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 1-3 of 3 results.

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

Views

Author

Antti Karttunen, Dec 05 2021

Keywords

Links

Crossrefs

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.

Programs

  • Mathematica
    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 *)
  • PARI
    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));

Formula

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

A347965 Dirichlet convolution of A003415 (arithmetic derivative) with A003961 (prime shift towards larger primes).

Original entry on oeis.org

0, 1, 1, 7, 1, 13, 1, 33, 11, 17, 1, 75, 1, 23, 20, 131, 1, 104, 1, 103, 26, 29, 1, 329, 17, 35, 82, 145, 1, 196, 1, 473, 32, 41, 30, 552, 1, 47, 38, 461, 1, 274, 1, 187, 181, 57, 1, 1259, 25, 194, 44, 229, 1, 682, 36, 659, 50, 65, 1, 1052, 1, 73, 247, 1611, 42, 352, 1, 271, 60, 366, 1, 2332, 1, 83, 245, 313, 42
Offset: 1

Views

Author

Antti Karttunen, Sep 25 2021

Keywords

Links

Crossrefs

Cf. A003415, A003961.
Cf. also A347130, A347964.

Programs

  • PARI
    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); };
A347965(n) = sumdiv(n,d,A003415(n/d)*A003961(d));

Formula

a(n) = Sum_{d|n} A003415(d) * A003961(n/d).

A348937 a(n) = A003961(n) - A003415(n), where A003961 shifts the prime factorization of n one step towards larger primes, and A003415 gives the arithmetic derivative of n.

Original entry on oeis.org

1, 2, 4, 5, 6, 10, 10, 15, 19, 14, 12, 29, 16, 24, 27, 49, 18, 54, 22, 39, 45, 26, 28, 91, 39, 36, 98, 67, 30, 74, 36, 163, 51, 38, 65, 165, 40, 48, 69, 121, 42, 124, 46, 69, 136, 62, 52, 293, 107, 102, 75, 97, 58, 294, 75, 205, 93, 62, 60, 223, 66, 78, 224, 537, 101, 134, 70, 99, 119, 172, 72, 519, 78, 84, 190, 127
Offset: 1

Views

Author

Antti Karttunen, Nov 06 2021

Keywords

Links

Crossrefs

Cf. A003415, A003961, A168036, A286385, A336853, A343224, A347964, A347965.

Programs

  • Mathematica
    f1[p_, e_] := e/p; f2[p_, e_] := NextPrime[p]^e; a[n_] := Times @@ f2 @@@ (f = FactorInteger[n]) - n * Plus @@ f1 @@@ f; Array[a, 100] (* Amiram Eldar, Nov 06 2021 *)
  • PARI
    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); };
A348937(n) = (A003961(n) - A003415(n));

Formula

a(n) = A003961(n) - A003415(n).
a(n) = A336853(n) - A168036(n).
a(n) = A286385(n) + A343224(n).
Showing 1-3 of 3 results.

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