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 1-5 of 5 results.

A342925 a(n) = A003415(sigma(n)), where A003415 is the arithmetic derivative, and sigma is the sum of divisors of n.

Original entry on oeis.org

0, 1, 4, 1, 5, 16, 12, 8, 1, 21, 16, 32, 9, 44, 44, 1, 21, 16, 24, 41, 80, 60, 44, 92, 1, 41, 68, 92, 31, 156, 80, 51, 112, 81, 112, 20, 21, 92, 92, 123, 41, 272, 48, 124, 71, 156, 112, 128, 22, 34, 156, 77, 81, 244, 156, 244, 176, 123, 92, 332, 33, 272, 164, 1, 124, 384, 72, 165, 272, 384, 156, 119, 39, 101, 128, 188
Offset: 1

Views

Author

Antti Karttunen, Apr 07 2021

Keywords

Links

Crossrefs

Cf. A023194 (positions of ones, which is a subsequence of prime powers, A000961).
Cf. A342922, A342923, A342924, A342926, A351568, A351569, A351570, A351571.
Cf. A342021 (fixed points), A343216 [positions k where a(k) < k], A343217 [a(k) >= k], A343218 [a(k) > k].
Cf. A347870 (parity of terms), A347872, A347873, A347877 (positions of odd terms), A347878 (of even terms), A343218, A343220, A344024.

Programs

  • Mathematica
    Array[If[# < 2, 0, # Total[#2/#1 & @@@ FactorInteger[#]]] &@ DivisorSigma[1, #] &, 76] (* Michael De Vlieger, Apr 08 2021 *)
  • PARI
    A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A342925(n) = A003415(sigma(n));

Formula

a(A023194(n)) = 1.
If gcd(m,n) = 1, a(m*n) = sigma(m)*A003415(sigma(n)) + sigma(n)*A003415(sigma(m)) = sigma(m)*a(n) + sigma(n)*a(m).
a(n) = (A351568(n)*A351571(n)) + (A351569(n)*A351570(n)). - Antti Karttunen, Feb 23 2022

A351568 Sum of the divisors of the largest unitary divisor of n that is a square.

Original entry on oeis.org

1, 1, 1, 7, 1, 1, 1, 1, 13, 1, 1, 7, 1, 1, 1, 31, 1, 13, 1, 7, 1, 1, 1, 1, 31, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 91, 1, 1, 1, 1, 1, 1, 1, 7, 13, 1, 1, 31, 57, 31, 1, 7, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 13, 127, 1, 1, 1, 7, 1, 1, 1, 13, 1, 1, 31, 7, 1, 1, 1, 31, 121, 1, 1, 7, 1, 1, 1, 1, 1, 13, 1, 7, 1, 1, 1, 1, 1, 57, 13
Offset: 1

Views

Author

Antti Karttunen, Feb 23 2022

Keywords

Comments

Obviously, all terms are odd.

Links

Crossrefs

Cf. A000203, A002117, A350388, A351569, A351570, A351575 (positions of primes).

Programs

  • Mathematica
    f[p_, e_] := If[EvenQ[e], (p^(e + 1) - 1)/(p - 1), 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 23 2022 *)
  • PARI
    A350388(n) = { my(m=1, f=factor(n)); for(k=1,#f~,if(0==(f[k,2]%2), m *= (f[k,1]^f[k,2]))); (m); };
A351568(n) = sigma(A350388(n));
  • Python
    from math import prod
from sympy import factorint
def A351568(n): return prod(1 if e % 2 else (p**(e+1)-1)//(p-1) for p, e in factorint(n).items()) # Chai Wah Wu, Feb 24 2022

Formula

Multiplicative with a(p^e) = (p^(e+1)-1)/(p-1) if e is even and 1 otherwise.
a(n) = A000203(A350388(n)).
a(n) = A000203(n) / A351569(n).
Sum_{k=1..n} a(k) ~ c * n^(3/2), where c = (zeta(3)/3) * Product_{p prime} (1 + 1/p^(3/2) + 1/p^2 - 1/p^(5/2)) = 1.008259499413... . - Amiram Eldar, Nov 20 2022
Dirichlet g.f.: zeta(2*s) * zeta(2*s-2) * Product_{p prime} (1 + 1/p^s + 1/p^(2*s-1) - 1/p^(3*s-2)). - Amiram Eldar, Sep 03 2023

A351571 Arithmetic derivative of the sum of the divisors of the largest unitary divisor of n that is an exponentially odd number.

