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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A230636 Numbers n such that m + (sum of digits in base-4 representation of m) = n has exactly four solutions.

Original entry on oeis.org

16777234, 33554451, 50331668, 83886099, 100663316, 117440533, 150994964, 167772181, 184549398, 218103829, 234881046, 251658263, 268435476, 268435478, 285212691, 301989908, 318767125, 352321556, 369098773, 385875990, 419430421, 436207638, 452984855, 486539286
Offset: 1

Views

Author

Donovan Johnson and N. J. A. Sloane, Oct 31 2013

Keywords

Links

Crossrefs

Related base-4 sequences: A053737, A230631, A230632, A010064, A230633, A230634, A230635, A230636, A230637, A230638, A010065 (trajectory of 1)

A230637 Leading power of 4 in A230638.

Original entry on oeis.org

2, 7, 12, 5468, 10924, 5597880, 16777229
Offset: 2

Views

Author

N. J. A. Sloane, Oct 31 2013

Keywords

Comments

a(9) = ( 4^5468 + 2*4^12 + 39 ) / 3.
a(10) = 4^5468 + 13.
a(11) = ( 4^10924 + 2*4^5468 + 16407 ) / 3.
a(12) = 4^10924 + 10925
a(13) = ( 4^5597880 + 3*4^10924 + 32793 ) / 3.
a(14) = ( 2*4^5597880 + 32812 ) / 3.
a(15) = ( 4^16777229 + 4^5597880 + 2*4^12 + 16427 ) / 3.
a(16) = ( 2*4^16777229 + 4^13 + 42 ) / 3.

Links

Crossrefs

Cf. A230638.
Related base-4 sequences: A053737, A230631, A230632, A010064, A230633, A230634, A230635, A230636, A230637, A230638, A010065 (trajectory of 1)

Extensions

Terms a(8) onward from Max Alekseyev, Oct 31 2013

A339215 Primorial-base self numbers: numbers not of the form k + A276150(k).

Original entry on oeis.org

1, 4, 11, 18, 25, 32, 35, 42, 49, 56, 63, 66, 73, 80, 87, 94, 97, 104, 111, 118, 125, 128, 135, 142, 149, 156, 159, 166, 173, 180, 187, 190, 197, 204, 229, 236, 243, 246, 253, 260, 267, 274, 277, 284, 291, 298, 305, 308, 315, 322, 329, 336, 339, 346, 353, 360
Offset: 1

Views

Author

Amiram Eldar, Nov 27 2020

Keywords

Comments

Analogous to self numbers (A003052) using primorial base representation (A049345) instead of decimal expansion.
The numbers of terms that do not exceed 10^k, for k = 0, 1, ..., are 1, 2, 17, 150, 1469, 14669, 146680, 1466723, 14667162, 146671527, 1466715137, ... . Apparently, the asymptotic density of this sequence exists and equals 0.1466715... . - Amiram Eldar, Aug 08 2025

References

  • József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 4, pp. 384-386.

Links

Crossrefs

Cf. A003052, A010061, A010064, A010067, A010070, A049345, A276150, A333426, A339211, A339212, A339213, A339214.

Programs

  • Mathematica
    max = 4; bases = Prime@Range[max, 1, -1]; m = Times @@ bases; s[n_] := n + Plus @@ IntegerDigits[n, MixedRadix[bases]]; Complement[Range[m], Array[s, m]]

A339211 Zeckendorf self numbers: numbers not of the form k + A007895(k).

Original entry on oeis.org

1, 5, 7, 10, 19, 21, 27, 29, 32, 36, 40, 42, 45, 54, 61, 63, 66, 75, 77, 83, 85, 88, 95, 97, 100, 109, 111, 117, 119, 122, 126, 130, 132, 135, 144, 146, 150, 152, 155, 164, 166, 172, 174, 177, 181, 185, 187, 190, 199, 206, 208, 211, 220, 222, 228, 230, 233, 239
Offset: 1

Views

Author

Amiram Eldar, Nov 27 2020

Keywords

Comments

Analogous to self numbers (A003052) using the Zeckendorf representation (A014417) instead of decimal expansion.
The numbers of terms that do not exceed 10^k, for k = 0, 1, ..., are 1, 4, 25, 236, 2351, 23495, 234949, 2349463, 23494586, 234945839, 2349458364, ... . Apparently, the asymptotic density of this sequence exists and equals 0.23494583... . - Amiram Eldar, Aug 08 2025

References

  • József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 4, p. 384-386.

Links

Crossrefs

Cf. A003052, A007895, A010061, A010064, A010067, A010070, A014417, A328208, A339212, A339213, A339214, A339215.

Programs

  • Mathematica
    z[n_] := n + Length[DeleteCases[NestWhileList[# - Fibonacci[Floor[Log[Sqrt[5]*# + 3/2]/Log[GoldenRatio]]] &, n, # > 1 &], 0]]; m = 250; Complement[Range[m], Array[z, m]] (* after Alonso del Arte at A007895 *)

A339212 Dual-Zeckendorf self numbers: numbers not of the form k + A112310(k).

