cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A346474 a(n) = A342414(A276086(n)).

Original entry on oeis.org

0, 1, 1, 5, 1, 7, 1, 7, 1, 31, 13, 41, 1, 9, 11, 37, 2, 47, 3, 11, 7, 43, 19, 53, 1, 13, 17, 49, 11, 59, 1, 3, 5, 41, 17, 55, 1, 59, 71, 247, 53, 317, 19, 73, 23, 289, 127, 359, 13, 29, 113, 331, 37, 401, 11, 101, 67, 373, 169, 443, 1, 11, 13, 47, 5, 61, 17, 23, 43, 277, 121, 347, 1, 83, 107, 319, 71, 389, 31, 97, 8, 361, 163
Offset: 0

Views

Author

Antti Karttunen, Jul 21 2021

Keywords

Comments

For n >= 1, each term a(n) is a divisor of A342002(n).

Links

Crossrefs

Cf. A000010, A003415, A276086, A324650, A327860, A342414.
Cf. also A342002, A345930, A346475 for sequences with similar scatter plots.

Programs

  • PARI
    A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A342414(n) = { my(u=A003415(n)); (u/gcd(eulerphi(n),u)); };
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A346474(n) = A342414(A276086(n));

Formula

a(n) = A342414(A276086(n)).
a(n) = A327860(n) / gcd(A327860(n), A324650(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

Views

Author

Antti Karttunen, Feb 28 2021

Keywords

Comments

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

Links

Crossrefs

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.

Programs

  • Mathematica
    Array[#1/#2 & @@ {If[# < 2, 0, # Total[#2/#1 & @@@ FactorInteger[#]]] &@ Abs[#], #/Times @@ FactorInteger[#][[All, 1]]} &, 91] (* Michael De Vlieger, Mar 11 2021 *)
  • 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]));
A342001(n) = (A003415(n) / A003557(n));
  • Python
    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

Formula

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

A342413 a(n) = gcd(phi(n), A003415(n)), where A003415(n) is the arithmetic derivative of n, and phi is Euler totient function.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 4, 6, 1, 1, 4, 1, 3, 8, 8, 1, 3, 1, 8, 2, 1, 1, 4, 10, 3, 9, 4, 1, 1, 1, 16, 2, 1, 12, 12, 1, 3, 8, 4, 1, 1, 1, 4, 3, 1, 1, 16, 14, 5, 4, 8, 1, 9, 8, 4, 2, 1, 1, 4, 1, 3, 3, 32, 6, 1, 1, 8, 2, 1, 1, 12, 1, 3, 5, 4, 6, 1, 1, 16, 54, 1, 1, 4, 2, 3, 8, 20, 1, 3, 4, 4, 2, 1, 24, 16, 1, 7, 15, 20
Offset: 1

Views

Author

Antti Karttunen, Mar 11 2021

Keywords

Links

Crossrefs

Cf. A000010, A003415, A003557, A166374, A342009, A342414, A342415, A342416.
Cf. also A009195, A085731.

Programs

  • Mathematica
    Array[GCD[If[# < 2, 0, # Total[#2/#1 & @@@ FactorInteger[#]]], EulerPhi[#]] &@ Abs[#] &, 100] (* Michael De Vlieger, Mar 11 2021 *)
  • PARI
    A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A342413(n) = gcd(eulerphi(n), A003415(n));

Formula

a(n) = gcd(A000010(n), A003415(n)).
a(n) = A003415(n) / A342414(n) = A000010(n) / A342415(n).
a(n) = A003557(n) * A342416(n).

