A081604 -id:A081604 - cp's OEIS frontend

A330358 a(n) = n mod 5 + n mod 25 + ... + n mod 5^k, where 5^k <= n < 5^(k+1).

0, 0, 0, 0, 0, 1, 2, 3, 4, 0, 1, 2, 3, 4, 0, 1, 2, 3, 4, 0, 1, 2, 3, 4, 0, 2, 4, 6, 8, 5, 7, 9, 11, 13, 10, 12, 14, 16, 18, 15, 17, 19, 21, 23, 20, 22, 24, 26, 28, 0, 2, 4, 6, 8, 5, 7, 9, 11, 13, 10, 12, 14, 16, 18, 15, 17, 19, 21, 23, 20, 22, 24, 26, 28, 0, 2, 4, 6, 8, 5, 7, 9, 11, 13, 10

Offset: 1

Petros Hadjicostas, Dec 12 2019

Conjecture: For b >= 2, consider the function s(n,b) = Sum_{1 <= b^j <= n} (n mod b^j) from p. 8 in Dearden et al. (2011). Then s(b*n + r, b) = b*s(n,b) + r*N(n,b) for 0 <= r <= b-1, where N(n,b) = floor(log_b(n)) + 1 is the number of digits in the base-b representation of n. As initial conditions, we have s(n,b) = 0 for 1 <= n <= b. (This is a generalization of a result by Robert Israel in A049802.)
Here b = 5 and a(n) = s(n,5).
We have N(n,2) = A070939(n), N(n,3) = A081604(n), N(n,4) = A110591(n), and N(n,5) = A110592(n).
If A_b(x) = Sum_{n >= 1} s(n,b)*x^n is the g.f. of the sequence (s(n,b): n >= 1) and the above conjecture is correct, then it can be proved that A_b(x) = b * A_b(x^b) * (1-x^b)/(1-x) + x * ((b-1)*x^b - b*x^(b-1) + 1)/((1-x)^2 * (1-x^b)) * Sum_{k >= 1} x^(b^k).

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A049802, A049803, A049804, A070939, A081604, A110591, A110592.

a:= n-> add(irem(n, 5^j), j=1..ilog[5](n)):
seq(a(n), n=1..105);  # Alois P. Heinz, Dec 13 2019

a[n_] := Sum[Mod[n, 5^j], {j, 1, Length[IntegerDigits[n, 5]] - 1}];
Array[a, 105] (* Jean-François Alcover, Dec 31 2021 *)

a(n) = sum(k=1, logint(n, 5), n % 5^k);
for(n=1, 100, print1(a(n), ", ")); \\ (after Michel Marcus's program in A049804)

Conjecture: a(5*n+r) = 5*a(n) + r*A110592(n) = 5*a(n) + r*(floor(log_5(n)) + 1) for n >= 1 and r = 0, 1, 2, 3, 4.

If the conjecture above is true, the g.f. A(x) satisfies A(x) = 5*(1 + x + x^2 + x^3 + x^4)*A(x^5) + x*(1 + 2*x + 3*x^2 + 4*x^3)/(1 - x^5) * Sum_{k >= 1} x^(5^k).

A107681 Repeat(first 2^k numbers with no 2 in ternary representation) for k>0.

Original entry on oeis.org

0, 1, 0, 1, 3, 4, 0, 1, 3, 4, 9, 10, 12, 13, 0, 1, 3, 4, 9, 10, 12, 13, 27, 28, 30, 31, 36, 37, 39, 40, 0, 1, 3, 4, 9, 10, 12, 13, 27, 28, 30, 31, 36, 37, 39, 40, 81, 82, 84, 85, 90, 91, 93, 94, 108, 109, 111, 112, 117, 118, 120, 121, 0, 1, 3, 4, 9, 10, 12, 13, 27, 28, 30, 31, 36, 37

Offset: 1

Reinhard Zumkeller, May 20 2005

let A032924(n) = Sum(d(i)*3^i: 0

then .... a(n) = Sum((d(i)-1)*3^i: 0<=i

A032924(n) = A107680(n) + a(n);

A081604 (a(n)) <= A081604(A107680(n)) = A081604(A032924(n)) = A000523(n+1).

			A032924(177) = A107680(177) + a(177),
....... 1420 = ....... 1093 + 327,
.. '1221121' = ... '1111111'+ '110010',
............ = . A003462(7) + A005836(51).

Ruud H.G. van Tol, Table of n, a(n) for n = 1..8190 (12 rows).

Cf. A000523, A005836, A003462, A007089, A032924, A062050, A081604, A107680.

a(n)= fromdigits(binary(n+1-1<Ruud H.G. van Tol, Nov 18 2024

def A107681(n): return int(bin(n+1)[3:],3) # Chai Wah Wu, May 06 2025

a(n) = A005836(A062050(n+1)).

Data corrected by Ruud H.G. van Tol, Nov 18 2024.

A343601 For any positive number n, the ternary representation of a(n) is obtained by right-rotating the ternary representation of n until a nonzero digit appears again as the leftmost digit; a(0) = 0.

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 6, 5, 8, 9, 12, 21, 10, 13, 22, 19, 14, 23, 18, 15, 24, 11, 16, 25, 20, 17, 26, 27, 36, 63, 30, 37, 64, 57, 38, 65, 28, 39, 66, 31, 40, 67, 58, 41, 68, 55, 42, 69, 32, 43, 70, 59, 44, 71, 54, 45, 72, 33, 46, 73, 60, 47, 74, 29, 48, 75, 34, 49

Offset: 0

Rémy Sigrist, Apr 21 2021

This sequence is a permutation of the nonnegative integers with inverse A343600.

			The first terms, in base 10 and in base 3, are:
  n   a(n)  ter(n)  ter(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      22         22
   9     9     100        100
  10    12     101        110
  11    21     102        210
  12    10     110        101
  13    13     111        111
  14    22     112        211

Cf. A053735, A081604, A139706 (binary variant), A160384, A343600 (inverse).

a(n, base=3) = { my (d=digits(n, base)); forstep (k=#d, 2, -1, if (d[k], return (fromdigits(concat(d[k..#d], d[1..k-1]), base)))); n }

A053735(a(n)) = A053735(n).
A081604(a(n)) = A081604(n).
a^k(n) = n for k = A160384(n) (where a^k denotes the k-th iterate of a).

A091093 In ternary representation: minimal number of editing steps (delete, insert or substitute) to transform n into n^2.

Original entry on oeis.org

0, 0, 2, 1, 1, 2, 3, 2, 3, 2, 2, 4, 2, 4, 3, 3, 4, 4, 4, 4, 5, 3, 4, 3, 4, 3, 4, 3, 3, 4, 3, 3, 3, 5, 3, 5, 3, 3, 5, 5, 5, 6, 4, 3, 4, 4, 4, 5, 5, 4, 4, 5, 5, 5, 5, 5, 6, 5, 5, 5, 6, 4, 5, 4, 4, 5, 5, 4, 5, 4, 5, 5, 5, 4, 6, 4, 5, 4, 5, 4, 5, 4, 4, 5, 4, 4, 4, 5, 5, 5, 4, 4, 6, 4, 5, 4, 4, 5, 5, 5, 5, 6

Offset: 0

Reinhard Zumkeller, Dec 18 2003

			a(12)=2: 12->'110', insert a 2 between the 1's and insert a 0 at the end: '12100'->144=12^2.

Robert Israel, Table of n, a(n) for n = 0..10000
Michael Gilleland, Levenshtein Distance [It has been suggested that this algorithm gives incorrect results sometimes. - _N. J. A. Sloane_]
Eric Weisstein's World of Mathematics, Square Number
Eric Weisstein's World of Mathematics, Ternary

Cf. A091092, A091091, A081604, A000290.

A091093:= proc(x) local L1, L2;
   L1:= convert(map(`+`,ListTools:-Reverse(convert(x,base,3)),48),bytes);
   L2:= convert(map(`+`,ListTools:-Reverse(convert(x^2,base,3)),48),bytes);
   StringTools:-Levenshtein(L1,L2)
end proc:
seq(A091093(i),i=0..1000); # Robert Israel, May 06 2014

a(n) = LevenshteinDistance(A007089(n), A001738(n)).

A138167 Numbers containing their length in ternary representation.

Original entry on oeis.org

1, 5, 6, 7, 8, 9, 10, 11, 12, 21, 31, 36, 37, 38, 39, 40, 41, 42, 43, 44, 49, 58, 66, 67, 68, 76, 86, 95, 96, 97, 98, 104, 113, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149

Offset: 1

Reinhard Zumkeller, Mar 03 2008

			42 ->'1120', length = 4 ->'11', therefore 42 is a term;
420 ->'120120', length = 6 ->'20', therefore 420 is a term.

R. Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A081604, A138166, A138168.

A276624 The infinite trunk of ternary beanstalk with reversed subsections.

Original entry on oeis.org

0, 2, 8, 4, 26, 20, 16, 12, 10, 80, 72, 68, 62, 56, 52, 46, 42, 38, 34, 30, 28, 242, 232, 224, 216, 212, 204, 198, 194, 188, 180, 176, 170, 164, 160, 152, 144, 140, 134, 126, 122, 116, 110, 106, 100, 96, 92, 88, 84, 82, 728, 716, 706, 698, 688, 680, 672, 664, 656, 648, 644, 634, 626, 618, 610, 602, 594, 590, 582, 576, 572

Offset: 0

Antti Karttunen, Sep 11 2016

See A276623.