Original entry on oeis.org

0, 1, 4, 0, 5, 16, 12, 8, 0, 21, 16, 4, 9, 44, 44, 0, 21, 1, 24, 5, 80, 60, 44, 92, 0, 41, 68, 12, 31, 156, 80, 51, 112, 81, 112, 0, 21, 92, 92, 123, 41, 272, 48, 16, 5, 156, 112, 4, 0, 1, 156, 9, 81, 244, 156, 244, 176, 123, 92, 44, 33, 272, 12, 0, 124, 384, 72, 21, 272, 384, 156, 8, 39, 101, 4, 24, 272, 332, 176, 5
Offset: 1

Views

Author

Antti Karttunen, Feb 23 2022

Keywords

Links

Crossrefs

Cf. A000203, A003415, A268335 (exponentially odd numbers), A342925, A350389, A351569, A351570, A351573.

Programs

  • PARI
    A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A350389(n) = { my(m=1, f=factor(n)); for(k=1,#f~,if(1==(f[k,2]%2), m *= (f[k,1]^f[k,2]))); (m); };
A351571(n) = A003415(sigma(A350389(n)));

Formula

a(n) = A003415(A351569(n)) = A342925(A350389(n)) = A003415(A000203(A350389(n))).

A351572 Arithmetic derivative of the largest unitary divisor of n that is a square.

Original entry on oeis.org

0, 0, 0, 4, 0, 0, 0, 0, 6, 0, 0, 4, 0, 0, 0, 32, 0, 6, 0, 4, 0, 0, 0, 0, 10, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 60, 0, 0, 0, 0, 0, 0, 0, 4, 6, 0, 0, 32, 14, 10, 0, 4, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 6, 192, 0, 0, 0, 4, 0, 0, 0, 6, 0, 0, 10, 4, 0, 0, 0, 32, 108, 0, 0, 4, 0, 0, 0, 0, 0, 6, 0, 4, 0, 0, 0, 0, 0, 14, 6, 140
Offset: 1

Views

Author

Antti Karttunen, Feb 23 2022

Keywords

Comments

All terms are even: see comments in A235992 and observe that the terms of A350388 are all either odd or multiples of 4.

Links

Crossrefs

Cf. A003415, A235992, A350388, A351570, A351573.

Programs

  • Mathematica
    f1[p_, e_] := If[EvenQ[e], p^e, 1]; f2[p_, e_] := If[EvenQ[e], e/p, 0]; a[1] = 0; a[n_] := (Times @@ f1 @@@ (f = FactorInteger[n]))*(Plus @@ f2 @@@ f); Array[a, 100] (* Amiram Eldar, Feb 23 2022 *)
  • PARI
    A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A350388(n) = { my(m=1, f=factor(n)); for(k=1,#f~,if(0==(f[k,2]%2), m *= (f[k,1]^f[k,2]))); (m); };
A351572(n) = A003415(A350388(n));

Formula

a(n) = A003415(A350388(n)).

A351575 Positions of primes in A351568.

Original entry on oeis.org

4, 9, 12, 16, 18, 20, 25, 28, 44, 45, 48, 50, 52, 60, 63, 64, 68, 72, 75, 76, 80, 84, 90, 92, 99, 108, 112, 116, 117, 124, 126, 132, 140, 148, 150, 153, 156, 164, 171, 172, 175, 176, 188, 192, 198, 200, 204, 207, 208, 212, 220, 228, 234, 236, 240, 244, 260, 261, 268, 272, 275, 276, 279, 284, 288, 289, 292, 304, 306
Offset: 1

Views

Author

Antti Karttunen, Feb 24 2022

Keywords

Comments

Numbers k such that A350388(k) is one of the terms of A023194.

Crossrefs

Positions of primes in A351568, positions of ones in A351570.
Cf. A000203, A023194, A350388.

Programs

  • Mathematica
    f[p_, e_] := If[EvenQ[e], (p^(e + 1) - 1)/(p - 1), 1]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[300], PrimeQ[s[#]] &] (* Amiram Eldar, Feb 25 2022 *)
  • PARI
    A350388(n) = { my(m=1, f=factor(n)); for(k=1,#f~,if(0==(f[k,2]%2), m *= (f[k,1]^f[k,2]))); (m); };
A351568(n) = sigma(A350388(n));
isA351575(n) = isprime(A351568(n));
Showing 1-5 of 5 results.

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