cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A279773 Numbers n such that the sum of digits of 3n equals 6.

Original entry on oeis.org

2, 5, 8, 11, 14, 17, 20, 35, 38, 41, 44, 47, 50, 68, 71, 74, 77, 80, 101, 104, 107, 110, 134, 137, 140, 167, 170, 200, 335, 338, 341, 344, 347, 350, 368, 371, 374, 377, 380, 401, 404, 407, 410, 434, 437, 440, 467, 470, 500, 668, 671, 674, 677, 680, 701
Offset: 1

Views

Author

M. F. Hasler, Dec 23 2016

Keywords

Comments

Inspired by A088405 = A052217/3 and A279769 (the analog for 9).

Crossrefs

Cf. A007953 (digital sum), A279772 (sumdigits(2n) = 4), A279774 (sumdigits(4n) = 8), A279775 (sumdigits(5n) = 10), A279776 (sumdigits(6n) = 12), A279770 (sumdigits(7n) = 14), A279768 (sumdigits(8n) = 16), A279769 (sumdigits(9n) = 18), A279777 (sumdigits(9n) = 27).
Digital sum of m*n equals m: A088404 = A069537/2, A088405 = A052217/3, A088406 = A063997/4, A088407 = A069540/5, A088408 = A062768/6, A088409 = A063416/7, A088410 = A069543/8.
Cf. A005349 (Niven or Harshad numbers), A245062 (arranged in rows by digit sums).
Numbers with given digital sum: A011557 (1), A052216 (2), A052217 (3), A052218 (4), A052219 (5), A052220 (6), A052221 (7), A052222 (8), A052223 (9), A052224 (10), A166311 (11), A235151 (12), A143164 (13), A235225 (14), A235226 (15), A235227 (16), A166370 (17), A235228 (18), A166459 (19), A235229 (20).

Programs

  • Mathematica
    Select[Range@ 720, Total@ IntegerDigits[3 #] == 6 &] (* Michael De Vlieger, Dec 23 2016 *)
  • PARI
    select( is(n)=sumdigits(3*n)==6, [1..999])

A279774 Numbers n such that the sum of digits of 4n equals 8.

Original entry on oeis.org

2, 11, 20, 29, 38, 56, 65, 83, 101, 110, 128, 155, 200, 254, 263, 281, 290, 308, 326, 335, 353, 380, 425, 506, 515, 533, 551, 560, 578, 605, 650, 758, 776, 785, 803, 830, 875, 1001, 1010, 1028, 1055, 1100, 1253, 1280, 1325, 1505, 1550, 1775
Offset: 1

Views

Author

M. F. Hasler, Dec 23 2016

Keywords

Comments

Inspired by A088406 = A063997/4 and A279769 (the analog for 9).

Crossrefs

Cf. A007953 (digital sum), A279772 (sumdigits(2n) = 4), A279773 (sumdigits(3n) = 6), A279775 (sumdigits(5n) = 10), A279776 (sumdigits(6n) = 12), A279770 (sumdigits(7n) = 14), A279768 (sumdigits(8n) = 16), A279769 (sumdigits(9n) = 18), A279777 (sumdigits(9n) = 27).
Digital sum of m*n equals m: A088404 = A069537/2, A088405 = A052217/3, A088406 = A063997/4, A088407 = A069540/5, A088408 = A062768/6, A088409 = A063416/7, A088410 = A069543/8.
Cf. A005349 (Niven or Harshad numbers), A245062 (arranged in rows by digit sums).
Numbers with given digital sum: A011557 (1), A052216 (2), A052217 (3), A052218 (4), A052219 (5), A052220 (6), A052221 (7), A052222 (8), A052223 (9), A052224 (10), A166311 (11), A235151 (12), A143164 (13), A235225 (14), A235226 (15), A235227 (16), A166370 (17), A235228 (18), A166459 (19), A235229 (20).

Programs

  • Mathematica
    Select[Range@ 2000, Total@ IntegerDigits[4 #] == 8 &] (* Michael De Vlieger, Dec 23 2016 *)
  • PARI
    select( is(n)=sumdigits(4*n)==8, [1..1999])

A279776 Numbers n such that the sum of digits of 6n equals 12.

