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.

A351231 Denominator of A003415(n) / A276086(n), where A003415 is the arithmetic derivative and A276086 is the primorial base exp-function.

Original entry on oeis.org

1, 1, 3, 6, 9, 18, 1, 10, 5, 5, 45, 90, 25, 50, 25, 75, 225, 450, 125, 250, 125, 75, 1125, 2250, 625, 125, 125, 1250, 5625, 11250, 7, 14, 21, 3, 63, 21, 7, 70, 5, 105, 315, 630, 175, 350, 175, 350, 63, 3150, 125, 125, 175, 525, 1125, 15750, 4375, 4375, 13125, 13125, 39375, 78750, 49, 98, 49, 98, 147, 49, 245, 490, 245
Offset: 0

Views

Author

Antti Karttunen, Feb 05 2022

Keywords

Links

Crossrefs

Cf. A003415, A276086, A327858, A351230 (numerators), A351232, A351233.

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); };
A351231(n) = denominator(A003415(n) / A276086(n));

Formula

a(n) = A276086(n) / A327858(n) = A276086(n) / gcd(A003415(n), A276086(n)).
a(n) = A276086(A351233(n)).

A351250 Numerator of n / A276086(n).

Original entry on oeis.org

0, 1, 2, 1, 4, 5, 6, 7, 8, 3, 2, 11, 12, 13, 14, 1, 16, 17, 18, 19, 4, 7, 22, 23, 24, 1, 26, 9, 28, 29, 30, 31, 32, 11, 34, 5, 36, 37, 38, 13, 8, 41, 6, 43, 44, 3, 46, 47, 48, 7, 2, 17, 52, 53, 54, 11, 8, 19, 58, 59, 60, 61, 62, 3, 64, 65, 66, 67, 68, 23, 2, 71, 72, 73, 74, 1, 76, 11, 78, 79, 16, 27, 82, 83, 12, 17
Offset: 0

Views

Author

Antti Karttunen, Feb 05 2022

Keywords

Links

Crossrefs

Cf. A276086, A324198, A328386, A328387 (positions of ones), A351251 (denominators).
Cf. A324583 (the positions of fixed points after the zero).
Cf. also A351230.

Programs

  • Mathematica
    Array[Block[{i, m, n = #, p}, m = i = 1; While[n > 0, p = Prime[i]; m *= p^Mod[n, p]; n = Quotient[n, p]; i++]; Numerator[#/m]] &, 86, 0] (* Michael De Vlieger, Feb 06 2022 *)
  • PARI
    A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A351250(n) = numerator(n/A276086(n));

Formula

a(n) = n / A324198(n) = n / gcd(n, A276086(n)).
a(n) = n / gcd(n, A328386(n)).

A351232 a(n) = floor(A276086(n) / A003415(n)), where A003415 is the arithmetic derivative and A276086 is the primorial base exp-function.

Original entry on oeis.org

3, 6, 2, 18, 1, 10, 1, 5, 6, 90, 1, 50, 8, 18, 7, 450, 5, 250, 15, 75, 86, 2250, 14, 125, 125, 138, 175, 11250, 0, 14, 0, 3, 3, 10, 0, 70, 5, 13, 4, 630, 4, 350, 10, 26, 63, 3150, 7, 125, 58, 262, 140, 15750, 54, 546, 142, 1193, 1270, 78750, 0, 98, 4, 5, 2, 49, 4, 490, 10, 56, 37, 4410, 7, 2450, 94, 133, 137, 1225
Offset: 2

Views

Author

Antti Karttunen, Feb 05 2022

Keywords

Links

Crossrefs

Cf. A003415, A276086, A351230, A351231.
Cf. A351228 (conjectured to give the positions of zeros from its second term onward).

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); };
A351232(n) = floor(A276086(n) / A003415(n));

Formula

a(n) = floor(A276086(n) / A003415(n)) = floor(A351231(n) / A351230(n)).

A369038 Numerator of ratio A003415(n) / A003415(A276086(n)), where A003415 is the arithmetic derivative and A276086 is the primorial base exp-function.

Original entry on oeis.org

0, 1, 1, 2, 1, 5, 1, 3, 6, 7, 1, 8, 1, 9, 8, 2, 1, 7, 1, 12, 2, 13, 1, 11, 2, 3, 27, 16, 1, 31, 1, 8, 14, 19, 4, 5, 1, 21, 16, 34, 1, 41, 1, 12, 39, 5, 1, 56, 14, 9, 4, 14, 1, 27, 16, 46, 22, 31, 1, 46, 1, 33, 51, 16, 6, 61, 1, 36, 26, 59, 1, 13, 1, 39, 1, 8, 6, 71, 1, 11, 108, 43, 1, 62, 22, 9, 32, 1, 1, 41, 20
Offset: 1

Views

Author

Antti Karttunen, Jan 20 2024

Keywords

Links

Crossrefs

Cf. A003415, A276086, A327860, A345000, A369039 (denominators).
Cf. also A351230, A351250.

Programs

  • PARI
    A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A327860(n) = { my(s=0, m=1, p=2, e); while(n, e = (n%p); m *= (p^e); s += (e/p); n = n\p; p = nextprime(1+p)); (s*m); };
A369038(n) = { my(u=A003415(n)); (u/gcd(u,A327860(n))); };

Formula

a(n) = A003415(n) / A345000(n) = A003415(n) / gcd(A003415(n), A327860(n)).
Showing 1-4 of 4 results.

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