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.

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A328768 The first primorial based variant of arithmetic derivative: a(prime(i)) = A002110(i-1), where prime(i) = A000040(i), a(u*v) = a(u)*v + u*a(v), with a(0) = a(1) = 0.

Original entry on oeis.org

0, 0, 1, 2, 4, 6, 7, 30, 12, 12, 17, 210, 20, 2310, 67, 28, 32, 30030, 33, 510510, 44, 104, 431, 9699690, 52, 60, 4633, 54, 148, 223092870, 71, 6469693230, 80, 652, 60077, 192, 84, 200560490130, 1021039, 6956, 108, 7420738134810, 229, 304250263527210, 884, 114, 19399403, 13082761331670030, 128, 420, 145, 90124, 9292, 614889782588491410, 135, 1116, 324
Offset: 0

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Author

Antti Karttunen, Oct 28 2019

Keywords

Links

Crossrefs

Cf. A002110, A108951, A276085, A276150, A328771, A373982, A374211.
Cf. A042965 (indices of even terms), A016825 (of odd terms), A152822 (antiparity of terms), A373992 (indices of multiples of 3), A374212 (2-adic valuation), A374213 (3-adic valuation), A374123 [a(n) mod 360].
Cf. A374031 [gcd(a(n), A276085(n))], A374116 [gcd(a(n), A328845(n))].
For variants of the same formula, see A003415, A258851, A328769, A328845, A328846, A371192.

Programs

  • PARI
    A002110(n) = prod(i=1,n,prime(i));
A328768(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]*A002110(primepi(f[i,1])-1)/f[i, 1]));
  • PARI
    A328768(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]*A002110(primepi(f[i, 1]))/(f[i, 1]^2)));

Formula

a(n) = n * Sum e_j * A276085(p_j)/p_j for n = Product p_j^e_j, where for primes p, A276085(p) = A002110(A000720(p)-1).
a(n) = n * Sum e_j * (p_j)#/(p_j^2) for n = Product p_j^e_j with (p_j)# = A034386(p_j).
For all n >= 0, A276150(a(n)) = A328771(n).

A374124 a(n) = A113177(n) mod 360, where A113177 is fully additive with a(p) = Fibonacci(p).

Original entry on oeis.org

0, 1, 2, 2, 5, 3, 13, 3, 4, 6, 89, 4, 233, 14, 7, 4, 157, 5, 221, 7, 15, 90, 217, 5, 10, 234, 6, 15, 149, 8, 229, 5, 91, 158, 18, 6, 17, 222, 235, 8, 301, 16, 77, 91, 9, 218, 73, 6, 26, 11, 159, 235, 293, 7, 94, 16, 223, 150, 161, 9, 161, 230, 17, 6, 238, 92, 293, 159, 219, 19, 289, 7, 73, 18, 12, 223, 102, 236, 301
Offset: 1

Views

Author

Antti Karttunen, Jun 30 2024

Keywords

Links

Crossrefs

Cf. A000045, A113177.
Cf. A373585 (antiparity of terms), A373586 (indices of even terms), A373587 (of odd terms), A374052 (of multiples of 3).
Cf. also A372576, A374123, A374125.

Programs

  • PARI
    A113177(n) = if(n<=1, 0, my(f=factor(n)); sum(i=1, #f~, f[i, 2]*fibonacci(f[i,1])));
A374124(n) = (A113177(n)%360);

A374125 a(n) = A328845(n) mod 360, where A328845 is the first Fibonacci-based variant of the arithmetic derivative.

Original entry on oeis.org

0, 0, 1, 2, 4, 5, 7, 13, 12, 12, 15, 89, 20, 233, 33, 25, 32, 157, 33, 221, 40, 53, 189, 217, 52, 50, 119, 54, 80, 149, 65, 229, 80, 289, 331, 100, 84, 17, 101, 5, 100, 301, 127, 77, 40, 105, 97, 73, 128, 182, 125, 145, 264, 293, 135, 140, 188, 341, 327, 161, 160, 161, 129, 201, 192, 150, 251, 293, 336, 337, 235, 289
Offset: 0

Views

Author

Antti Karttunen, Jun 30 2024

Keywords

Links

Crossrefs

Cf. A000045, A328845.
Cf. A374045 (antiparity of terms), A374046 (indices of even terms), A374047 (of odd terms), A374122 (of multiples of 3).
Cf. also A372576, A374123, A374124.

Programs

  • Mathematica
    A374125[n_] := If[n <= 1, 0, Mod[n*Total[MapApply[#2*Fibonacci[#]/# &, FactorInteger[n]]], 360]];
Array[A374125, 100, 0] (* Paolo Xausa, Dec 16 2024 *)
  • PARI
    A328845(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]*fibonacci(f[i,1])/f[i, 1]));
A374125(n) = (A328845(n)%360);
Showing 1-3 of 3 results.

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