Original entry on oeis.org

1, 4, 8, 10, 14, 17, 19, 28, 31, 33, 39, 41, 50, 53, 55, 59, 63, 66, 68, 74, 76, 85, 88, 90, 97, 106, 109, 111, 115, 119, 122, 124, 130, 132, 141, 144, 146, 153, 156, 158, 164, 166, 175, 178, 180, 187, 196, 199, 201, 205, 209, 212, 214, 220, 222, 231, 234, 236
Offset: 1

Views

Author

Amiram Eldar, Nov 27 2020

Keywords

Comments

Analogous to self numbers (A003052) using the dual Zeckendorf representation (A104326) instead of decimal expansion.

References

  • József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 4, p. 384-386.

Links

Crossrefs

Cf. A003052, A010061, A010064, A010067, A010070, A104326, A112310, A328212, A339211, A339213, A339214, A339215.

Programs

  • Mathematica
    fibTerms[n_] := Module[{k = Ceiling[Log[GoldenRatio, n*Sqrt[5]]], t = n, fr = {}}, While[k > 1, If[t >= Fibonacci[k], AppendTo[fr, 1]; t = t - Fibonacci[k], AppendTo[fr, 0]]; k--]; fr]; dzs[n_] := n + Module[{v = fibTerms[n]}, nv = Length[v]; i = 1; While[i <= nv - 2, If[v[[i]] == 1 && v[[i + 1]] == 0 && v[[i + 2]] == 0, v[[i]] = 0; v[[i + 1]] = 1; v[[i + 2]] = 1; If[i > 2, i -= 3]]; i++]; i = Position[v, _?(# > 0 &)]; If[i == {}, 0, Total[v[[i[[1, 1]] ;; -1]]]]]; m = 240; Complement[Range[m], Array[dzs, m]]

A339213 Phi-base self numbers: positive numbers not of the form k + A055778(k).

Original entry on oeis.org

1, 3, 6, 10, 12, 15, 19, 23, 26, 30, 32, 38, 41, 43, 52, 55, 59, 61, 64, 68, 72, 75, 79, 81, 86, 89, 91, 97, 101, 104, 108, 110, 115, 118, 120, 126, 131, 135, 137, 140, 144, 148, 151, 155, 157, 163, 166, 168, 177, 180, 184, 186, 189, 193, 197, 200, 204, 206, 213
Offset: 1

Views

Author

Amiram Eldar, Nov 27 2020

Keywords

Comments

Analogous to self numbers (A003052) using base phi (A130600) instead of base 10.

References

  • József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 4, p. 384-386.

Links

Crossrefs

Cf. A003052, A010061, A010064, A010067, A010070, A055778, A130600, A331083, A334308, A339211, A339212, A339214, A339215.

Programs

  • Mathematica
    s[1] = 2; s[n_] := n + Plus @@ RealDigits[n, GoldenRatio, 2*Ceiling[Log[GoldenRatio, n]]][[1]]; m = 220; Complement[Range[m], Array[s, m]]

A339214 Factorial-base self numbers: numbers not of the form k + A034968(k).

Original entry on oeis.org

1, 4, 11, 18, 36, 43, 61, 68, 86, 93, 111, 118, 125, 132, 139, 157, 164, 182, 189, 207, 214, 232, 239, 246, 253, 260, 278, 285, 303, 310, 328, 335, 353, 360, 367, 374, 381, 399, 406, 424, 431, 449, 456, 474, 481, 488, 495, 502, 520
Offset: 1

Views

Author

Amiram Eldar, Nov 27 2020

Keywords

Comments

Analogous to self numbers (A003052) using factorial base representation (A007623) instead of decimal expansion.
The numbers of terms that do not exceed 10^k, for k = 0, 1, ..., are 1, 2, 10, 90, 878, 8749, 87455, 874499, 8744934, 87449296, 874492907, ... . Apparently, the asymptotic density of this sequence exists and equals 0.08744929... . - Amiram Eldar, Aug 08 2025

References

  • József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 4, p. 384-386.

Links

Crossrefs

Cf. A003052, A007623, A010061, A010064, A010067, A010070, A034968, A118363, A339211, A339212, A339213, A339215.
Cf. also A219650, A219658, A230412.

Programs

  • Mathematica
    max = 6; s[n_] := n + Plus @@ IntegerDigits[n, MixedRadix[Range[max, 2, -1]]]; m = max!; Complement[Range[m], Array[s, m]]

A342729 Self numbers in base i-1: numbers not of the form k + A066323(k).