Original entry on oeis.org

8, 11, 14, 23, 26, 29, 32, 38, 41, 44, 47, 53, 56, 59, 62, 65, 68, 71, 74, 77, 80, 86, 89, 92, 95, 101, 104, 107, 110, 119, 122, 125, 134, 137, 140, 152, 155, 173, 176, 179, 182, 188, 191, 194, 197, 203, 206, 209, 212, 215, 218, 221, 224, 227, 230, 236
Offset: 1

Views

Author

M. F. Hasler, Dec 23 2016

Keywords

Comments

Inspired by A088408 = A062768/6 and A279769 (the analog for 9).

Crossrefs

Cf. A007953 (digital sum), A279772 (sumdigits(2n) = 4), A279773 (sumdigits(3n) = 6), A279774 (sumdigits(4n) = 8), A279775 (sumdigits(5n) = 10), A279770 (sumdigits(7n) = 14), A279768 (sumdigits(8n) = 16), A279769 (sumdigits(9n) = 18), A279777 (sumdigits(9n) = 27).
Digital sum of m*n equals m: A088404 = A069537/2, A088405 = A052217/3, A088406 = A063997/4, A088407 = A069540/5, A088408 = A062768/6, A088409 = A063416/7, A088410 = A069543/8.
Cf. A005349 (Niven or Harshad numbers), A245062 (arranged in rows by digit sums).
Numbers with given digital sum: A011557 (1), A052216 (2), A052217 (3), A052218 (4), A052219 (5), A052220 (6), A052221 (7), A052222 (8), A052223 (9), A052224 (10), A166311 (11), A235151 (12), A143164 (13), A235225 (14), A235226 (15), A235227 (16), A166370 (17), A235228 (18), A166459 (19), A235229 (20).

Programs

  • Mathematica
    Select[Range@ 240, Total@ IntegerDigits[6 #] == 12 &] (* Michael De Vlieger, Dec 23 2016 *)
  • PARI
    is(n)=sumdigits(6*n)==12

A055237 Sums of two powers of 5.

Original entry on oeis.org

2, 6, 10, 26, 30, 50, 126, 130, 150, 250, 626, 630, 650, 750, 1250, 3126, 3130, 3150, 3250, 3750, 6250, 15626, 15630, 15650, 15750, 16250, 18750, 31250, 78126, 78130, 78150, 78250, 78750, 81250, 93750, 156250, 390626, 390630, 390650, 390750, 391250, 393750, 406250, 468750, 781250
Offset: 0

Views

Author

Henry Bottomley, Jun 22 2000

Keywords

Links

Crossrefs

Cf. A052216.

Programs

  • Mathematica
    t = 5^Range[0, 9]; Select[Union[Flatten[Table[i + j, {i, t}, {j, t}]]], # <= t[[-1]] + 1 &] (* T. D. Noe, Oct 09 2011 *)
Total/@Tuples[5^Range[0,9],2]//Union (* Harvey P. Dale, Jan 29 2017 *)
  • Python
    from math import isqrt
def A055237(n): return 5**(a:=(k:=isqrt(m:=n<<1))+(m>k*(k+1))-1)+5**(n-1-(a*(a+1)>>1)) # Chai Wah Wu, Apr 08 2025

Formula

a(n) = 5^(n-trinv(n))+5^trinv(n), where trinv(n) = floor((1+sqrt(1+8*n))/2) = A002262(n) and n-trinv(n) = A003056(n).
Regarded as a triangle T(n, k) = 5^n + 5^k, so as a sequence a(n) = 5^A002262(n) + 5^A003056(n).

A055258 Sums of two powers of 7.

Original entry on oeis.org

2, 8, 14, 50, 56, 98, 344, 350, 392, 686, 2402, 2408, 2450, 2744, 4802, 16808, 16814, 16856, 17150, 19208, 33614, 117650, 117656, 117698, 117992, 120050, 134456, 235298, 823544, 823550, 823592, 823886, 825944, 840350, 941192, 1647086, 5764802, 5764808, 5764850, 5765144, 5767202, 5781608, 5882450, 6588344, 11529602
Offset: 0

Views

Author

Henry Bottomley, Jun 22 2000

Keywords

Links

Crossrefs

Cf. A052216.
Equals 2*A073218.

Programs

  • Mathematica
    t = 7^Range[0, 9]; Select[Union[Flatten[Table[i + j, {i, t}, {j, t}]]], # <= t[[-1]] + 1 &] (* T. D. Noe, Oct 09 2011 *)
Total/@Tuples[7^Range[0,10],2]//Union (* Harvey P. Dale, Dec 31 2017 *)
  • Python
    from math import isqrt
def A055258(n): return 7**(a:=(k:=isqrt(m:=n<<1))+(m>k*(k+1))-1)+7**(n-1-(a*(a+1)>>1)) # Chai Wah Wu, Apr 08 2025

Formula

a(n) = 7^(n-trinv(n))+7^trinv(n), where trinv(n) = floor((1+sqrt(1+8*n))/2) = A002262(n) and n-trinv(n) = A003056(n)
Regarded as a triangle T(n, k) = 7^n + 7^k, so as a sequence a(n) = 7^A002262(n) + 7^A003056(n).

