A052216 -id:A052216 - cp's OEIS frontend

A073211 Sum of two powers of 11.

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

A194887 Numbers that are the sum of two powers of 12.

Original entry on oeis.org

2, 13, 24, 145, 156, 288, 1729, 1740, 1872, 3456, 20737, 20748, 20880, 22464, 41472, 248833, 248844, 248976, 250560, 269568, 497664, 2985985, 2985996, 2986128, 2987712, 3006720, 3234816, 5971968, 35831809, 35831820, 35831952, 35833536, 35852544, 36080640

Offset: 1

Jeremy Gardiner, Oct 09 2011

Parity of this sequence is A073424.

			12^0 + 12^2 = 145

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A001021, A033048, A052216, A073424.

t = 12^Range[0, 9]; Select[Union[Flatten[Table[i + j, {i, t}, {j, t}]]], # <= t[[-1]] + 1 &] (* T. D. Noe, Oct 09 2011 *)
Total/@Tuples[12^Range[0,10],2]//Union (* Harvey P. Dale, Jul 20 2019 *)

Typo in example corrected by Zak Seidov, Oct 23 2011

A262711 Numbers k such that sum of digits of k^2 is 7.

Original entry on oeis.org

4, 5, 32, 40, 49, 50, 149, 320, 400, 490, 500, 1049, 1490, 3200, 4000, 4900, 5000, 10490, 14900, 32000, 40000, 49000, 50000, 104900, 149000, 320000, 400000, 490000, 500000, 1049000, 1490000, 3200000, 4000000, 4900000, 5000000, 10490000, 14900000

Offset: 1

Vincenzo Librandi, Sep 28 2015

Subsequence of A156638. [Bruno Berselli, Sep 28 2015]

			4 is in sequence because 4^2 = 16 and 1+6 = 7.

Cf. sum of digits of n^2 is k: A052216 (k=4), this sequence (k=7), A262712 (k=9), A262713 (k=10).
Cf. A004159, A061910, A156638.
Cf. A215614.

[n: n in [1..2*10^7] | &+Intseq(n^2) eq 7];

Select[Range[10^7], Total[IntegerDigits[#^2]] == 7 &]

for(n=1, 1e8, if (sumdigits(n^2) == 7, print1(n", "))) \\ Altug Alkan, Sep 28 2015

A340063 The primes appear in their natural order and the absolute difference between two successive primes is the sum of the digits between them.

Original entry on oeis.org

2, 1, 3, 10, 100, 5, 20, 7, 4, 11, 110, 13, 12, 1000, 17, 200, 19, 21, 10000, 23, 6, 29, 1001, 31, 14, 100000, 37, 22, 41, 1010, 43, 30, 1000000, 47, 15, 53, 24, 59, 1100, 61, 32, 10000000, 67, 40, 71, 2000, 73, 33, 79, 102, 100000000, 83, 42, 89, 8, 97, 111, 1000000000, 101, 10001, 103, 112, 107, 10010, 109

Offset: 1

Eric Angelini, Dec 27 2020

Lexicographically earliest sequence of distinct positive terms with this property. It is conjectured that the sequence is a permutation of the integers > 1.

			prime 2 + (1) = prime 3;
prime 3 + (1+0 + 1+0+0) = prime 5; (we do not put 2 between 5 and 7 as 2 is in the sequence already and not 20 as 10 is lexicographically earlier along with 100 gives the digital sum 2).
prime 5 + (2+0) = prime 7;
prime 7 + (4) = prime 11;
prime 11 + (1+1+0) = prime 13;
prime 13 + (1+2 + 1+0+0+0) = 17; etc.

David A. Corneth, Table of n, a(n) for n = 1..8564 (terms < 10^999; first 515 terms from Carole Dubois)

Cf. A000040 (the prime numbers), A001223 (prime gaps), A052216, A052217.

A135293 Differences between successive numbers whose sum of digits in base 3 is 2.

Original entry on oeis.org

2, 2, 2, 4, 2, 6, 10, 2, 6, 18, 28, 2, 6, 18, 54, 82, 2, 6, 18, 54, 162, 244, 2, 6, 18, 54, 162, 486, 730, 2, 6, 18, 54, 162, 486, 1458, 2188, 2, 6, 18, 54, 162, 486, 1458, 4374, 6562, 2, 6, 18, 54, 162, 486, 1458, 4374, 13122

Offset: 0

Adam Shelly (adam.shelly(AT)gmail.com), Dec 04 2007, Dec 05 2007

First differences of A052216 when the entries in that sequence are interpreted as base 3 numbers.

Can be regarded as a triangle, where T(0,0)=2, T(n+1,0) = T(n,0)+T(n,n), T(n+1,m) = T(n,m) for 0 < m <= n and T(n+1,n+1) = sum of T(n+1,0..n)

			triangle begins:
2
2 2
4 2 6
10 2 6 18
28 2 6 18 54
82 2 6 18 54 162
244 2 6 18 54 162 486.

G. C. Greubel, Table of n, a(n) for the first 50 rows

Cf. A052216.

