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

A373147 a(n) = A003415(n) mod A276085(n), where A003415 is the arithmetic derivative and A276085 is the primorial base log-function.

Original entry on oeis.org

0, 1, 0, 1, 2, 1, 0, 2, 0, 1, 0, 1, 9, 0, 0, 1, 1, 1, 0, 10, 13, 1, 4, 10, 15, 3, 0, 1, 4, 1, 0, 14, 19, 12, 0, 1, 21, 16, 5, 1, 8, 1, 48, 9, 25, 1, 4, 14, 6, 20, 56, 1, 4, 16, 26, 22, 31, 1, 2, 1, 33, 17, 0, 18, 61, 1, 72, 26, 22, 1, 2, 1, 39, 13, 80, 18, 71, 1, 6, 4, 43, 1, 22, 22, 45, 32, 140, 1, 2, 20, 96, 34, 49
Offset: 2

Views

Author

Antti Karttunen, May 27 2024

Keywords

Links

Crossrefs

Cf. A002110, A003415, A276085.
Cf. also A373145, A373148, and also A369971.

Programs

  • PARI
    A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A002110(n) = prod(i=1,n,prime(i));
A276085(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*A002110(primepi(f[k, 1])-1)); };
A373147(n) = (A003415(n)%A276085(n));

A369970 Numbers k such that A003415(k) is a multiple of A276086(k), where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

Original entry on oeis.org

0, 1, 6, 2315, 510510
Offset: 1

Views

Author

Antti Karttunen, Feb 07 2024

Keywords

Comments

For the general dynamics of this phenomenon, see the scatter plots of A351231 and A351233.
Question: Are the terms by necessity all squarefree?
As a subsequence this sequence includes all primorials with indices k such that A024451(k) is a multiple of A000040(1+k). See A369972 and A369973.
872415232 < a(6) <= 13082761331670030 [= A369973(4)].

Examples

			2315 is included as A003415(2315) = 5+463 = 468 = 2^2 * 3^2 * 13 (note that 2315 is a semiprime = 5*463, thus its arithmetic derivative is the sum of its two prime factors), and because that 468 is a multiple of A276086(2315) = 234 = 2 * 3^2 * 13 [the exponents of primes are here read from the primorial base expansion of 2315, A049345(2315) = 100021].
510510 is included because A003415(510510) = 19*37693, which is a multiple of A276086(510510) = 19.

Links

Crossrefs

Cf. A000040, A003415, A024451, A276086, A369972, A369973 (subsequence).
Positions of 1's in A351231, positions of 0's in A351233 and in A369971.
After the two initial terms, a subsequence of A351228.
Cf. also A358221.

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); };
isA369970(n) = !(A003415(n)%A276086(n));

A373849 Difference between the primorial base exp-function and the arithmetic derivative.

Original entry on oeis.org

1, 2, 2, 5, 5, 17, 0, 9, 3, 24, 38, 89, 9, 49, 66, 142, 193, 449, 104, 249, 351, 740, 1112, 2249, 581, 1240, 1860, 3723, 5593, 11249, -24, 13, -59, 28, 44, 114, -25, 69, 84, 194, 247, 629, 134, 349, 477, 1011, 1550, 3149, 763, 1736, 2580, 5230, 7819, 15749, 4294, 8734, 13033, 26228, 39344, 78749, -43, 97, 114, 243
Offset: 0

Views

Author

Antti Karttunen, Jun 23 2024

Keywords

Links

Crossrefs

Cf. A003415, A276086, A327858 [= gcd(a(n), A003415(n))], A351228 (indices of nonpositive terms).
Cf. A328382, A369971, A373603.
Cf. A359821 (positions of even terms), A359822 (of odd terms).

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); };
A373849(n) = (A276086(n)-A003415(n));

Formula

a(n) = A276086(n) - A003415(n).
Showing 1-4 of 4 results.

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