A381579 -id:A381579 - cp's OEIS frontend

A381581 Numbers divisible by the sum of the digits in their Chung-Graham representation (A381579).

1, 2, 3, 4, 6, 8, 12, 16, 20, 21, 22, 24, 27, 28, 30, 40, 42, 44, 45, 48, 55, 56, 57, 58, 60, 66, 70, 72, 75, 76, 80, 84, 90, 92, 95, 96, 100, 102, 110, 111, 112, 115, 116, 120, 132, 135, 138, 140, 144, 150, 152, 153, 156, 168, 170, 175, 176, 180, 186, 190, 195, 198

Offset: 1

Amiram Eldar, Feb 28 2025

Numbers k such that A291711(k) divides k.
Analogous to Niven numbers (A005349) with the Chung-Graham representation (A381579) instead of the decimal representation.
A001906(k) = Fibonacci(2*k) is a term for all k >= 1.
If k is not divisible by 3 (A001651), then Fibonacci(2*k) + 1 is a term.

			4 is a term since A291711(4) = 1 divides 4.
6 is a term since A291711(6) = 2 divides 6.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000045, A001651, A001906, A291711.
Subsequences: A381582, A381583, A381584, A381585.
Similar sequences: A005349, A049445, A064150, A328208, A328212.

f[n_] := f[n] = Fibonacci[2*n]; q[n_] := Module[{s = 0, m = n, k}, While[m > 0, k = 1; While[m > f[k], k++]; If[m < f[k], k--]; If[m >= 2*f[k], s += 2; m -= 2*f[k], s++; m -= f[k]]]; Divisible[n, s]]; Select[Range[200], q]

mx = 20; fvec = vector(mx, i, fibonacci(2*i)); f(n) = if(n <= mx, fvec[n], fibonacci(2*n));
isok(n) = {my(s = 0, m = n, k); while(m > 0, k = 1; while(m > f(k), k++); if(m < f(k), k--); if(m >= 2*f(k), s += 2; m -= 2*f(k), s++; m -= f(k))); !(n % s);}

A381580 Numbers whose Chung-Graham representation (A381579) is palindromic.

Original entry on oeis.org

0, 1, 2, 4, 9, 12, 15, 18, 22, 33, 44, 56, 64, 72, 80, 88, 96, 104, 112, 120, 128, 136, 145, 174, 203, 232, 261, 290, 319, 348, 378, 399, 420, 441, 462, 483, 504, 525, 546, 567, 588, 609, 630, 651, 672, 693, 714, 735, 756, 777, 798, 819, 840, 861, 882, 903, 924, 945, 966, 988

Offset: 1

Amiram Eldar, Feb 28 2025

The numbers of the form Fibonacci(2*k) + 1 (A055588) are all terms since A381579(A055588(0)) = 1, A381579(A055588(1)) = 2, and A381579(A055588(k)) = 10^(k-1)+1 (i.e., two 1's with k-2 0's between them) for k >= 2.

			The first 10 terms are:
   n  a(n) A381579(a(n))
   ---------------------
   1   0               0
   2   1               1
   3   2               2
   4   4              11
   5   9             101
   6  12             111
   7  15             121
   8  18             202
   9  22            1001
  10  33            1111

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000045, A381579.
Subsequence: A055588.
Similar sequences: A002113, A006995, A094202, A331191.

f[n_] := f[n] = Fibonacci[2*n]; q[n_] := Module[{s = 0, m = n, k}, While[m > 0, k = 1; While[m > f[k], k++]; If[m < f[k], k--]; If[m >= 2*f[k], s += 2*10^(k-1); m -= 2*f[k], s += 10^(k-1); m -= f[k]]]; PalindromeQ[s]]; Select[Range[0, 1000], q]

mx = 20; fvec = vector(mx, i, fibonacci(2*i)); f(n) = if(n <= mx, fvec[n], fibonacci(2*n));
isok(n) = {my(s = 0, m = n, k, d); while(m > 0, k = 1; while(m > f(k), k++); if(m < f(k), k--); if(m >= 2*f(k), s += 2*10^(k-1); m -= 2*f(k), s += 10^(k-1); m -= f(k))); d = digits(s); Vecrev(d) == d;}

A291711 The minimum number of coins needed to pay for n units in the currency system of values 1, 3, 8, 21, 55, 144, ..., Fibonacci(2k), ...

