs[#], #/Times @@ FactorInteger[#][[All, 1]]} &, 91] (* Michael De Vlieger, Mar 11 2021 *)

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]));
A342001(n) = (A003415(n) / A003557(n));

from math import prod
from sympy import factorint
def A342001(n):
    q = prod(f:=factorint(n))
    return sum(q*e//p for p, e in f.items()) # Chai Wah Wu, Nov 04 2022

a(n) = A003415(n) / A003557(n).
For all n >= 0, a(A276086(n)) = A342002(n).
a(n) = A342414(n) * A342416(n) = A342459(n) * A342919(n). - Antti Karttunen, Apr 30 2022
Dirichlet g.f.: Dirichlet g.f. of A007947 * Sum_{p prime} p^s/((p^s-1)*(p^s+p-1)) = zeta(s) * Product_{p prime} (1+p^(1-s)-p^(-s)) * Sum_{p prime} p^s/((p^s-1)*(p^s+p-1)). - Sebastian Karlsson, May 05 2022
Sum_{k=1..n} a(k) ~ c * A065464 * Pi^2 * n^2 / 12, where c = Sum_{j>=2} (1/2 + (-1)^j * (Fibonacci(j) - 1/2))*PrimeZetaP(j) = 0.4526952873143153104685540856936425315834753528741817723313791528384... - Vaclav Kotesovec, May 09 2022

A349905 Arithmetic derivative of A003961(n), where A003961 is fully multiplicative with a(p) = nextprime(p).

Original entry on oeis.org

0, 1, 1, 6, 1, 8, 1, 27, 10, 10, 1, 39, 1, 14, 12, 108, 1, 55, 1, 51, 16, 16, 1, 162, 14, 20, 75, 75, 1, 71, 1, 405, 18, 22, 18, 240, 1, 26, 22, 216, 1, 103, 1, 87, 95, 32, 1, 621, 22, 91, 24, 111, 1, 350, 20, 324, 28, 34, 1, 318, 1, 40, 135, 1458, 24, 119, 1, 123, 34, 131, 1, 945, 1, 44, 119, 147, 24, 151, 1, 837

Offset: 1

Antti Karttunen, Dec 05 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences computed from indices in prime factorization

Cf. A003415, A003961, A026424 (positions of odd terms), A028260 (of even terms), A066829 (parity of a(n)).
Cf. also A343221, A343222, A347964, A347965, A348937, A353571.
Cf. A358760, A358761, A358762, A358763 for indices of terms that of the form 4k+j, for j=0..3, and A358750, A358751, A358752, A358753 for their characteristic functions.

f1[p_, e_] := e/p; d[1] = 0; d[n_] := n * Plus @@ f1 @@@ FactorInteger[n]; f2[p_, e_] := NextPrime[p]^e; s[1] = 1; s[n_] := Times @@ f2 @@@ FactorInteger[n]; a[n_] := d[s[n]]; Array[a, 100] (* Amiram Eldar, Dec 05 2021 *)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A349905(n) = A003415(A003961(n));

a(n) = A003415(A003961(n)).

A353574 a(n) = A342002(n) / gcd(A342002(n), A342002(A276154(n))).

Original entry on oeis.org

1, 1, 5, 1, 7, 1, 7, 2, 31, 13, 41, 1, 9, 11, 37, 2, 47, 1, 11, 7, 43, 19, 53, 1, 13, 17, 49, 11, 59, 1, 9, 5, 41, 17, 55, 2, 59, 71, 247, 53, 317, 19, 73, 46, 289, 127, 359, 13, 87, 113, 331, 74, 401, 11, 101, 67, 373, 169, 443, 1, 11, 13, 47, 5, 61, 17, 69, 43, 277, 121, 347, 2, 83, 107, 319, 71, 389, 31, 97, 16

Offset: 1

Antti Karttunen, Apr 29 2022

Compare the scatter plot to that of A342002.

Cf. A003415, A003557, A003961, A342001, A342002, A353572, A353573, A353575.

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); };
A342001(n) = (A003415(n) / A003557(n));
A353571(n) = A342001(A003961(n));
A353574(n) = { my(u=A342001(A276086(n))); u/gcd(u, A353571(A276086(n))); };

a(n) = A342002(n) / A353573(n) = A342002(n) / gcd(A342002(n), A353572(n)).

A353523 Lexicographically earliest infinite sequence such that a(i) = a(j) => A349905(i) = A349905(j) and A003557(i) = A003557(j), for all i, j >= 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Apr 27 2022

nonn

Restricted growth sequence transform of the ordered pair [A003557(n), A349905(n)], or equally, of the ordered pair [A003415(A003961(n)), A003557(A003961(n))].
This is a prime-shifted variant of A344025, as this is the restricted growth sequence transform of A344025(A003961(n)).
For all i, j:
  A305800(i) = A305800(j) => a(i) = a(j),
  a(i) = a(j) => A349905(i) = A349905(j) => A008836(i) = A008836(j),
  a(i) = a(j) => A353571(i) = A353571(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A003415, A003557, A003961, A008836, A305800, A344025, A349905, A353571.

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; };
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
Aux353523(n) = { my(s=A003961(n)); [A003415(s), A003557(s)]; };
v353523 = rgs_transform(vector(up_to, n, Aux353523(n)));
A353523(n) = v353523[n];

A353573 Greatest common divisor of A342002 and its shifted variant, where A342002(n) = A003415(A276086(n)) / A003557(A276086(n)) and A276086 is the primorial base exp-function.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Apr 29 2022

nonn

Cf. A003415, A003557, A003961, A342001, A342002, A353572, A353574.

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); };
A342001(n) = (A003415(n) / A003557(n));
A353571(n) = A342001(A003961(n));
A353573(n) = gcd(A353571(A276086(n)), A342001(A276086(n)));

a(n) = gcd(A342002(n), A353572(n)).
a(n) = gcd(A353571(A276086(n)), A342001(A276086(n))).
For all n >= 1, a(n) = A342002(n) / A353574(n).

cp's OEIS Frontend

A353572 Shifted variant of A342002: a(n) = A353571(A276086(n)), where A353571(x) = A003415(A003961(x)) / A003557(A003961(x)) and A276086 is the primorial base exp-function.

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A342001 Arithmetic derivative without its inherited divisor; the arithmetic derivative of n divided by A003557(n), which is a common divisor of both n and A003415(n).

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A349905 Arithmetic derivative of A003961(n), where A003961 is fully multiplicative with a(p) = nextprime(p).

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A353574 a(n) = A342002(n) / gcd(A342002(n), A342002(A276154(n))).

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A353523 Lexicographically earliest infinite sequence such that a(i) = a(j) => A349905(i) = A349905(j) and A003557(i) = A003557(j), for all i, j >= 1.

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A353573 Greatest common divisor of A342002 and its shifted variant, where A342002(n) = A003415(A276086(n)) / A003557(A276086(n)) and A276086 is the primorial base exp-function.

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