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.

A289838 a(n) = A289815(n) * A289816(n).

Original entry on oeis.org

1, 2, 2, 3, 6, 6, 3, 6, 6, 4, 10, 10, 12, 30, 30, 12, 30, 30, 4, 10, 10, 12, 30, 30, 12, 30, 30, 5, 14, 14, 15, 42, 42, 15, 42, 42, 20, 70, 70, 60, 210, 210, 60, 210, 210, 20, 70, 70, 60, 210, 210, 60, 210, 210, 5, 14, 14, 15, 42, 42, 15, 42, 42, 20, 70, 70
Offset: 0

Views

Author

Rémy Sigrist, Jul 13 2017

Keywords

Comments

Each number k > 0 appears 2^omega(k) times (where omega = A001221).
a(A004488(n)) = a(n) for any n >= 0.
The number of distinct prime factors of a(n) equals the number of nonzero digits in the ternary representation of n.

Examples

			a(42) = A289815(42) * A289816(42) = 20 * 3 = 60.

Links

Crossrefs

Cf. A001221, A004488, A289815, A289816.

Programs

  • PARI
    a(n) = { my (v=1);
       for (o=2, oo,
           if (n==0, return (v));
           if (gcd(v, o)==1 && omega(o)==1,
               if (n % 3, v *= o);
               n \= 3;
           );
       );}
  • Python
    from sympy import gcd, primefactors
def omega(n): return 0 if n==1 else len(primefactors(n))
def a(n):
    v, o = 1, 2
    while True:
        if n==0: return v
        if gcd(v, o)==1 and omega(o)==1:
            if n%3: v*=o
            n //= 3
        o+=1
print([a(n) for n in range(101)]) # Indranil Ghosh, Aug 02 2017

A289905 Square array T(n,k) (n>0, k>0) read by antidiagonals: if gcd(n,k)>1 then T(n,k)=-1, otherwise T(n,k) = the unique m such that A289815(m) = n and A289816(m) = k.

Original entry on oeis.org

0, 1, 2, 3, -1, 6, 9, 5, 7, 18, 27, -1, -1, -1, 54, 4, 11, 15, 21, 19, 8, 81, -1, 33, -1, 57, -1, 162, 243, 29, -1, 45, 63, -1, 55, 486, 729, -1, 87, -1, -1, -1, 165, -1, 1458, 10, 83, 249, 99, 22, 17, 171, 489, 163, 20, 2187, -1, -1, -1, 135, -1, 189, -1, -1
Offset: 1

Views

Author

Rémy Sigrist, Jul 14 2017

Keywords

Comments

This sequence, when restricted to the pairs of coprime numbers, is the inverse of the function n -> (A289815(n), A289816(n)).
If n and k are coprime, then the number of nonzero digits of the ternary representation of T(n,k) equals the number of distinct prime factors of n*k.

Examples

			The table begins:
x\y:    1       2       3       4       5       6       7       8   ...
1:      0       2       6       18      54      8       162     486 ...
2:      1       -1      7       -1      19      -1      55      -1  ...
3:      3       5       -1      21      57      -1      165     489 ...
4:      9       -1      15      -1      63      -1      171     -1  ...
5:      27      11      33      45      -1      17      189     513 ...
6:      4       -1      -1      -1      22      -1      58      -1  ...
7:      81      29      87      99      135     35      -1      567 ...
8:      243     -1      249     -1      297     -1      405     -1  ...
...

Links

Crossrefs

Cf. A289815, A289816.

Programs

  • PARI
    \\ See Links section.

A004488 Tersum n + n.

Original entry on oeis.org

0, 2, 1, 6, 8, 7, 3, 5, 4, 18, 20, 19, 24, 26, 25, 21, 23, 22, 9, 11, 10, 15, 17, 16, 12, 14, 13, 54, 56, 55, 60, 62, 61, 57, 59, 58, 72, 74, 73, 78, 80, 79, 75, 77, 76, 63, 65, 64, 69, 71, 70, 66, 68, 67, 27, 29, 28, 33, 35, 34, 30, 32, 31, 45, 47, 46, 51
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Could also be described as "Write n in base 3, then replace each digit with its base-3 negative" as with A048647 for base 4. - Henry Bottomley, Apr 19 2000
a(a(n)) = n, a self-inverse permutation of the nonnegative integers. - Reinhard Zumkeller, Dec 19 2003
First 3^n terms of the sequence form a permutation s(n) of 0..3^n-1, n>=1; the number of inversions of s(n) is A016142(n-1). - Gheorghe Coserea, Apr 23 2018

Links

Crossrefs

Column k=0 of A253586, A253587.
Column k=3 of A248813.
Row / column 2 of A325820.
Main diagonal of A004489.
Cf. A048647, A055115, A055116, A055120, A059249, A117966, A117967, A117968, A225901, A242399, A244042, A263273, A289813, A289814, A289815, A289816, A289831, A289838, A300222, A321464.

