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 1-5 of 5 results.

A324198 a(n) = gcd(n, A276086(n)), where A276086 is the primorial base exp-function.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 5, 1, 1, 1, 1, 15, 1, 1, 1, 1, 5, 3, 1, 1, 1, 25, 1, 3, 1, 1, 1, 1, 1, 3, 1, 7, 1, 1, 1, 3, 5, 1, 7, 1, 1, 15, 1, 1, 1, 7, 25, 3, 1, 1, 1, 5, 7, 3, 1, 1, 1, 1, 1, 21, 1, 1, 1, 1, 1, 3, 35, 1, 1, 1, 1, 75, 1, 7, 1, 1, 5, 3, 1, 1, 7, 5, 1, 3, 1, 1, 1, 7, 1, 3, 1, 1, 1, 1, 49, 3, 5, 1, 1, 1, 1, 105
Offset: 0

Views

Author

Antti Karttunen, Feb 25 2019

Keywords

Links

Crossrefs

Cf. A003989, A276086, A323878, A324284, A324350, A324351, A324384, A324577, A324580, A324646, A327858, A328323, A327168, A328316, A328323, A328386, A328584, A346242 (Dirichlet inverse), A351254.
Cf. A324583 (positions of ones), A324584 (and terms larger than one).
Cf. A371098 (odd bisection), A371099 [= a(36n+9)].
Cf. also A328231.

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++]; GCD[#, m]] &, 106, 0] (* Michael De Vlieger, Feb 04 2022 *)
  • 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; };
A324198(n) = gcd(n,A276086(n));
  • PARI
    A324198(n) = { my(m=1, p=2, orgn=n); while(n, m *= (p^min(n%p,valuation(orgn,p))); n = n\p; p = nextprime(1+p)); (m); }; \\ Antti Karttunen, Oct 21 2019

Formula

a(n) = gcd(n, A276086(n)).
From Antti Karttunen, Oct 21 2019: (Start)
A000005(a(n)) = A327168(n).
a(A328316(n)) = A328323(n).
a(n) = A324580(n) / A328584(n).
(End)

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

Original entry on oeis.org

0, 2, 6, 18, 36, 90, 30, 70, 120, 270, 450, 990, 300, 650, 1050, 2250, 3600, 7650, 2250, 4750, 7500, 15750, 24750, 51750, 15000, 31250, 48750, 101250, 157500, 326250, 210, 434, 672, 1386, 2142, 4410, 1260, 2590, 3990, 8190, 12600, 25830, 7350, 15050, 23100, 47250, 72450, 148050, 42000, 85750, 131250, 267750, 409500
Offset: 0

Views

Author

Antti Karttunen, Mar 09 2019

Keywords

Links

Crossrefs

Cf. A002110, A276086, A324198, A324577, A324578, A324579, A324582.

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; };
A324580(n) = n*A276086(n);

Formula

a(n) = n * A276086(n).
For n >= 0, a(A002110(n)) = A002110(1+n).

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)).

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).

A324887 a(n) = A108951(n) * A276086(A108951(n)).

Original entry on oeis.org

2, 6, 30, 36, 210, 300, 2310, 120, 1260, 2940, 30030, 15000, 510510, 50820, 21176820, 3600, 9699690, 88200, 223092870, 288120, 2232166860, 780780, 6469693230, 42000, 645668100, 17357340, 11880, 12298440, 200560490130, 66555720, 7420738134810, 672, 66899572740, 368588220, 228227900600700, 216090000, 304250263527210
Offset: 1

Views

Author

Antti Karttunen, Mar 30 2019

Keywords

Links

Crossrefs

Cf. A034386, A108951, A276086, A324580, A324886, A324888.
Permutation of A324577.

Programs

Formula

a(n) = A324580(A108951(n)) = A108951(n) * A324886(n).
Showing 1-5 of 5 results.

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