A055235 -id:A055235 - cp's OEIS frontend

A073216 The terms of A055235 (sums of two powers of 3) divided by 2.

1, 2, 3, 5, 6, 9, 14, 15, 18, 27, 41, 42, 45, 54, 81, 122, 123, 126, 135, 162, 243, 365, 366, 369, 378, 405, 486, 729, 1094, 1095, 1098, 1107, 1134, 1215, 1458, 2187, 3281, 3282, 3285, 3294, 3321, 3402, 3645, 4374, 6561, 9842, 9843, 9846, 9855, 9882, 9963, 10206, 10935, 13122, 19683

Offset: 0

Jeremy Gardiner, Jul 21 2002

n such that 3 is the largest power of 3 dividing binomial(3n,n). - Benoit Cloitre, Jan 01 2004
Equals A023745 + 1.
This sequence is A007051 together with its (successive) multiples by (powers of) 3. - R. K. Guy, Oct 08 2011

			T(2,0) = 5 = (3^2 + 3^0) / 2.
Triangle begins:
     1;
     2,    3;
     5,    6,    9;
    14,   15,   18,   27;
    41,   42,   45,   54,   81;
   122,  123,  126,  135,  162,  243;
   365,  366,  369,  378,  405,  486,  729;
  1094, 1095, 1098, 1107, 1134, 1215, 1458, 2187;
  ...

C. Armana, Coefficients of Drinfeld modular forms and Hecke operators, Journal of Number Theory 131 (2011), 1435-1460.

Cf. A000244 (main diagonal), A055235, A007051 (first column), A023745.

T(2n,n) gives A025551.

from math import isqrt
def A073216(n): return 3**(a:=(k:=isqrt(m:=n<<1))+(m>k*(k+1))-1)+3**(n-1-(a*(a+1)>>1))>>1 # Chai Wah Wu, Apr 08 2025

T(n,m) = (3^n + 3^m) / 2, n = 0, 1, 2, 3 ..., m = 0, 1, 2, 3, ... n.

Edited by Jeremy Gardiner, Oct 08 2011

Offset changed by Alois P. Heinz, Apr 08 2025

A226636 Numbers whose base-3 sum of digits is 3.

Original entry on oeis.org

5, 7, 11, 13, 15, 19, 21, 29, 31, 33, 37, 39, 45, 55, 57, 63, 83, 85, 87, 91, 93, 99, 109, 111, 117, 135, 163, 165, 171, 189, 245, 247, 249, 253, 255, 261, 271, 273, 279, 297, 325, 327, 333, 351, 405, 487, 489, 495, 513, 567, 731, 733, 735, 739, 741, 747, 757

Offset: 1

Tom Edgar, Aug 31 2013

All of the entries are odd.
Subsequence of A005408. - Michel Marcus, Sep 03 2013
In general, the set of numbers with sum of base-b digits equal to b is a subset of { (b-1)*k + 1; k = 2, 3, 4, ... }. - M. F. Hasler, Dec 23 2016

			The ternary expansion of 5 is (1,2), which has sum of digits 3.
The ternary expansion of 31 is (1,0,0,2), which has sum of digits 3.
10 is not on the list since the ternary expansion of 10 is (1,0,1), which has sum of digits 2 not 3.

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

Cf. A018900, A023694, A055235, A187813.

Cf. A226969 (b = 4), A227062 (b = 5), A227080 (b = 6), A227092 (b = 7), A227095 (b = 8), A227238 (b = 9), A052224 (b = 10).

N:= 10: # for all terms < 3^(N+1)
[seq(seq(seq(3^a+3^b+3^c, c=0..`if`(b=a, b-1,b)),b = 0..a),a=0..N)]; # Robert Israel, Jun 05 2018

Select[Range@ 757, Total@ IntegerDigits[#, 3] == 3 &] (* Michael De Vlieger, Dec 23 2016 *)

select( is(n)=sumdigits(n,3)==3, [1..999]) \\ M. F. Hasler, Dec 23 2016

from itertools import islice
def nextsod(n, base):
    c, b, w = 0, base, 0
    while True:
        d = n%b
        if d+1 < b and c:
            return (n+1)*b**w + ((c-1)%(b-1)+1)*b**((c-1)//(b-1))-1
        c += d; n //= b; w += 1
def A226636gen(sod=3, base=3): # generator of terms for any sod, base
    an = (sod%(base-1)+1)*base**(sod//(base-1))-1
    while True: yield an; an = nextsod(an, base)
print(list(islice(A226636gen(), 57))) # Michael S. Branicky, Jul 10 2022, generalizing the code by M. F. Hasler in A052224

[i for i in [0..1000] if sum(Integer(i).digits(base=3))==3]

a(k^3/6 + k^2 + 5*k/6 + j) = 3^(k+1) + A055235(j-1) for 1 <= j <= k^2/2+5*k/2+2. - Robert Israel, Jun 05 2018

A073211 Sum of two powers of 11.

Original entry on oeis.org

2, 12, 22, 122, 132, 242, 1332, 1342, 1452, 2662, 14642, 14652, 14762, 15972, 29282, 161052, 161062, 161172, 162382, 175692, 322102, 1771562, 1771572, 1771682, 1772892, 1786202, 1932612, 3543122, 19487172, 19487182, 19487292, 19488502, 19501812, 19648222, 21258732, 38974342

Offset: 0

Jeremy Gardiner, Jul 20 2002

			T(2,0) = 11^2 + 11^0 = 122.
Table T(n,m) begins:
      2;
     12,    22;
    122,   132,   242;
   1332,  1342,  1452,  2662;
  14642, 14652, 14762, 15972, 29282;
  ...

T. D. Noe, Rows n = 0..100 of triangle, flattened

Cf. A001020 (powers of 11).
Equals twice A073219.
Sums of two powers of n: A073423 (0), A007395 (1), A173786 (2), A055235 (3), A055236 (4), A055237 (5), A055257 (6), A055258 (7), A055259 (8), A055260 (9), A052216 (10), A194887 (12), A072390 (13), A055261 (16), A073213 (17), A073214 (19), A073215 (23).

t = 11^Range[0, 9]; Select[Union[Flatten[Table[i + j, {i, t}, {j, t}]]], # <= t[[-1]] + 1 &] (* T. D. Noe, Oct 09 2011 *)

from math import isqrt
def A073211(n): return 11**(a:=(k:=isqrt(m:=n<<1))+(m>k*(k+1))-1)+11**(n-1-(a*(a+1)>>1)) # Chai Wah Wu, Apr 09 2025

T(n,m) = 11^n + 11^m, n = 0, 1, 2, 3, ..., m = 0, 1, 2, 3, ... n.

Bivariate g.f.: (2 - 12*x)/((1 - x)*(1 - 11*x)*(1 - 11*x*y)). - J. Douglas Morrison, Jul 26 2021

A073213 Sum of two powers of 17.

Original entry on oeis.org

2, 18, 34, 290, 306, 578, 4914, 4930, 5202, 9826, 83522, 83538, 83810, 88434, 167042, 1419858, 1419874, 1420146, 1424770, 1503378, 2839714, 24137570, 24137586, 24137858, 24142482, 24221090, 25557426, 48275138, 410338674, 410338690, 410338962, 410343586, 410422194, 411758530, 434476242, 820677346

Offset: 0

Jeremy Gardiner, Jul 20 2002

			T(2,0) = 17^2 + 17^0 = 290.
Table T(n,m) begins:
      2;
     18,    34;
    290,   306,   578;
   4914,  4930,  5202,  9826;
  83522, 83538, 83810, 88434, 167042;
  ...

T. D. Noe, Rows n = 0..100 of triangle, flattened

Cf. A001026 (powers of 17).
Equals twice A073221.
Sums of two powers of n: A073423 (0), A007395 (1), A173786 (2), A055235 (3), A055236 (4), A055237 (5), A055257 (6), A055258 (7), A055259 (8), A055260 (9), A052216 (10), A073211 (11), A194887 (12), A072390 (13), A055261 (16), A073214 (19), A073215 (23).