Programs

  • Haskell
    a004488 0 = 0
a004488 n = if d == 0 then 3 * a004488 n' else 3 * a004488 n' + 3 - d
            where (n', d) = divMod n 3
-- Reinhard Zumkeller, Mar 12 2014
  • Maple
    a:= proc(n) local t, r, i;
      t, r:= n, 0;
      for i from 0 while t>0 do
        r:= r+3^i *irem(2*irem(t, 3, 't'), 3)
      od; r
    end:
seq(a(n), n=0..80);  # Alois P. Heinz, Sep 07 2011
  • Mathematica
    a[n_] := FromDigits[Mod[3-IntegerDigits[n, 3], 3], 3]; Table[a[n], {n, 0, 66}] (* Jean-François Alcover, Mar 03 2014 *)
  • PARI
    a(n) = my(b=3); fromdigits(apply(d->(b-d)%b, digits(n, b)), b);
vector(67, i, a(i-1))  \\ Gheorghe Coserea, Apr 23 2018
  • Python
    from sympy.ntheory.factor_ import digits
def a(n): return int("".join([str((3 - i)%3) for i in digits(n, 3)[1:]]), 3) # Indranil Ghosh, Jun 06 2017

Formula

Tersum m + n: write m and n in base 3 and add mod 3 with no carries, e.g., 5 + 8 = "21" + "22" = "10" = 1.
a(n) = Sum(3-d(i)-3*0^d(i): n=Sum(d(i)*3^d(i): 0<=d(i)<3)). - Reinhard Zumkeller, Dec 19 2003
a(3*n) = 3*a(n), a(3*n+1) = 3*a(n)+2, a(3*n+2) = 3*a(n)+1. - Robert Israel, May 09 2014

A289816 The second of a pair of coprime numbers whose factorizations depend on the ternary representation of n (See Comments for precise definition).

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 3, 3, 6, 1, 1, 2, 1, 1, 2, 3, 3, 6, 4, 5, 10, 4, 5, 10, 12, 15, 30, 1, 1, 2, 1, 1, 2, 3, 3, 6, 1, 1, 2, 1, 1, 2, 3, 3, 6, 4, 5, 10, 4, 5, 10, 12, 15, 30, 5, 7, 14, 5, 7, 14, 15, 21, 42, 5, 7, 14, 5, 7, 14, 15, 21, 42, 20, 35, 70, 20, 35, 70
Offset: 0

Views

Author

Rémy Sigrist, Jul 12 2017

Keywords

Comments

For n >= 0, with ternary representation Sum_{i=1..k} t_i * 3^e_i (all t_i in {1, 2} and all e_i distinct and in increasing order):
- let S(0) = A000961 \ { 1 },
- and S(i) = S(i-1) \ { p^(f + j), with p^f = the (e_i+1)-th term of S(i-1) and j > 0 } for any i=1..k,
- then a(n) = Product_{i=1..k such that t_i=2} "the (e_i+1)-th term of S(k)".
See A289815 for the first coprime number and additional comments.
The number of distinct prime factors of a(n) equals the number of twos in the ternary representation of n.

Examples

			For n=42:
- 42 = 2*3^1 + 1*3^2 + 1*3^3,
- S(0) = { 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, ... },
- S(1) = S(0) \ { 3^(1+j) with j > 0 }
       = { 2, 3, 4, 5, 7, 8,    11, 13, 16, 17, 19, 23, 25,     29, ... },
- S(2) = S(1) \ { 2^(2+j) with j > 0 }
       = { 2, 3, 4, 5, 7,       11, 13,     17, 19, 23, 25,     29, ... },
- S(3) = S(2) \ { 5^(1+j) with j > 0 }
       = { 2, 3, 4, 5, 7,       11, 13,     17, 19, 23,         29, ... },
- a(42) = 3.

Links

Crossrefs

Cf. A000961, A004488, A289815.

Programs

  • PARI
    a(n) = my (v=1, x=1);                   \
       for (o=2, oo,                           \
           if (n==0, return (v));              \
           if (gcd(x,o)==1 && omega(o)==1,     \
               if (n % 3,    x *= o);          \
               if (n % 3==2, v *= o);          \
               n \= 3;                         \
           );                                  \
       );
  • Python
    from sympy import gcd, primefactors
def omega(n): return 0 if n==1 else len(primefactors(n))
def a(n):
    v, x, o = 1, 1, 2
    while True:
        if n==0: return v
        if gcd(x, o)==1 and omega(o)==1:
            if n%3: x*=o
            if n%3==2:v*=o
            n //= 3
        o+=1
print([a(n) for n in range(101)]) # Indranil Ghosh, Aug 02 2017

Formula

a(n) = A289815(A004488(n)) for any n >= 0.
a(A005836(n)) = 1 for any n > 0.
a(2 * A005836(n)) = A289272(n-1) for any n > 0.
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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