A010061 -id:A010061 - cp's OEIS frontend

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

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

Amiram Eldar, Nov 27 2020

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

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

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Self Number.
Wikipedia, Primorial number system.
Wikipedia, Self number.
Index entries for Colombian or self numbers and related sequences

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

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

Amiram Eldar, Nov 27 2020

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

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

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Self Number.
Eric Weisstein's World of Mathematics, Zeckendorf Representation.
Wikipedia, Self number.
Index entries for Colombian or self numbers and related sequences

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

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

Amiram Eldar, Nov 27 2020

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

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

Amiram Eldar, Table of n, a(n) for n = 1..10000
J. L. Brown, Jr., A new characterization of the Fibonacci numbers, Fibonacci Quarterly, Vol. 3, No. 1 (1965) pp. 1-8.
Eric Weisstein's World of Mathematics, Self Number.
Wikipedia, Self number.
Index entries for Colombian or self numbers and related sequences

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

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

Amiram Eldar, Nov 27 2020

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

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

Amiram Eldar, Table of n, a(n) for n = 1..10000
F. Michel Dekking, The sum of digits function of the base phi expansion of the natural numbers, arXiv:1911.10705 [math.NT], 2019.
Ron Knott, Using Powers of Phi to represent Integers (Base Phi).
Eric Weisstein's World of Mathematics, Self Number.
Wikipedia, Golden ratio base.
Wikipedia, Self number.
Index entries for Colombian or self numbers and related sequences

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

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

Amiram Eldar, Nov 27 2020

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

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

Amiram Eldar, Table of n, a(n) for n = 1..10000
Rosalind Guaraldo, On the Density of the Image Sets of Certain Arithmetic Functions - III, The Fibonacci Quarterly, Vol. 16, No. 6 (1978), pp. 481-488.
Eric Weisstein's World of Mathematics, Self Number.
Wikipedia, Factorial number system.
Wikipedia, Self number.
Index entries for Colombian or self numbers and related sequences

Cf. A003052, A007623, A010061, A010064, A010067, A010070, A034968, A118363, A339211, A339212, A339213, A339215.

Cf. also A219650, A219658, A230412.

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

A228083 Table of binary Self-numbers and their descendants; square array T(r,c), with row r>=1, column c>=1, read by antidiagonals.

Original entry on oeis.org

1, 2, 4, 3, 5, 6, 5, 7, 8, 13, 7, 10, 9, 16, 15, 10, 12, 11, 17, 19, 18, 12, 14, 14, 19, 22, 20, 21, 14, 17, 17, 22, 25, 22, 24, 23, 17, 19, 19, 25, 28, 25, 26, 27, 30, 19, 22, 22, 28, 31, 28, 29, 31, 34, 32, 22, 25, 25, 31, 36, 31, 33, 36, 36, 33, 37

Offset: 1

Antti Karttunen, Aug 09 2013

			The top-left corner of the square array:
   1,  2,  3,  5,  7, 10, 12, 14, ...
   4,  5,  7, 10, 12, 14, 17, 19, ...
   6,  8,  9, 11, 14, 17, 19, 22, ...
  13, 16, 17, 19, 22, 25, 28, 31, ...
  15, 19, 22, 25, 28, 31, 36, 38, ...
  18, 20, 22, 25, 28, 31, 36, 38, ...
  21, 24, 26, 29, 33, 35, 38, 41, ...
  23, 27, 31, 36, 38, 41, 44, 47, ...
  ...
The non-initial terms on each row are obtained by adding to the preceding term the number of 1-bits in its binary representation (A000120).

First column: A010061. First row: A010062. Transpose: A228084. See A151942 for decimal analog.

Cf. also A000120, A092391, A227643, A228082, A228085, A228086, A228087, A228088, A228091, A218254.

nmax0 = 100;
nmax := Length[col[1]];
col[1] = Table[n + DigitCount[n, 2, 1], {n, 0, nmax0}] // Complement[Range[Last[#]], #]&;
col[k_] := col[k] = col[k - 1] + DigitCount[col[k-1], 2, 1];
T[n_, k_] := col[k][[n]];
Table[T[n-k+1, k], {n, 1, nmax}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Nov 28 2020 *)

T(r,1) are those numbers not of form n + sum of binary digits of n (binary Self numbers) = A010061(r);

T(r,c) = T(r,c-1) + sum of binary digits of T(r,c-1) = A092391(T(r,c-1)).

A230058 Numbers of the form k + wt(k) for at least two distinct k, where wt(k) = A000120(k) is the binary weight of k.

Original entry on oeis.org

5, 14, 17, 19, 22, 31, 33, 36, 38, 47, 50, 52, 55, 64, 67, 70, 79, 82, 84, 87, 96, 98, 101, 103, 112, 115, 117, 120, 129, 131, 132, 134, 143, 146, 148, 151, 160, 162, 165, 167, 176, 179, 181, 184, 193, 196, 199, 208, 211, 213, 216, 225, 227, 230, 232, 241, 244

Offset: 1

Matthew C. Russell, Oct 07 2013

The positions of entries greater than 1 in A228085, or numbers that appear multiple times in A092391.

Numbers that can be expressed as the sum of distinct terms of the form 2^n+1, n=0,1,... in multiple ways.

			5 = 3 + 2 = 4 + 1, so 5 is in this list.

Vincenzo Librandi and Donovan Johnson, Table of n, a(n) for n = 1..10000 (first 1000 terms from Vincenzo Librandi)
Index entries for Colombian or self numbers and related sequences

Cf. A000120, A010061, A228088, A230091, A230092.

Sort[Transpose[Select[Tally[Table[k + Total[IntegerDigits[k, 2]], {k, 0, 300}]], #[[2]] > 1 &]][[1]]] (* T. D. Noe, Oct 09 2013 *)

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

Amiram Eldar, Mar 19 2021

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?

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

Amiram Eldar, Table of n, a(n) for n = 1..10000
Walter Penney, A "binary" system for complex numbers, Journal of the ACM, Vol. 12, No. 2 (1965), pp. 247-248.
Eric Weisstein's World of Mathematics, Self Number.
Wikipedia, Self number.

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).

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

Jean-François Alcover, May 13 2014

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

			0.2526602590088829221550627143278941418252...

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.

G. Troi and U. Zannier, Note on the density constant in the distribution of self-numbers - II, Bollettino dell'Unione Matematica Italiana, Serie 8, Vol. 2-B, No. 2 (1999), pp. 397-399; alternative link.
Umberto Zannier, On the distribution of self-numbers, Proc. Amer. Math. Soc., Vol. 85, No. 1 (1982), pp. 10-14.
Index entries for transcendental numbers

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

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]]

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

N. J. A. Sloane, Jan 07 2022

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

Cf. A010061, A228082.

Complement of A010064.

g:= n -> n+convert(convert(n,base,4),`+`):
select(`<=`,map(g, {$0..200}),200); # Robert Israel, Jan 11 2022

cp's OEIS Frontend

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

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

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

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

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

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A228083 Table of binary Self-numbers and their descendants; square array T(r,c), with row r>=1, column c>=1, read by antidiagonals.

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A230058 Numbers of the form k + wt(k) for at least two distinct k, where wt(k) = A000120(k) is the binary weight of k.

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

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