Flatten[Table[Table[17^n + 17^m, {m, 0, n}], {n, 0, 7}]] (* T. D. Noe, Jun 18 2013 *)
Union[Total/@Tuples[17^Range[0,10],2]] (* Harvey P. Dale, Apr 09 2015 *)

from math import isqrt
def A073213(n): return 17**(a:=(k:=isqrt(m:=n<<1))+(m>k*(k+1))-1)+17**(n-1-(a*(a+1)>>1)) # Chai Wah Wu, Apr 09 2025

T(n,m) = 17^n + 17^m, n = 0, 1, 2, 3, ..., m = 0, 1, 2, 3, ... n.

Bivariate g.f.: (2 - 18*x)/((1 - x)*(1 - 17*x)*(1 - 17*x*y)). - J. Douglas Morrison, Jul 26 2021

A073214 Sum of two powers of 19.

Original entry on oeis.org

2, 20, 38, 362, 380, 722, 6860, 6878, 7220, 13718, 130322, 130340, 130682, 137180, 260642, 2476100, 2476118, 2476460, 2482958, 2606420, 4952198, 47045882, 47045900, 47046242, 47052740, 47176202, 49521980, 94091762, 893871740, 893871758, 893872100, 893878598, 894002060, 896347838, 940917620, 1787743478

Offset: 0

Jeremy Gardiner, Jul 20 2002

			T(2,0) = 19^2 + 19^0 = 362.
Table begins:
       2;
      20,     38;
     362,    380,    722;
    6860,   6878,   7220,  13718;
  130322, 130340, 130682, 137180, 260642;
  ...

T. D. Noe, Rows n = 0..100 of triangle, flattened

Cf. A001029.
Equals twice A073222.
Sums of two powers of n: A073423 (0), A007395 (1), A173786 (2), A055235 (3), A055236 (4), A055237 (5), A055257 (6), A055258 (7), A055259 (8), A055260 (9), A052216 (10), A073211 (11), A194887 (12), A072390 (13), A055261 (16), A073213 (17), A073215 (23).

Flatten[Table[Table[19^n + 19^m, {m, 0, n}], {n, 0, 7}]] (* T. D. Noe, Jun 18 2013 *)
Total/@Tuples[19^Range[0,10],2]//Union (* Harvey P. Dale, Jan 04 2019 *)

from math import isqrt
def A073214(n): return 19**(a:=(k:=isqrt(m:=n<<1))+(m>k*(k+1))-1)+19**(n-1-(a*(a+1)>>1)) # Chai Wah Wu, Apr 09 2025

T(n,m) = 19^n + 19^m for n >= 0 and m in [0..n].

Bivariate g.f.: (2 - 20*x) / ((1 - x) * (1 - 19*x) * (1 - 19*x*y)). - J. Douglas Morrison, Jul 28 2021

A073215 Sum of two powers of 23.

Original entry on oeis.org

2, 24, 46, 530, 552, 1058, 12168, 12190, 12696, 24334, 279842, 279864, 280370, 292008, 559682, 6436344, 6436366, 6436872, 6448510, 6716184, 12872686, 148035890, 148035912, 148036418, 148048056, 148315730, 154472232, 296071778

Offset: 0

Jeremy Gardiner, Jul 20 2002

			T(2,0) = 23^2 + 23^0 = 530.
Table begins:
       2;
      24,     46;
     530,    552,   1058;
   12168,  12190,  12696,  24334;
  279842, 279864, 280370, 292008, 559682;
  ...

G. C. Greubel, Table of n, a(n) for n = 0..999 [Offset changed to 0 by _Georg Fischer_, Mar 01 2022]

Cf. A009967.
Equals twice A072822.
Sums of two powers of n: A073423 (0), A007395 (1), A173786 (2), A055235 (3), A055236 (4), A055237 (5), A055257 (6), A055258 (7), A055259 (8), A055260 (9), A052216 (10), A073211 (11), A194887 (12), A072390 (13), A055261 (16), A073213 (17), A073214 (19).

With[{nn=30},Take[Union[Total/@Tuples[23^Range[0,nn],2]],nn]] (* Harvey P. Dale, Oct 16 2017 *)

from math import isqrt
def A073215(n): return 23**(a:=(k:=isqrt(m:=n<<1))+(m>k*(k+1))-1)+23**(n-1-(a*(a+1)>>1)) # Chai Wah Wu, Apr 09 2025

T(n, m) = 23^n + 23^m, for n >= 0 and m in [0..n].

Bivariate g.f.: (2 - 24*x) / ((1 - x) * (1 - 23*x) * (1 - 23*x*y)). - J. Douglas Morrison, Jul 29 2021

A325984 Lexicographically earliest sequence of distinct nonnegative terms such that for any n >= 0, the sum of digits of n in base 2 equals the sum of digits of a(n) in base 3.

Original entry on oeis.org

0, 1, 3, 2, 9, 4, 6, 5, 27, 10, 12, 7, 18, 11, 13, 8, 81, 28, 30, 15, 36, 19, 21, 14, 54, 29, 31, 16, 33, 20, 22, 17, 243, 82, 84, 37, 90, 39, 45, 24, 108, 55, 57, 32, 63, 34, 38, 23, 162, 83, 85, 40, 87, 42, 46, 25, 91, 48, 56, 35, 58, 41, 43, 26, 729, 244

Offset: 0

Rémy Sigrist, May 30 2019

This sequence is a permutation of the nonnegative integers, with inverse A325985.
The first known fixed points are: 0, 1, 6 and 129936.
We can generalize this sequence for any pair of bases > 1, say u and v:
- let f_{u,v} be the lexicographically earliest sequence of distinct nonnegative terms such that for any n >= 0, the sum of digits of n in base u equals the sum of digits of f_{u,v}(n) in base v,
- in particular f_{2,3} = a (this sequence) and f_{3,2} = A325985,
- f_{u,v} is a permutation of the nonnegative integers, with inverse f_{v,u},
- f_{u,v}(u^k) = v^k for any k >= 0,
- f_{u,u} is the identity function,
- f_{u,v} o f_{v,w} = f_{u,w} (where o denotes function composition).

			The first terms, alongside the binary representation of n and the ternary representation of a(n), are:
  n   a(n)  bin(n)  ter(a(n))
  --  ----  ------  ---------
   0     0       0          0
   1     1       1          1
   2     3      10         10
   3     2      11          2
   4     9     100        100
   5     4     101         11
   6     6     110         20
   7     5     111         12
   8    27    1000       1000
   9    10    1001        101
  10    12    1010        110
  11     7    1011         21
  12    18    1100        200
  13    11    1101        102
  14    13    1110        111
  15     8    1111         22
  16    81   10000      10000

Rémy Sigrist, Table of n, a(n) for n = 0..8192
Rémy Sigrist, Colored logarithmic scatterplot of the sequence for n = 0..2^16 (where the color is function of A000120(n)).
Rémy Sigrist, PARI program for A325984.
Index entries for sequences that are permutations of the natural numbers.

Cf. A000120, A018900, A053735, A055235, A325985 (inverse).

a[0] = 0; a[n_] := a[n] = Module[{s = DigitCount[n, 2, 1], k = 1}, While[! FreeQ[Array[a, n - 1], k] || Plus @@ IntegerDigits[k, 3] != s, k++]; k]; Array[a, 66, 0] (* Amiram Eldar, Jul 25 2023 *)

PARI
```
See Links section.
```

a(2^k) = 3^k for any k >= 0.
A000120(n) = A053735(a(n)).
a(A018900(k)) = A055235(k-1) for any k > 0.

A102861 Numbers which in base 5 have digit-sum 4.

