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 1-4 of 4 results.

A327858 Greatest common divisor of the arithmetic derivative and the primorial base exp-function: a(n) = gcd(A003415(n), A276086(n)).

Original entry on oeis.org

1, 2, 1, 1, 1, 1, 5, 1, 3, 6, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 3, 10, 1, 1, 1, 10, 15, 3, 1, 1, 1, 1, 1, 14, 1, 6, 5, 1, 21, 2, 1, 1, 1, 1, 3, 3, 25, 1, 7, 14, 15, 10, 7, 1, 1, 2, 1, 2, 1, 1, 1, 1, 3, 3, 3, 18, 1, 1, 3, 2, 1, 1, 1, 1, 3, 5, 5, 18, 1, 1, 1, 6, 1, 1, 1, 2, 15, 2, 35, 1, 1, 2, 3, 2, 49, 6, 1, 1, 7, 15, 35, 1, 7, 1, 1, 1
Offset: 0

Views

Author

Antti Karttunen, Sep 30 2019

Keywords

Comments

Sequence contains only terms of A048103.
Proof that A046337 gives the positions of even terms: see Charlie Neder's Feb 25 2019 comment in A235992 and recall that A276086 is never a multiple of 4, as it is a permutation of A048103, and furthermore it toggles the parity. See also comment in A327860. - Antti Karttunen, May 01 2022

Links

Crossrefs

Cf. A003415, A048103, A235992, A276086, A327859, A328382, A351234, A354348, A356299, A358669, A359423, A359589 (Dirichlet inverse of a(n)-1), A369971, A373849.
Cf. A046337 (positions of even terms), A356311 (positions of 1's), A356310 (their characteristic function).
Cf. also A085731, A324198, A328572 [= gcd(A276086(n), A327860(n))], A345000, A373145, A373843.

Programs

  • Mathematica
    Block[{b = MixedRadix[Reverse@ Prime@ Range@ 12], f, g}, f[n_] := If[Abs@ n < 2, 0, n Total[#2/#1 & @@@ FactorInteger[Abs@ n]]]; g[n_] := Times @@ Power @@@ # &@ Transpose@ {Prime@ Range@ Length@ #, Reverse@ #} &@ IntegerDigits[n, b]; Array[GCD[f@ #, g@ #] &, 105]] (* Michael De Vlieger, Sep 30 2019 *)
  • PARI
    A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A327858(n) = gcd(A003415(n),A276086(n));

Formula

a(n) = gcd(A003415(n), A276086(n)).
a(p) = 1 for all primes p.
a(n) = A276086(A351234(n)). - Antti Karttunen, May 01 2022
From Antti Karttunen, Dec 05 2022: (Start)
For n >= 2, a(n) = gcd(A003415(n), A328382(n)).
(End)
For n >= 2, a(n) = A358669(n) / A359423(n). For n >= 1, A356299(n) | a(n). - Antti Karttunen, Jan 09 2023
a(n) = gcd(A003415(n), A373849(n)) = gcd(A276086(n), A369971(n)) = A373843(A276086(n)). - Antti Karttunen, Jun 21 & 23 2024

Extensions

Verbal description added to the definition by Antti Karttunen, May 01 2022

A327864 Numbers whose arithmetic derivative is a multiple of 4, cf. A003415.

Original entry on oeis.org

0, 1, 4, 8, 12, 15, 16, 20, 24, 28, 32, 35, 36, 39, 40, 44, 48, 51, 52, 55, 56, 60, 64, 68, 72, 76, 80, 81, 84, 87, 88, 91, 92, 95, 96, 100, 104, 108, 111, 112, 115, 116, 119, 120, 123, 124, 128, 132, 136, 140, 143, 144, 148, 152, 155, 156, 159, 160, 164, 168, 172, 176, 180, 183, 184, 187, 188, 189, 192, 196, 200, 203
Offset: 1

Views

Author

Antti Karttunen, Sep 30 2019

Keywords

Comments

Also k such that A359423(k) is a multiple of 4. - Antti Karttunen, Jan 02 2023
A multiplicative semigroup; if m and n are in the sequence then so is m*n. - Antti Karttunen, Feb 01 2023

Links

Crossrefs

Subsequence of A235992.
Union of A008586 and A360110.
Cf. A003415, A327862, A327863, A327865, A353493, A353494 (characteristic function), A359423.

Programs

  • PARI
    A003415(n) = {my(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1]))}; \\ From A003415
isA327864(n) = !(A003415(n)%4);
k=1; n=0; while(k<105, if(isA327864(n), print1(n, ", "); k++); n++);

Formula

{k | A353493(k)=0}. - Antti Karttunen, Jan 02 2023

A358669 Pointwise product of the arithmetic derivative and the primorial base exp-function.

Original entry on oeis.org

0, 0, 3, 6, 36, 18, 25, 10, 180, 180, 315, 90, 400, 50, 675, 1200, 7200, 450, 2625, 250, 9000, 7500, 14625, 2250, 27500, 12500, 28125, 101250, 180000, 11250, 217, 14, 1680, 588, 1197, 1512, 2100, 70, 2205, 3360, 21420, 630, 7175, 350, 25200, 40950, 39375, 3150, 98000, 24500, 118125, 105000, 441000
Offset: 0

Views

Author

Antti Karttunen, Dec 05 2022

Keywords

Links

Crossrefs

Cf. A003415, A059841, A121262, A152822, A276086, A327858, A353558, A358680, A358765 (= a(n) mod 60), A359423, A359603 [Dirichlet inverse of 1+a(n)].
Cf. A016825 (positions of odd terms), A042965 (of even terms), A235992 (of multiples of 4), A067019 (of terms of the form 4k+2), A358748 (of the form 4k+1), A358749 (of the form 4k+3).

Programs

  • PARI
    A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A358669(n) = (A003415(n)*A276086(n));

Formula

a(n) = A003415(n) * A276086(n).
From Antti Karttunen, Jan 09 2023: (Start)
a(n) = A327858(n) * A359423(n).
For all n >= 0, A059841(a(n)) = A152822(n).
For all n >= 1, 1-A152822(a(n)) = A353558(n).
For all n >= 0, A121262(a(n)) = A358680(n).
(End)

A359424 The least common multiple of the arithmetic derivative and the primorial base exp-function, reduced modulo 60.

Original entry on oeis.org

0, 0, 3, 6, 36, 18, 5, 10, 0, 30, 15, 30, 40, 50, 45, 0, 0, 30, 45, 10, 0, 30, 45, 30, 20, 50, 15, 30, 0, 30, 37, 14, 0, 42, 57, 12, 0, 10, 45, 0, 0, 30, 35, 50, 0, 30, 15, 30, 20, 10, 15, 0, 0, 30, 15, 40, 0, 30, 45, 30, 8, 38, 57, 18, 24, 42, 5, 10, 0, 30, 15, 30, 0, 50, 15, 30, 0, 30, 55, 10, 0, 0, 15
Offset: 0

Views

Author

Antti Karttunen, Jan 02 2023

Keywords

Links

Crossrefs

Cf. A003415, A276086, A327858, A358669, A359423.
Cf. A016825 (positions of odd terms), A042965 (of even terms), A327864 (of multiples of 4).
Cf. also A358765.

Programs

  • PARI
    A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A359423(n) = lcm(A003415(n), A276086(n));
A359424(n) = (A359423(n)%60);
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.