Original entry on oeis.org

1, 2, 1, 2, 3, 2, 3, 1, 2, 3, 2, 3, 4, 3, 4, 2, 3, 4, 3, 4, 1, 2, 3, 2, 3, 4, 3, 4, 2, 3, 4, 3, 4, 5, 4, 5, 3, 4, 5, 4, 5, 2, 3, 4, 3, 4, 5, 4, 5, 3, 4, 5, 4, 5, 1, 2, 3, 2, 3, 4, 3, 4, 2, 3, 4, 3, 4, 5, 4, 5, 3, 4, 5, 4, 5, 2, 3, 4, 3, 4, 5, 4, 5, 3, 4, 5, 4, 5, 6, 5, 6, 4, 5, 6

Offset: 1

Yuriko Suwa, Aug 30 2017

nonn

It has been proved that there is a unique way to pay any price n with a(n) coins having values of the form Fibonacci(2k).

			a(7) = 3 because 7 = 3 + 3 + 1 is a minimal sum using 3 coins.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A007895 (for Fibonacci(k)), A245588 (for Fibonacci(2k-1)), A007953, A381579.

x1:=1: x2:=3: L:=[x1,x2]: nn:=12: LS:=[]: for k from 1 to nn-2 do:z:=3*x2-x1: L:=[op(L),z]: x1:=x2: x2:=z: od:
for n from 1 to 200 do: m:=n: ct:=0: for s from 1 to nn while m>0 do:
for j from 1 to nn-1 do:if m n then
            return i-1 ;
        end if;
    end do:
end proc:
A291711 := proc(n)
    local fibm,gf,e,gfe ;
    fibm := fibIdx(n) ;
    gf := add( x^combinat[fibonacci](2*m),m=1..fibm/2) ;
    gfe := gf ;
    for e from 1 do
        expand(coeftayl(gfe,x=0,n)) ;
        if % > 0 then
            return e ;
        end if;
        gfe := expand(gfe*gf) ;
    end do:
end proc:
seq(A291711(n),n=1..100) ; # R. J. Mathar, Nov 11 2017

f[n_] := f[n] = Fibonacci[2*n]; a[n_] := Module[{s = 0, m = n, k}, While[m > 0, k = 1; While[m > f[k], k++]; If[m < f[k], k--]; If[m >= 2*f[k], s += 2; m -= 2*f[k], s++; m -= f[k]]]; s]; Array[a, 100] (* Amiram Eldar, Feb 28 2025 *)

mx = 20; fvec = vector(mx, i, fibonacci(2*i)); f(n) = if(n <= mx, fvec[n], fibonacci(2*n));
a(n) = {my(s = 0, m = n, k); while(m > 0, k = 1; while(m > f(k), k++); if(m < f(k), k--); if(m >= 2*f(k), s += 2; m -= 2*f(k), s++; m -= f(k))); s;} \\ Amiram Eldar, Feb 28 2025

a(n) = A007953(A381579(n)). - Amiram Eldar, Feb 28 2025

A381607 For any nonnegative integer n with ternary expansion Sum_{k >= 0} t_k * 3^k, a(n) = Sum_{k >= 0} t_k * A000045(2*k + 2).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 8, 9, 10, 11, 12, 13, 14, 15, 16, 16, 17, 18, 19, 20, 21, 22, 23, 24, 21, 22, 23, 24, 25, 26, 27, 28, 29, 29, 30, 31, 32, 33, 34, 35, 36, 37, 37, 38, 39, 40, 41, 42, 43, 44, 45, 42, 43, 44, 45, 46, 47, 48, 49, 50, 50, 51, 52, 53, 54

Offset: 0

Rémy Sigrist, Mar 01 2025

Every nonnegative integer appears in the sequence, a finite number of times.

			42 = 3^3 + 3^2 + 2*3^1, so a(42) = A000045(8) + A000045(6) + 2*A000045(4) = 21 + 8 + 2*3 = 35.

Rémy Sigrist, Table of n, a(n) for n = 0..6560

See A022290 for a similar sequence.

Cf. A000045, A028898, A381579.

a(n) = { my (t = Vecrev(digits(n, 3))); sum(k = 1, #t, t[k] * fibonacci(2*k)); }

a(A028898(A381579(n))) = n.