Original entry on oeis.org

4, 8, 12, 16, 20, 28, 32, 36, 40, 52, 56, 60, 76, 80, 100, 128, 132, 136, 140, 152, 156, 160, 176, 180, 200, 252, 256, 260, 276, 280, 300, 376, 380, 400, 500, 628, 632, 636, 640, 652, 656, 660, 676, 680, 700, 752, 756, 760, 776, 780, 800, 876, 880, 900, 1000

Offset: 1

Sergi Adamchuk (adamchuk(AT)gmail.com), Mar 15 2005

Or, sums of four powers of 5.

			In base 5 the numbers are 4, 13, 31, 40, 103, 112, 121, 130, 202, 211, 220, 301, 310, 400, 1003, ...

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A055235.

for n from 0 to 2500 do if coeff(series(add(x^(5^i),i=0..4)^4,x,2500),x,n)<>0 then printf(`%d,`,n);fi:od:

Select[Range[1000],Total[IntegerDigits[#,5]]==4&] (* Harvey P. Dale, Jul 13 2023 *)

The author would like a formula for the n-th term.

More terms from Joshua Zucker, May 15 2006

A281228 Expansion of (Sum_{k>=0} x^(3^k))^2 [even terms only].

Original entry on oeis.org

0, 1, 2, 1, 0, 2, 2, 0, 0, 1, 0, 0, 0, 0, 2, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Ilya Gutkovskiy, Jan 18 2017

nonn

Number of ways to write 2n as an ordered sum of two powers of 3.

First bisection of self-convolution of characteristic function of powers of 3.

			G.f. = x^2 + 2*x^4 + x^6 + 2*x^10 + 2*x^12 + x^18 + 2*x^28 + 2*x^30 + 2*x^36 + ...
a(2) = 2 because we have [3, 1] and [1, 3].

Index entries for sequences related to compositions

Cf. A000244, A055235, A073267, A078932.

Take[CoefficientList[Series[Sum[x^3^k, {k, 0, 15}]^2, {x, 0, 260}], x], {1, -1, 2}]

G.f.: (Sum_{k>=0} x^(3^k))^2 [even terms only].

A362861 Positive integers n such that 2*n cannot be written as a sum of distinct elements of the set {5^a + 5^b: a,b = 0,1,2,...}.

Original entry on oeis.org

2, 7, 10, 11, 12, 27, 35, 50, 51, 52, 55, 60, 135, 255

Offset: 1

Zhi-Wei Sun, May 05 2023

If a(15) exists, it should be greater than 10290.
Conjecture 1: (i) The current sequence only has the listed 14 terms. Also, each positive even number can be written as a sum of distinct elements of the set {3^a + 3^b: a,b = 0,1,2,...}.
(ii) Each positive even number can be written as a sum of distinct elements of the set {3^a + 7^b: a,b = 0,1,2,...}. Also, any positive even number not equal to 12 can be written as a sum of numbers of the form 3^a + 5^b (a,b >= 0) with no summand dividing another.
Conjecture 2: Let k and m be positive odd numbers greater than one. Then, any sufficiently large even numbers can be written as a sum of distinct elements of the set {k^a + m^b: a,b = 0,1,2,...}.
Conjecture 3: Let k and m be positive odd numbers greater than one. Then, any sufficiently large even numbers can be written as a sum of some numbers of the form k^a + m^b (a,b >= 0) with no summand dividing another.
Clearly, Conjecture 3 is stronger than Conjecture 2.
See also A362743 for similar conjectures.
a(15) >= 10^6. - Martin Ehrenstein, May 16 2023

			a(1) = 2, since 2*1 = 5^0 + 5^0 but 2*2 cannot be written as a sum of distinct numbers of the form 5^a + 5^b (a,b >= 0).

Cf. A055235, A055237, A226809, A226816, A362743.

cp's OEIS Frontend

A073216 The terms of A055235 (sums of two powers of 3) divided by 2.

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A226636 Numbers whose base-3 sum of digits is 3.

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A073211 Sum of two powers of 11.

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A073213 Sum of two powers of 17.

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A073214 Sum of two powers of 19.

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A073215 Sum of two powers of 23.

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A325984 Lexicographically earliest sequence of distinct nonnegative terms such that for any n >= 0, the sum of digits of n in base 2 equals the sum of digits of a(n) in base 3.

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