A342919 a(n) = A003415(n) / gcd(A001615(n), A003415(n)), where A001615 is Dedekind psi, and A003415 is the arithmetic derivative of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 5, 1, 1, 1, 7, 1, 2, 1, 3, 1, 4, 1, 7, 1, 2, 5, 13, 1, 11, 1, 5, 3, 2, 1, 31, 1, 5, 7, 19, 1, 5, 1, 7, 2, 17, 1, 41, 1, 2, 13, 25, 1, 7, 1, 1, 5, 2, 1, 3, 2, 23, 11, 31, 1, 23, 1, 11, 17, 2, 3, 61, 1, 2, 13, 59, 1, 13, 1, 13, 11, 2, 3, 71, 1, 11, 1, 43, 1, 31, 11, 15, 4, 35, 1, 41, 5, 2, 17, 49, 1, 17, 1, 11, 25
Offset: 1

Views

Author

Antti Karttunen, Mar 29 2021

Keywords

Links

Crossrefs

Cf. A001615, A003415, A342001, A342458, A342459.
Cf. also A342414.

Programs

  • PARI
    A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))); \\ After code in A001615
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A342919(n) = { my(u=A003415(n)); (u/gcd(u, A001615(n))); };

Formula

a(n) = A003415(n) / A342458(n) = A003415(n) / gcd(A001615(n), A003415(n)).
a(n) = A342001(n) / A342459(n).

A342008 Numbers k such that Euler totient phi(k) is a multiple of the arithmetic derivative of k.

Original entry on oeis.org

2, 3, 5, 7, 9, 11, 13, 15, 17, 19, 23, 25, 29, 31, 35, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 83, 89, 95, 97, 101, 103, 107, 109, 113, 119, 121, 127, 131, 137, 139, 143, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 209, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281
Offset: 1

Views

Author

Antti Karttunen, Mar 12 2021

Keywords

Comments

Numbers k for which A000010(k) is a multiple of A003415(k), or equally, k for which A173557(k) is a multiple of A342001(k).

Crossrefs

Cf. A000010, A003415, A173557, A342001, A342009.
Subsequences: A000040, A166374, A342418 (composite terms).
Positions of ones in A342414.

Programs

  • Mathematica
    Select[Range[2, 281], Mod[EulerPhi[#], If[# < 2, 0, # Total[#2/#1 & @@@ FactorInteger[#]] ] &@ Abs[#]] == 0 &] (* Michael De Vlieger, Mar 12 2021 *)
  • PARI
    A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
isA342008(n) = ((n>1)&&!(eulerphi(n)%A003415(n)));
for(n=2,2^8,if(isA342008(n),print1(n,", ")));

A342415 a(n) = phi(n) / gcd(phi(n),A003415(n)), where A003415(n) is the arithmetic derivative of n, and phi is Euler totient function.

Original entry on oeis.org

1, 1, 2, 1, 4, 2, 6, 1, 1, 4, 10, 1, 12, 2, 1, 1, 16, 2, 18, 1, 6, 10, 22, 2, 2, 4, 2, 3, 28, 8, 30, 1, 10, 16, 2, 1, 36, 6, 3, 4, 40, 12, 42, 5, 8, 22, 46, 1, 3, 4, 8, 3, 52, 2, 5, 6, 18, 28, 58, 4, 60, 10, 12, 1, 8, 20, 66, 4, 22, 24, 70, 2, 72, 12, 8, 9, 10, 24, 78, 2, 1, 40, 82, 6, 32, 14, 7, 2, 88, 8, 18, 11, 30
Offset: 1

Views

Author

Antti Karttunen, Mar 11 2021

Keywords

Links

Crossrefs

Cf. A000010, A003415, A173557, A342009 (positions of ones), A342413, A342414, A342416.

Programs

  • Mathematica
    Array[#2/GCD[#1, #2] & @@ {If[# < 2, 0, # Total[#2/#1 & @@@ FactorInteger[#]]] &@ Abs[#], EulerPhi[#]} &, 93] (* Michael De Vlieger, Mar 11 2021 *)
  • PARI
    A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A342415(n) = { my(u=eulerphi(n)); (u/gcd(u,A003415(n))); };

Formula

a(n) = A000010(n) / A342413(n) = A000010(n) / gcd(A000010(n),A003415(n)).
a(n) = A173557(n) / A342416(n).
Showing 1-6 of 6 results.

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