A055237 -id:A055237 - cp's OEIS frontend

A073217 The terms of A055237 (sums of two powers of 5) divided by 2.

1, 3, 5, 13, 15, 25, 63, 65, 75, 125, 313, 315, 325, 375, 625, 1563, 1565, 1575, 1625, 1875, 3125, 7813, 7815, 7825, 7875, 8125, 9375, 15625, 39063, 39065, 39075, 39125, 39375, 40625, 46875, 78125

Offset: 1

Jeremy Gardiner, Jul 21 2002

			a(4) = 13 = (5^2+5^0) / 2

Cf. A055237.

(5^n + 5^m) / 2, n = 0, 1, 2, 3 ..., m = 0, 1, 2, 3, ... n.

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

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

A073217 The terms of A055237 (sums of two powers of 5) divided by 2.

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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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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,...}.

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