T[0, 0] := 2; T[n_, 0] := 3^(n - 1) + 1; T[n_, m_] := 2*3^(m - 1); Table[T[n, m], {n, 0, 5}, {m, 0, n}] (* G. C. Greubel, Oct 09 2016 *)
Join[{2},Differences[Select[Range[50000],Total[IntegerDigits[#,3]]==2&]]] (* Harvey P. Dale, Jul 04 2019 *)

T(n,m) = 2*3^(m-1) = A025192(m) for m>0. T(n,0) = 2*A124302(n). - Franklin T. Adams-Watters, Sep 29 2011

Edited by Franklin T. Adams-Watters, Sep 29 2011

A375460 Lexicographically earliest sequence of distinct nonnegative terms arranged in successive chunks whose digitsum = 10.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 10, 11, 20, 6, 12, 100, 7, 21, 8, 101, 9, 1000, 13, 14, 10000, 15, 22, 16, 30, 17, 110, 18, 100000, 19, 23, 31, 1000000, 24, 40, 25, 102, 26, 200, 27, 10000000, 28, 32, 41, 33, 103, 34, 111, 35, 1001, 36, 100000000, 37, 42, 112, 43, 120, 44, 1010, 45, 1000000000

Offset: 1

Eric Angelini, Aug 15 2024

The first integer that will never appear in the sequence is 29, as its digitsum exceeds 10.
From Michael S. Branicky, Aug 16 2024: (Start)
Infinite since A052224 is infinite (as are all sequences with digital sum 1..10).
a(6492) has 1001 digits. (End)

			The first chunk of integers with digitsum 10 is (0,1,2,3,4);
the next one is (5,10,11,20),
the next one is (6,12,100),
the next one is (7,21),
the next one is (8,101),
the next one is (9,1000),
the next one is (13,14,10000), etc.
The concatenation of the above chunks produce the sequence.

Michael S. Branicky, Table of n, a(n) for n = 1..6491

Cf. A166311, A051885, A228915.

Numbers with digital sum 1..10: A011557 (1), A052216 (2), A052217 (3), A052218 (4), A052219 (5), A052220 (6), A052221 (7), A052222 (8), A052223 (9), A052224 (10).

from itertools import islice
def bgen(ds): # generator of terms with digital sum ds
    def A051885(n): return ((n%9)+1)*10**(n//9)-1 # due to Chai Wah Wu
    def A228915(n): # due to M. F. Hasler
        p = r = 0
        while True:
            d = n % 10
            if d < 9 and r: return (n+1)*10**p + A051885(r-1)
            n //= 10; r += d; p += 1
    k = A051885(ds)
    while True: yield k; k = A228915(k)
def agen(): # generator of terms
    an, ds_block = 0, 0
    dsg = [None] + [bgen(i) for i in range(1, 11)]
    dsi = [None] + [(next(dsg[i]), i) for i in range(1, 11)]
    while True:
        yield an
        an, ds_an = min(dsi[j] for j in range(1, 11-ds_block))
        ds_block = (ds_block + ds_an)%10
        dsi[ds_an] = (next(dsg[ds_an]), ds_an)
print(list(islice(agen(), 61))) # Michael S. Branicky, Aug 16 2024

a(46) and beyond from Michael S. Branicky, Aug 16 2024.

A384094 Numbers whose square has digit sum 9 and no trailing zero.

Original entry on oeis.org

3, 6, 9, 12, 15, 18, 21, 39, 45, 48, 51, 102, 105, 111, 201, 249, 318, 321, 348, 351, 501, 549, 1002, 1005, 1011, 1101, 1149, 1761, 2001, 4899, 5001, 10002, 10005, 10011, 10101, 10149, 11001, 14499, 20001, 50001, 100002, 100005, 100011, 100101, 101001, 110001, 200001, 375501, 500001, 1000002

Offset: 1

M. F. Hasler, Jun 15 2025

All numbers of the form 10^a + 10^b + 1 (i.e., A052216+1 = 3*A237424) and of the form 10^a + 5*10^b with min(a, b) = 0 (i.e., A133472 U A199685), are in this sequence. Terms not of this form are (9, 18, 39, 45, 48, 249, 318, 321, 348, 351, 549, 1149, 1761, 4899, 10149, 14499, 375501, ...), see subsequence A384095. (Is this sequence finite? What is the next term?)

Is it true that no number > 1049 = A215614(6) has a square with digit sum less than 9, other than the trivial 1 and 4?

Cf. A004159 (sum of digits of n^2), A215614 (sumdigits(n^2) = 7), A133472 (10^n + 5), A199685 (5*10^n + 1), A052216 (10^a + 10^b), A237424 ((10^a + 10^b + 1)/3).

See also: A058414 (digits(n^2) in {0,1,4}).

select( {is_A384094(n)=n%10 && sumdigits(n^2)==9}, [1..10^5])

cp's OEIS Frontend

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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A194887 Numbers that are the sum of two powers of 12.

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A262711 Numbers k such that sum of digits of k^2 is 7.

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A340063 The primes appear in their natural order and the absolute difference between two successive primes is the sum of the digits between them.

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A135293 Differences between successive numbers whose sum of digits in base 3 is 2.

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