A381608 Nonnegative integers whose ternary expansion does not contain pairs of 2's only separated by (zero or more) 1's, with offset 0.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22, 27, 28, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 54, 55, 56, 57, 58, 59, 60, 61, 63, 64, 65, 66, 67, 81, 82, 83, 84, 85, 86, 87, 88, 90, 91, 92, 93

Offset: 0

Rémy Sigrist, Mar 01 2025

The ternary expansion of a(n) equals the decimal expansion of A381579(n).

			The ternary expansion of 20, "211", has no pairs of 2's, so 20 belongs to the sequence.The ternary expansion of 21, "212", has a pair of 2s only separated by 1's, so 21 does not belong to the sequence.

Rémy Sigrist, Table of n, a(n) for n = 0..6560

Cf. A028898, A381579, A381607.

a(n) = { for (k = 1, oo, my (f = fibonacci(2*k)); if (f >= n, my (v = 0); while (n, while (n >= f, n -= f; v += 3^(k-1);); f = fibonacci(2*k--);); return (v););); }

a(n) = A028898(A381579(n)).

A381607(a(n)) = n.

A381609 a(n) is the number of occurrences of n in A381607.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 2, 3, 2, 2, 2, 2, 3, 2, 2, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 2, 1, 1

Offset: 0

Rémy Sigrist, Mar 01 2025

All terms are positive (see A381579).

			The number 8 appears twice in A381607, so a(8) = 2.

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Rémy Sigrist, PARI program

Cf. A381579, A381607.

PARI
```
\\ See Links section.
```

a(n) > 0.

A381618 Reverse the Chung-Graham representation of n while preserving its trailing zeros: a(n) = A381607(A263273(A381608(n))).

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 6, 5, 8, 9, 17, 11, 12, 20, 19, 15, 16, 10, 18, 14, 13, 21, 22, 43, 24, 30, 51, 45, 38, 29, 25, 46, 32, 33, 54, 53, 41, 50, 28, 49, 40, 36, 42, 23, 44, 27, 31, 52, 48, 39, 37, 26, 47, 35, 34, 55, 56, 111, 58, 77, 132, 113, 98, 63, 64, 119, 79

Offset: 0

Rémy Sigrist, Mar 02 2025

This sequence is a self-inverse permutation of the nonnegative integers.

			The first terms, alongside their Chung-Graham representation, are:
  n   a(n)  A381579(n)  A381579(a(n))
  --  ----  ----------  -------------
   0     0           0              0
   1     1           1              1
   2     2           2              2
   3     3          10             10
   4     4          11             11
   5     7          12             21
   6     6          20             20
   7     5          21             12
   8     8         100            100
   9     9         101            101
  10    17         102            201
  11    11         110            110
  12    12         111            111
  13    20         112            211
  14    19         120            210
  15    15         121            121
  16    16         200            200

See A345201 for a similar sequence.

Cf. A000045, A263273, A381579, A381607, A381608.

A381607(n) = { my (t = Vecrev(digits(n, 3))); sum(k = 1, #t, t[k] * fibonacci(2*k)); }
A263273(n) = { my (t = 3^if (n, valuation(n, 3), 0)); t * fromdigits(Vecrev(digits(n / t, 3)), 3) }
A381608(n) = { for (k = 1, oo, my (f = fibonacci(2*k)); if (f >= n, my (v = 0); while (n, while (n >= f, n -= f; v += 3^(k-1);); f = fibonacci(2*k--);); return (v););); }
a(n) = A381607(A263273(A381608(n)))

a(n) <= A000045(2*k) iff n <= A000045(2*k).

cp's OEIS Frontend

A381581 Numbers divisible by the sum of the digits in their Chung-Graham representation (A381579).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A381580 Numbers whose Chung-Graham representation (A381579) is palindromic.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A291711 The minimum number of coins needed to pay for n units in the currency system of values 1, 3, 8, 21, 55, 144, ..., Fibonacci(2k), ...

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A381607 For any nonnegative integer n with ternary expansion Sum_{k >= 0} t_k * 3^k, a(n) = Sum_{k >= 0} t_k * A000045(2*k + 2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A381608 Nonnegative integers whose ternary expansion does not contain pairs of 2's only separated by (zero or more) 1's, with offset 0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A381609 a(n) is the number of occurrences of n in A381607.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A381618 Reverse the Chung-Graham representation of n while preserving its trailing zeros: a(n) = A381607(A263273(A381608(n))).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula