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.

Previous Showing 41-47 of 47 results.

A230301 Positive numbers not of the form m + wt(m-1), m >= 1.

Original entry on oeis.org

2, 5, 7, 14, 16, 19, 22, 24, 31, 33, 38, 40, 47, 49, 52, 55, 57, 64, 72, 79, 81, 84, 87, 89, 96, 98, 103, 105, 112, 114, 117, 120, 122, 129, 131, 134, 136, 143, 145, 148, 151, 153, 160, 162, 167, 169, 176, 178, 181, 184, 186, 193, 201, 208, 210, 213, 216, 218, 225, 227, 232, 234, 241, 243, 246, 249, 251, 271, 273, 276
Offset: 1

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Author

N. J. A. Sloane, Oct 23 2013

Keywords

Comments

wt(m) = A000120(m).
These are numbers k such that A228085(2^k) = A228085(k-1) = 0, or numbers k such that 2^k is a binary self number (A010061). - Amiram Eldar, Feb 23 2021

Links

Crossrefs

Cf. A000120, A010061, A228085, A230300, A232491.

Formula

a(n) = A010061(n) + 1.

A232229 a(1)=9; thereafter a(n) = 8*10^(n-1) + 8 + a(n-1).

Original entry on oeis.org

9, 97, 905, 8913, 88921, 888929, 8888937, 88888945, 888888953, 8888888961, 88888888969, 888888888977, 8888888888985, 88888888888993, 888888888889001, 8888888888889009, 88888888888889017, 888888888888889025, 8888888888888889033, 88888888888888889041, 888888888888888889049, 8888888888888888889057
Offset: 1

Views

Author

N. J. A. Sloane, Nov 22 2013

Keywords

Comments

An infinite subsequence of A003052.

Links

Crossrefs

Cf. A003052, A010061.

Programs

  • Maple
    f:=proc(n) option remember; if n=1 then 9 else
8*10^(n-1)+8+f(n-1); fi; end;
[seq(f(n),n=1..40)];
  • Mathematica
    RecurrenceTable[{a[1]==9,a[n]==8*10^(n-1)+8+a[n-1]},a,{n,30}] (* Harvey P. Dale, May 19 2018 *)

Formula

G.f.: x*(-9+11*x+70*x^2) / ( (10*x-1)*(x-1)^2 ). a(n) = (8*10^n-71)/9+8*n. - R. J. Mathar, Nov 24 2013

Extensions

Definition corrected by Harvey P. Dale, May 19 2018

A344512 a(n) is the least number larger than 1 which is a self number in all the bases 2 <= b <= n.

Original entry on oeis.org

4, 13, 13, 13, 287, 287, 2971, 2971, 27163, 27163, 90163, 90163, 5940609, 5940609, 6069129, 6069129, 276404649, 276404649
Offset: 2

Views

Author

Amiram Eldar, May 21 2021

Keywords

Comments

Since the sequence of base-b self numbers for odd b is the sequence of the odd numbers (A005408) (Joshi, 1973), all the terms beyond a(2) are odd numbers.
For the corresponding sequence with only even bases, see A344513.
a(20) > 1.5*10^10, if it exists.

Examples

			a(2) = 4 since the least binary self number after 1 is A010061(2) = 4.
a(3) = 13 since the least binary self number after 1 which is also a self number in base 3 is A010061(4) = 13.

References

  • Vijayshankar Shivshankar Joshi, Contributions to the theory of power-free integers and self-numbers, Ph.D. dissertation, Gujarat University, Ahmedabad (India), October, 1973.
  • 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, A339211, A339212, A339213, A339214, A339215, A342729, A344513.
Similar sequences: A016038, A217705, A225427, A226320, A228768, A258107.

Programs

  • Mathematica
    s[n_, b_] := n + Plus @@ IntegerDigits[n, b]; selfQ[n_, b_] := AllTrue[Range[n, n - (b - 1) * Ceiling @ Log[b, n], -1], s[#, b] != n &]; a[2] = 4; a[b_] := a[b] = Module[{n = a[b - 1]}, While[! AllTrue[Range[2, b], selfQ[n, #] &], n++]; n]; Array[a, 10, 2]

Formula

a(2*n+1) = a(2*n) for n >= 2.

A344513 a(n) is the least number larger than 1 which is a self number in all the even bases b = 2*k for 1 <= k <= n.

Original entry on oeis.org

4, 13, 287, 294, 6564, 90163, 1136828, 3301262, 276404649, 5643189146
Offset: 1

Views

Author

Amiram Eldar, May 21 2021

Keywords

Comments

Joshi (1973) proved that for all odd b the sequence of base-b self numbers is the sequence of odd numbers (A005408). Therefore, in this sequence the bases are restricted to even values. For the corresponding sequence with both odd and even bases, see A344512.

Examples

			a(1) = 4 since the least binary self number after 1 is A010061(2) = 4.
a(2) = 13 since the least binary self number after 1 which is also a self number in base 2*2 = 4 is A010061(4) = A010064(4) = 13.

References

  • Vijayshankar Shivshankar Joshi, Contributions to the theory of power-free integers and self-numbers, Ph.D. dissertation, Gujarat University, Ahmedabad (India), October, 1973.
  • 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, A339211, A339212, A339213, A339214, A339215, A342729, A344512.
Similar sequences: A016038, A217705, A225427, A226320, A228768, A258107.

Programs

  • Mathematica
    s[n_, b_] := n + Plus @@ IntegerDigits[n, b]; selfQ[n_, b_] := AllTrue[Range[n, n - (b - 1) * Ceiling @ Log[b, n], -1], s[#, b] != n &]; a[1] = 4; a[n_] := a[n] = Module[{k = a[n - 1]}, While[! AllTrue[Range[1, n], selfQ[k, 2*#] &], k++]; k]; Array[a, 7]

A386568 The number of binary self numbers not exceeding 10^n.

Original entry on oeis.org

1, 3, 26, 254, 2526, 25270, 252666, 2526605, 25266035, 252660259, 2526602596, 25266025903, 252660259016, 2526602590093
Offset: 0

Views

Author

Amiram Eldar, Jul 26 2025

Keywords

Examples

			There are 3 binary self numbers that do no exceed 10: 1, 4 and 6. Hence a(1) = 3.

Crossrefs

Cf. A010061, A242403, A382452 (decimal analog), A386569.

Programs

  • Mathematica
    selfQ[n_] := AllTrue[Range[n, n - Floor@Log2[n], -1], # + DigitCount[#, 2, 1] != n &]; a[n_] := Count[Range[10^n], _?selfQ]; Array[a, 6, 0]

Formula

Limit_{n->oo} a(n)/10^n = A242403.

A386569 The number of binary self numbers not exceeding 2^n.

Original entry on oeis.org

1, 1, 2, 3, 5, 10, 18, 34, 67, 131, 261, 520, 1037, 2073, 4143, 8283, 16562, 33121, 66237, 132471, 264938, 529870, 1059740, 2119473, 4238941, 8477878, 16955748, 33911492, 67822978, 135645949, 271291894, 542583782, 1085167557, 2170335106, 4340670206, 8681340402
Offset: 0

Views

Author

Amiram Eldar, Jul 26 2025

Keywords

Examples

			There are 3 binary self numbers that do no exceed 2^3 = 8: 1, 4 and 6. Hence a(3) = 3.

Crossrefs

Cf. A010061, A242403, A337078, A382452, A386568.

Programs

  • Mathematica
    selfQ[n_] := AllTrue[Range[n, n - Floor@Log2[n], -1], # + DigitCount[#, 2, 1] != n &]; a[n_] := Count[Range[2^n], _?selfQ]; Array[a, 16, 0]

Formula

Limit_{n->oo} a(n)/2^n = A242403.

A354254 a(n) is the least m >= 0 such that n = f^k(m) for some k >= 0 (where f^k denotes the k-th iterate of A092391).

Original entry on oeis.org

0, 1, 1, 1, 4, 1, 6, 1, 6, 6, 1, 6, 1, 13, 1, 15, 13, 1, 18, 1, 18, 21, 1, 23, 21, 1, 21, 23, 1, 21, 30, 1, 32, 21, 30, 21, 1, 37, 1, 39, 37, 1, 37, 39, 1, 37, 46, 1, 48, 37, 46, 51, 1, 46, 54, 1, 56, 46, 54, 56, 1, 46, 54, 63, 1, 1, 46, 1, 46, 63, 1, 71, 63
Offset: 0

Views

Author

Rémy Sigrist, May 21 2022

Keywords

Examples

			The first terms, alongside f(n), are:
  n   a(n)  f(n)
  --  ----  ----
   0     0     0
   1     1     2
   2     1     3
   3     1     5
   4     4     5
   5     1     7
   6     6     8
   7     1    10
   8     6     9
   9     6    11
  10     1    12
  11     6    14
  12     1    14
  13    13    16
  14     1    17

Crossrefs

Cf. A010061, A010062, A092391, A179016.

Programs

  • PARI
    a = vector(73, n, n-1); for (n=0, #a-1, m=n+hammingweight(n); if (m<#a, a[1+m]=min(a[1+n],a[1+m]))); print (a)

Formula

a(n) = n iff n = 0 or n belongs to A010061.
a(n) = 1 iff n belongs to A010062.
Previous Showing 41-47 of 47 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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