Original entry on oeis.org

1, 3, 5, 7, 9, 22, 24, 26, 39, 41, 43, 56, 58, 60, 73, 75, 77, 90, 92, 94, 107, 109, 111, 136, 138, 140, 153, 155, 157, 170, 172, 174, 199, 201, 203, 216, 218, 220, 233, 235, 237, 262, 264, 266, 279, 281, 283, 296, 298, 300, 313, 315, 317, 330, 332, 334, 347, 349
Offset: 1

Views

Author

Amiram Eldar, Mar 19 2021

Keywords

Comments

Equivalently, self numbers in base -4, since A066323(k) is also the sum of the digits of k in base -4.
Analogous to self numbers (A003052) using base i-1 representation (A271472) instead of decimal expansion.
The number of terms not exceeding 10^k, for k=1,2,..., is 5, 20, 155, 1507, 15008, 150007, 1500014, 15000011. Is the asymptotic density of this sequence exactly 3/20?

References

  • József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 4, p. 384-386.

Links

Crossrefs

Cf. A007608, A066323, A066321, A271472, A342725, A342726, A342727, A342728.
Similar sequences: A003052 (decimal), A010061 (binary), A010064 (base 4), A010067 (base 6), A010070 (base 8), A339211 (Zeckendorf), A339212 (dual Zeckendorf), A339213 (base phi), A339214 (factorial base), A339215 (primorial base).

Programs

  • Mathematica
    s[n_] := Module[{v = {{0, 0, 0, 0}, {0, 0, 0, 1}, {1, 1, 0, 0}, {1, 1, 0, 1}}}, Plus @@ Flatten @ v[[1 + Reverse @ Most[Mod[NestWhileList[(# - Mod[#, 4])/-4 &, n, # != 0 &], 4]]]]]; f[n_] := n + s[n]; m = 1000; Complement[Range[m], Select[Union@Array[f, m], # <= m &]]

A242403 Decimal expansion of the binary self-numbers density constant.

Original entry on oeis.org

2, 5, 2, 6, 6, 0, 2, 5, 9, 0, 0, 8, 8, 8, 2, 9, 2, 2, 1, 5, 5, 0, 6, 2, 7, 1, 4, 3, 2, 7, 8, 9, 4, 1, 4, 1, 8, 2, 5, 2, 1, 9, 3, 3, 9, 6, 2, 9, 7, 8, 4, 6, 1, 3, 0, 1, 6, 8, 6, 2, 1, 7, 2, 2, 9, 2, 2, 8, 0, 5, 4, 8, 4, 4, 7, 6, 6, 3, 2, 5, 6, 6, 9, 5, 9, 1, 4, 2, 4, 4, 7, 9, 3, 8, 6, 8, 8, 9, 4, 9
Offset: 0

Views

Author

Jean-François Alcover, May 13 2014

Keywords

Comments

This constant is transcendental (Troi and Zannier, 1999). - Amiram Eldar, Nov 28 2020

Examples

			0.2526602590088829221550627143278941418252...

References

  • Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 179.
  • József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 4, p. 384-386.
  • G. Troi and U. Zannier, Note on the density constant in the distribution of self-numbers, Bollettino dell'Unione Matematica Italiana, Serie 7, Vol. 9-A, No. 1 (1995), pp. 143-148.

Links

Crossrefs

Cf. A010061 (binary self numbers), A003052 (decimal self numbers), A010064, A010067, A010070, A092391, A228082.

Programs

  • Mathematica
    m0 = 100; dm = 100; digits = 100; Clear[lambda]; lambda[m_] := lambda[m] = Total[1/2^Union[Table[n + Total[IntegerDigits[n, 2]], {n, 0, m}]]]^2/8 // N[#, 2*digits]& // RealDigits[#, 10, 2*digits]& // First; lambda[m0]; lambda[m = m0 + dm]; While[lambda[m] != lambda[m - dm], Print["m = ", m]; m = m + dm]; lambda[m][[1 ;; digits]]

Formula

Equals (1/8)*(Sum_{n not a binary self-number} 1/2^n)^2.

A349829 Numbers k such that there is a number m with m + s_4(m) = k, where s_b(m) = sum of digits in base-b expansion of m.

Original entry on oeis.org

0, 2, 4, 5, 6, 7, 9, 10, 11, 12, 14, 15, 16, 17, 19, 21, 22, 23, 24, 26, 27, 28, 29, 31, 32, 33, 34, 36, 38, 39, 40, 41, 43, 44, 45, 46, 48, 49, 50, 51, 53, 55, 56, 57, 58, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 74, 75, 76, 77, 79, 80, 81, 82, 84, 86, 87, 88, 89, 91, 92, 93, 94, 96, 97, 98, 99
Offset: 1

Views

Author

N. J. A. Sloane, Jan 07 2022

Keywords

Links

Crossrefs

Cf. A010061, A228082.
Complement of A010064.

Programs

  • Maple
    g:= n -> n+convert(convert(n,base,4),`+`):
select(`<=`,map(g, {$0..200}),200); # Robert Israel, Jan 11 2022
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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