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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A328572 Primorial base expansion of n converted into its prime product form, but with 1 subtracted from all nonzero digits: a(n) = A003557(A276086(n)).

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 3, 3, 5, 5, 5, 5, 15, 15, 25, 25, 25, 25, 75, 75, 125, 125, 125, 125, 375, 375, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 3, 3, 5, 5, 5, 5, 15, 15, 25, 25, 25, 25, 75, 75, 125, 125, 125, 125, 375, 375, 7, 7, 7, 7, 21, 21, 7, 7, 7, 7, 21, 21, 35, 35, 35, 35, 105, 105, 175, 175, 175, 175, 525, 525, 875, 875, 875, 875
Offset: 0

Views

Author

Antti Karttunen, Oct 20 2019

Keywords

Links

Crossrefs

Cf. A003557, A051903, A085731, A276086, A327860, A328114, A328475, A328571, A328573, A328575, A328577.

Programs

  • Mathematica
    Block[{b = MixedRadix[Reverse@ Prime@ Range@ 12]}, Array[#1/(Times @@ #2[[All, 1]]) & @@ {#1, FactorInteger[#]} &[Times @@ Power @@@ #] &@ Transpose@ {Prime@ Range@ Length@ #, Reverse@ #} &@ IntegerDigits[#, b] &, 87, 0]] (* Michael De Vlieger, Mar 12 2021 *)
  • PARI
    A328572(n) = { my(m=1, p=2); while(n, if(n%p, m *= p^((n%p)-1)); n = n\p; p = nextprime(1+p)); (m); };

Formula

a(n) = A003557(A276086(n)).
a(n) = A276086(n) / A328571(n).
a(n) = A328475(n) / A328573(n).
For all n >= 1, 1+A051903(a(n)) = A328114(n).
a(n) = A085731(A276086(n)) = gcd(A276086(n), A327860(n)). - Antti Karttunen, Feb 28 2021

A329029 a(n) = A069359(A276086(n)), where A276086 is the primorial base exp-function and A069359(n) = n * Sum_{p|n} 1/p.

Original entry on oeis.org

0, 1, 1, 5, 3, 15, 1, 7, 8, 31, 24, 93, 5, 35, 40, 155, 120, 465, 25, 175, 200, 775, 600, 2325, 125, 875, 1000, 3875, 3000, 11625, 1, 9, 10, 41, 30, 123, 12, 59, 71, 247, 213, 741, 60, 295, 355, 1235, 1065, 3705, 300, 1475, 1775, 6175, 5325, 18525, 1500, 7375, 8875, 30875, 26625, 92625, 7, 63, 70, 287, 210, 861, 84, 413, 497, 1729
Offset: 0

Views

Author

Antti Karttunen, Nov 07 2019

Keywords

Comments

A380535 gives the indices n where a(n) is a multiple of A053669(n). This can be seen from the formula a(n) = A003557(A276086(n)) * A069359(A328571(n)). The left hand side of the product is a multiple of A053669(n) if and only if A276088(n) > 1, while the right hand side is never a multiple of A053669(n), as it is equal to A329031(n) = A003415(A007947(A276086(n))). - Antti Karttunen, Feb 11 2025

Links

Crossrefs

Cf. A003415, A003557, A053669, A069359, A276086, A276088, A328571, A328572, A329031, A380535.
Coincides with A327860 on the positions given by A276156.

Programs

  • PARI
    A329029(n) = { my(s=0, m=1, p=2, e); while(n, e = (n%p); if(e, m *= (p^e); s += (1/p)); n = n\p; p = nextprime(1+p)); (s*m); };
  • PARI
    A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A069359(n) = (n*sumdiv(n, d, isprime(d)/d));
A329029(n) = A069359(A276086(n));

Formula

a(n) = A069359(A276086(n)).
a(n) = A328572(n) * A329031(n) = A003557(A276086(n)) * A069359(A328571(n)). - Antti Karttunen, Feb 11 2025

A328841 Substitute ones for all nonzero digits in primorial base expansion of n, then convert back to decimal.

Original entry on oeis.org

0, 1, 2, 3, 2, 3, 6, 7, 8, 9, 8, 9, 6, 7, 8, 9, 8, 9, 6, 7, 8, 9, 8, 9, 6, 7, 8, 9, 8, 9, 30, 31, 32, 33, 32, 33, 36, 37, 38, 39, 38, 39, 36, 37, 38, 39, 38, 39, 36, 37, 38, 39, 38, 39, 36, 37, 38, 39, 38, 39, 30, 31, 32, 33, 32, 33, 36, 37, 38, 39, 38, 39, 36, 37, 38, 39, 38, 39, 36, 37, 38, 39, 38, 39, 36, 37, 38, 39, 38, 39, 30
Offset: 0

Views

Author

Antti Karttunen, Oct 30 2019

Keywords

Links

Crossrefs

Cf. A002110, A007947, A257993, A267263, A276085, A276086, A328570, A328571, A328620, A328842, A328843.
Cf. A276156 (fixed points).
Cf. A276008 for analogous sequence.

Programs

  • PARI
    A328841(n) = { my(p=2, r=1, s=0); while(n, s += ((!!(n%p))*r); r *= p; n = n\p; p = nextprime(1+p)); (s); };

Formula

a(n) = n - A328842(n).
a(n) = A276085(A328571(n)) = A276085(A007947(A276086(n))).
For all n>= 0, a(A276086(n)) = A328843(n).
For all n >= 1, A257993(a(n)) = A257993(n).
For all n >= 0, A328570(a(n)) = A328570(n), A328620(a(n)) = A328620(n), and A267263(a(n)) = A267263(n).

A329031 a(n) = A327860(A328841(n)).

Original entry on oeis.org

0, 1, 1, 5, 1, 5, 1, 7, 8, 31, 8, 31, 1, 7, 8, 31, 8, 31, 1, 7, 8, 31, 8, 31, 1, 7, 8, 31, 8, 31, 1, 9, 10, 41, 10, 41, 12, 59, 71, 247, 71, 247, 12, 59, 71, 247, 71, 247, 12, 59, 71, 247, 71, 247, 12, 59, 71, 247, 71, 247, 1, 9, 10, 41, 10, 41, 12, 59, 71, 247, 71, 247, 12, 59, 71, 247, 71, 247, 12, 59, 71, 247, 71, 247, 12, 59
Offset: 0

Views

Author

Antti Karttunen, Nov 07 2019

Keywords

Links

Crossrefs

Cf. A003415, A069359, A276086, A327860, A328571, A328841, A329032.
Cf. A060735 (the positions of ones).

Programs

  • PARI
    A329031(n) = { my(s=0, m=1, p=2); while(n, if(n%p, m *= p; s += (1/p)); n = n\p; p = nextprime(1+p)); (s*m); };

Formula

a(n) = A327860(A328841(n)) = A003415(A276086(A328841(n))).
a(n) = A003415(A328571(n)) = A069359(A328571(n)).
Previous Showing 11-14 of 14 results.

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