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.

A324582 a(n) = A002182(n) * A324581(n) = A002182(n) * A276086(A002182(n)).

Original entry on oeis.org

2, 6, 36, 30, 300, 15000, 1260, 42000, 2940, 288120, 21176820, 18480, 66555720, 328703760, 12298440, 2232166860, 360122920080, 360360, 103062960, 22107004920, 4215068938080, 129290917072196880, 3525159950945805332160, 90107494796113466546674800, 645822919595173320, 72532204477502449680, 1648012277067163992784800
Offset: 1

Views

Author

Antti Karttunen, Mar 09 2019

Keywords

Comments

Note that gcd(A002182(n), A324581(n)) = A324198(A002182(n)) = 1 for all n because each term of A002182 is a product of primorial numbers (A002110).
See also comments in A324382.

Links

Crossrefs

Subsequence of A324577.
Cf. A002110, A002182, A276086, A324198, A324382, A324580, A324581.

Programs

  • Mathematica
    Block[{b = MixedRadix[Reverse@ Prime@ Range@ 20], s = DivisorSigma[0, Range[10^5]], t}, t = Map[FirstPosition[s, #][[1]] &, Union@ FoldList[Max, s]]; Array[#1 (Times @@ Power @@@ Transpose@ {Prime@ Range@ Length@ #2, Reverse@ #2}) & @@ {#, IntegerDigits[#, b]} &@ t[[#]] &, Length@ t]] (* Michael De Vlieger, Mar 18 2019 *)
  • PARI
    \\ A002182 assumed to be precomputed
A276086(n) = { my(i=0,m=1,pr=1,nextpr); while((n>0),i=i+1; nextpr = prime(i)*pr; if((n%nextpr),m*=(prime(i)^((n%nextpr)/pr));n-=(n%nextpr));pr=nextpr); m; };
A324582(n) = A002182(n)*A276086(A002182(n));

Formula

a(n) = A002182(n) * A324581(n) = A002182(n) * A276086(A002182(n)).
a(n) = A324580(A002182(n)).

A329902 Primorial deflation of the n-th highly composite number: the unique integer k such that A108951(k) = A002182(n).

Original entry on oeis.org

1, 2, 4, 3, 6, 12, 9, 24, 10, 20, 15, 40, 30, 60, 28, 21, 56, 42, 84, 63, 168, 126, 336, 140, 66, 189, 280, 132, 99, 264, 198, 528, 220, 396, 297, 440, 792, 156, 117, 312, 234, 624, 260, 468, 351, 520, 936, 390, 1040, 1872, 780, 585, 306, 1560, 340, 612, 459, 680, 1224, 510, 1360, 2448, 1020, 765, 342, 2040, 1530, 684, 513
Offset: 1

Views

Author

Antti Karttunen, Dec 22 2019

Keywords

Links

Crossrefs

Cf. A002182, A002183, A056239, A108951, A112778, A112779, A306802, A319626, A324381, A324382, A324581, A324582, A324888, A329040, A329605, A329900, A330743, A330748.
Subsequence of A181815.

Programs

  • Mathematica
    Map[Times @@ Prime@(TakeWhile[Reap[FixedPointList[Block[{k = 1}, While[Mod[#, Prime@ k] == 0, k++]; Sow[k - 1]; #/Product[Prime@ i, {i, k - 1}]] &, #]][[-1, 1]], # > 0 &]) &, Take[Import["https://oeis.org/b002182.txt", "Data"][[All, -1]], 69] ] (* Michael De Vlieger, Jan 13 2020, imports b-file at A002182 *)

Formula

a(n) = A329900(A002182(n)) = A319626(A002182(n)).
a(n) = A181815(A306802(n)).
A108951(a(n)) = A002182(n). [Highly composite numbers (undeflated)]
A056239(a(n)) = A112778(n). [Number of prime factors, counted with multiplicity]
A001222(a(n)) = A112779(n). [Largest exponent in the prime factorization]
A329605(a(n)) = A002183(n). [Number of divisors]
A329040(a(n)) = A324381(n).
A324888(a(n)) = A324382(n).
a(A330748(n)) = A330743(n).

Extensions

More linking formulas added by Antti Karttunen, Jan 13 2020

A324576 a(n) = A276086(A025487(n)).

Original entry on oeis.org

2, 3, 9, 5, 15, 25, 225, 625, 7, 21, 35, 875, 49, 441, 1225, 1715, 2401, 36015, 1500625, 117649, 2941225, 11, 55, 77, 17325, 67375, 184877, 115548125, 121, 3025, 5929, 124509, 10168235, 456533, 399466375, 14641, 9150625, 717409, 35153041, 15502491081, 1127357, 1381012325, 1771561, 62004635, 208422380089, 4774356895, 214358881
Offset: 1

Views

Author

Antti Karttunen, Mar 09 2019

Keywords

Links

Crossrefs

Cf. A025487, A276086, A324387, A324577.
Cf. A324581 (a subsequence).

Programs

  • PARI
    A276086(n) = { my(i=0,m=1,pr=1,nextpr); while((n>0),i=i+1; nextpr = prime(i)*pr; if((n%nextpr),m*=(prime(i)^((n%nextpr)/pr));n-=(n%nextpr));pr=nextpr); m; };
A324576(n) = A276086(A025487(n));

Formula

a(n) = A276086(A025487(n)).
A001222(a(n)) = A324387(n).
Showing 1-3 of 3 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.