Antti Karttunen, Table of n, a(n) for n = 0..6250

Cf. A000244, A053735, A054861, A081604, A276623.

(definec (A276624 n) (cond ((zero? n) n) ((= n 1) 2) (else (let ((maybe_next (* 2 (A054861 (A276624 (- n 1)))))) (if (not (= 1 (A053735 (+ 1 maybe_next)))) maybe_next (+ -1 (A000244 (+ 1 (A081604 (+ 1 maybe_next))))))))))

a(0) = 0; a(1) = 2; for n > 1, if 2*A054861(a(n-1))+1 is not a power of 3, then a(n) = 2*A054861(a(n-1)), otherwise a(n) = A000244(1+A081604(1+2*A054861(a(n-1)))) - 1.

A343600 For any positive number n, the ternary representation of a(n) is obtained by left-rotating the ternary representation of n until a nonzero digit appears again as the leftmost digit; a(0) = 0.

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 6, 5, 8, 9, 12, 21, 10, 13, 16, 19, 22, 25, 18, 15, 24, 11, 14, 17, 20, 23, 26, 27, 36, 63, 30, 39, 48, 57, 66, 75, 28, 31, 34, 37, 40, 43, 46, 49, 52, 55, 58, 61, 64, 67, 70, 73, 76, 79, 54, 45, 72, 33, 42, 51, 60, 69, 78, 29, 32, 35, 38, 41

Offset: 0

Rémy Sigrist, Apr 21 2021

This sequence is a permutation of the nonnegative integers with inverse A343601.

			The first terms, in base 10 and in base 3, are:
  n   a(n)  ter(n)  ter(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      22         22
   9     9     100        100
  10    12     101        110
  11    21     102        210
  12    10     110        101
  13    13     111        111
  14    16     112        121

Cf. A053735, A081604, A139708 (binary variant), A160384, A343601 (inverse).

a(n, base=3) = { my (d=digits(n, base)); for (k=2, #d, if (d[k], return (fromdigits(concat(d[k..#d], d[1..k-1]), base)))); n }

A053735(a(n)) = A053735(n).
A081604(a(n)) = A081604(n).
a^k(n) = n for k = A160384(n) (where a^k denotes the k-th iterate of a).

A110595 a(1)=5. For n > 1, a(n) = 4*5^(n-1) = A005054(n).

Original entry on oeis.org

5, 20, 100, 500, 2500, 12500, 62500, 312500, 1562500, 7812500, 39062500, 195312500, 976562500, 4882812500, 24414062500, 122070312500, 610351562500, 3051757812500, 15258789062500, 76293945312500, 381469726562500

Offset: 1

Jonathan Vos Post, Jul 29 2005

a(n) is the number of n-digit integers that contain only even digits (A014263). - Bernard Schott, Nov 11 2022

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (5).

Cf. A000351, A007091, A014263, A081604, A110591, A110592, A110593, A110594.

Join[{5},NestList[5#&,20,20]] (* Harvey P. Dale, Jun 19 2013 *)
Rest[CoefficientList[Series[5 x (1 - x)/(1 - 5 x), {x,0,50}], x]] (* G. C. Greubel, Sep 01 2017 *)

my(x='x+O('x^50)); Vec(5*x*(1-x)/(1-5*x)) \\ G. C. Greubel, Sep 01 2017

O.g.f.: 5*x*(1-x)/(1-5*x). - Better definition from R. J. Mathar, May 13 2008

Sum_{n>=1} 1/a(n) = 21/80. - Bernard Schott, Nov 11 2022

Better definition from R. J. Mathar, May 13 2008

Incorrect comment removed by Michel Marcus, Nov 11 2022

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cp's OEIS Frontend

A110593 a(1) = 3, a(n+1) = 2*(3^n).

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A330358 a(n) = n mod 5 + n mod 25 + ... + n mod 5^k, where 5^k <= n < 5^(k+1).

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A107680 Repeating k-th ternary repunit (A003462) 2^k times, k >= 0.

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A107681 Repeat(first 2^k numbers with no 2 in ternary representation) for k>0.

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A343601 For any positive number n, the ternary representation of a(n) is obtained by right-rotating the ternary representation of n until a nonzero digit appears again as the leftmost digit; a(0) = 0.

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A091093 In ternary representation: minimal number of editing steps (delete, insert or substitute) to transform n into n^2.

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A138167 Numbers containing their length in ternary representation.

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A276624 The infinite trunk of ternary beanstalk with reversed subsections.

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