A055236 Sums of two powers of 4.

Original entry on oeis.org

2, 5, 8, 17, 20, 32, 65, 68, 80, 128, 257, 260, 272, 320, 512, 1025, 1028, 1040, 1088, 1280, 2048, 4097, 4100, 4112, 4160, 4352, 5120, 8192, 16385, 16388, 16400, 16448, 16640, 17408, 20480, 32768, 65537, 65540, 65552, 65600, 65792, 66560, 69632, 81920, 131072
Offset: 0

Views

Author

Henry Bottomley, Jun 22 2000

Keywords

Links

Crossrefs

Cf. A052216.
T(2n,n) gives 2*A026244.
T(n,n) gives A004171 = 2*A000302.
T(n,0) gives A052539.

Programs

  • Mathematica
    t = 4^Range[0, 9]; Select[Union[Flatten[Table[i + j, {i, t}, {j, t}]]], # <= t[[-1]] + 1 &] (* T. D. Noe, Oct 09 2011 *)
Union[Total/@Tuples[4^Range[0,9], 2]] (* Harvey P. Dale, Mar 25 2012 *)
  • Python
    from math import isqrt
def A055236(n): return (1<<((a:=(k:=isqrt(m:=n<<1))+(m>k*(k+1))-1)<<1))+(1<<(n-1-(a*(a+1)>>1)<<1)) # Chai Wah Wu, Apr 08 2025

Formula

a(n) = 4^(n-trinv(n))+4^trinv(n), where trinv(n) = floor((1+sqrt(1+8*n))/2) = A002262(n) and n-trinv(n) = A003056(n).
Regarded as a triangle T(n, k) = 4^n + 4^k, so as a sequence a(n) = 4^A002262(n) + 4^A003056(n).

A055257 Sums of two powers of 6.

Original entry on oeis.org

2, 7, 12, 37, 42, 72, 217, 222, 252, 432, 1297, 1302, 1332, 1512, 2592, 7777, 7782, 7812, 7992, 9072, 15552, 46657, 46662, 46692, 46872, 47952, 54432, 93312, 279937, 279942, 279972, 280152, 281232, 287712, 326592, 559872, 1679617, 1679622, 1679652, 1679832
Offset: 0

Views

Author

Henry Bottomley, Jun 22 2000

Keywords

Links

Crossrefs

Cf. A052216.

Programs

  • Mathematica
    t = 6^Range[0, 9]; Select[Union[Flatten[Table[i + j, {i, t}, {j, t}]]], # <= t[[-1]] + 1 &] (* T. D. Noe, Oct 09 2011 *)
  • Python
    from math import isqrt
def A055257(n): return 6**(a:=(k:=isqrt(m:=n<<1))+(m>k*(k+1))-1)+6**(n-1-(a*(a+1)>>1)) # Chai Wah Wu, Apr 08 2025

Formula

a(n) = 6^(n-trinv(n))+6^trinv(n), where trinv(n) = floor((1+sqrt(1+8*n))/2) = A002262(n) and n-trinv(n) = A003056(n)
Regarded as a triangle T(n, k) = 6^n + 6^k, so as a sequence a(n) = 6^A002262(n) + 6^A003056(n).

A055259 Sums of two powers of 8.

Original entry on oeis.org

2, 9, 16, 65, 72, 128, 513, 520, 576, 1024, 4097, 4104, 4160, 4608, 8192, 32769, 32776, 32832, 33280, 36864, 65536, 262145, 262152, 262208, 262656, 266240, 294912, 524288, 2097153, 2097160, 2097216, 2097664, 2101248, 2129920, 2359296, 4194304, 16777217
Offset: 0

Views

Author

Henry Bottomley, Jun 22 2000

Keywords

Links

Crossrefs

Cf. A052216.

Programs

  • Mathematica
    Union[Total/@Tuples[8^Range[0,10], {2}]]  (* Harvey P. Dale, Mar 13 2011 *)
  • Python
    def valuation(n, b):
  v = 0
  while n > 1: n //= b; v += 1
  return v
def aupto(lim):
  pows8 = [8**i for i in range(valuation(lim-1, 8) + 1)]
  sum_pows8 = sorted([a+b for i, a in enumerate(pows8) for b in pows8[i:]])
  return [s for s in sum_pows8 if s <= lim]
print(aupto(16777217)) # Michael S. Branicky, Feb 09 2021
  • Python
    from math import isqrt
def A055259(n): return (1<<3*(a:=(k:=isqrt(m:=n<<1))+(m>k*(k+1))-1))+(1<<3*(n-1-(a*(a+1)>>1))) # Chai Wah Wu, Apr 08 2025

Formula

a(n) = 8^(n-trinv(n))+8^trinv(n), where trinv(n) = floor((1+sqrt(1+8*n))/2) = A002262(n) and n-trinv(n) = A003056(n)
Regarded as a triangle T(n, k) = 8^n + 8^k, so as a sequence a(n) = 8^A002262(n) + 8^A003056(n).

A055260 Sums of two powers of 9.

Original entry on oeis.org

2, 10, 18, 82, 90, 162, 730, 738, 810, 1458, 6562, 6570, 6642, 7290, 13122, 59050, 59058, 59130, 59778, 65610, 118098, 531442, 531450, 531522, 532170, 538002, 590490, 1062882, 4782970, 4782978, 4783050, 4783698, 4789530, 4842018, 5314410, 9565938, 43046722
Offset: 0

Views

Author

Henry Bottomley, Jun 22 2000

Keywords

Links

Crossrefs

Cf. A001019, A052216.

Programs

  • Mathematica
    t = 9^Range[0, 9]; Select[Union[Flatten[Table[i + j, {i, t}, {j, t}]]], # <= t[[-1]] + 1 &] (* T. D. Noe, Oct 09 2011 *)
Total/@Tuples[9^Range[0,10],2]//Union (* Harvey P. Dale, Jul 03 2019 *)
  • Python
    def valuation(n, b):
  v = 0
  while n > 1: n //= b; v += 1
  return v
def aupto(lim):
  pows = [9**i for i in range(valuation(lim-1, 9) + 1)]
  sum_pows = sorted([a+b for i, a in enumerate(pows) for b in pows[i:]])
  return [s for s in sum_pows if s <= lim]
print(aupto(43046722)) # Michael S. Branicky, Feb 10 2021
  • Python
    from math import isqrt
def A055260(n): return 9**(a:=(k:=isqrt(m:=n<<1))+(m>k*(k+1))-1)+9**(n-1-(a*(a+1)>>1)) # Chai Wah Wu, Apr 09 2025

Formula

a(n) = 9^(n-trinv(n))+9^trinv(n), where trinv(n) = floor((1+sqrt(1+8*n))/2) = A002262(n) and n-trinv(n) = A003056(n)
Regarded as a triangle T(n, k) = 9^n + 9^k, so as a sequence a(n) = 9^A002262(n) + 9^A003056(n).

A055261 Sums of two powers of 16.

Original entry on oeis.org

2, 17, 32, 257, 272, 512, 4097, 4112, 4352, 8192, 65537, 65552, 65792, 69632, 131072, 1048577, 1048592, 1048832, 1052672, 1114112, 2097152, 16777217, 16777232, 16777472, 16781312, 16842752, 17825792, 33554432, 268435457
Offset: 1

Views

Author

Henry Bottomley, Jun 22 2000

Keywords

Examples

			a(4) = 272 = 16^2+16^1.

Links

Crossrefs

Cf. A052216.

Programs

  • Maple
    A055261:= proc(n)
     local p1, p2;
     p1:= floor((sqrt(8*n-7)-1)/2);
     p2:= n - 1 - p1*(p1+1)/2;
     16^p1 + 16^p2
end proc; # Robert Israel, Apr 07 2014
  • Python
    from math import isqrt
def A055261(n): return (1<<((a:=(k:=isqrt(m:=n<<1))+(m>k*(k+1))-1)<<2))+(1<<(n-1-(a*(a+1)>>1)<<2)) # Chai Wah Wu, Apr 08 2025

Formula

a(n) = 16^(n-trinv(n))+16^trinv(n), where trinv(n) = floor((1+sqrt(1+8*n))/2) = A002262(n) and n-trinv(n) = A003056(n).
Regarded as a triangle T(n, k)=16^n+16^k, so as a sequence a(n) =16^A002262(n)+16^A003056(n).
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