Colin Rose - cp's OEIS frontend

A193069 Pretty wild narcissistic numbers - numbers that pwn: - an integer n that can be expressed using just the digits of n (each digit used once only and in order from left to right) and the operators + - * / ^ ! and the radical symbol. Concatenation is allowed.

24, 36, 71, 119, 120, 127, 143, 144, 145, 216, 240, 343, 354, 355, 360, 384, 456, 595, 660, 693, 713, 715, 719, 720, 721, 722, 723, 724, 725, 726, 727, 728, 729, 733, 736, 744, 799, 936

Offset: 1

Colin Rose, Aug 08 2011

Unlike 'radical narcissistic numbers', the pretty wild variety are allowed to use factorials.

Pretty wild narcissistic numbers nest both radical narcissistic numbers and ordered (nice) Friedman numbers.

			24 = (2 Sqrt[4])! , 36 = 3! 6,  127 = -1 + 2^7

J. S. Madachy, Mathematics on Vacation, Thomas Nelson and Sons, (1966), pp. 163 - 175.
Colin Rose, "Radical Narcissistic numbers", J. Recreational Mathematics, vol. 33, (2004-2005), pp. 250-254.

C. Rose, Pretty Wild Narcissistic Numbers - numbers that pwn
Inder J. Taneja, Selfie Numbers-IV: Addition, Subtraction and Factorial, RGMIA Research Report Collection, 19 (2016), #163, pp. 1-42.
Inder J. Taneja, Selfie Numbers, Fibonacci Sequence and Selfie Fractions, RGMIA Collection, 19 (2016), #167, pp. 1-24.
Wikipedia, Friedman number

Cf. A080035, A119710

A119710 Radical narcissistic numbers: numbers n that can be expressed using just the digits of n (each digit used once only and in order from left to right) and the operators + - * / ^ and the radical symbol, but which are not already 'Good' Friedman numbers (i.e., the radical is needed for the solution to exist). Concatenation is allowed.

Original entry on oeis.org

729, 1296, 1764, 2378, 2744, 2746, 3645, 4372, 4374, 4913, 5184, 6495, 6859, 8192

Offset: 1

Colin Rose, Jun 10 2006

There are only 14 radical narcissistic integers n < 10000.

			729 = (7 + 2)^sqrt(9);
2378 = -23 + sqrt(7^8).

J. S. Madachy, Mathematics on Vacation, Thomas Nelson and Sons, (1966), pp. 163 - 175.
Colin Rose, "Radical Narcissistic numbers", J. Recreational Mathematics, vol. 33, (2004-2005), pp. 250-254.

C. Rose, Pretty Wild Narcissistic Numbers - numbers that pwn (2006)
Wikipedia, Friedman number

Cf. A080035.

Mathematica code to be available at: http://www.numq.com/pwn/

A118067 Number of (directed) Hamiltonian paths in the 3 X n knight graph.

Original entry on oeis.org

0, 0, 0, 16, 0, 0, 104, 792, 1120, 6096, 21344, 114496, 257728, 1292544, 3677568, 17273760, 46801984, 211731376, 611507360, 2645699504, 7725948608, 32451640000, 97488160384, 397346625760, 1214082434112, 4835168968464, 15039729265856, 58641619298000

Offset: 1

Colin Rose, May 11 2006

nonn

1. Jelliss computes the number of tour diagrams (which is equal to half the number of tours). 2. Sequence A079137 computes the number of tour DIAGRAMS for a 4 X k board (again, equal to half the number of tours). 3. Kraitchik (1942) incorrectly reports 376 tour diagrams for the 3 X 8 case; the correct number is 396 (i.e., 792 tours) [cf. Rose, Jelliss].

Kraitchik, M., Mathematical Recreations. New York: W. W. Norton, pp. 264-5, 1942.

Seiichi Manyama, Table of n, a(n) for n = 1..1861
G. Jelliss, Open Knight's Tours of Three-Rank Boards
Seiichi Manyama calculated a(14)-a(21) by yoh2's code
C. Rose, The Distribution of the Knight.
Eric Weisstein's World of Mathematics, Hamiltonian Path
Eric Weisstein's World of Mathematics, Knight Graph

Cf. A169696, A079137, A083386, A165134.

Cf. A158074. - Eric W. Weisstein, Mar 13 2009

Mathematica notebook available at: http://www.tri.org.au/knightframe.html

a(n) = 2 * A169696(n). - Andrew Howroyd, Jul 01 2017

a(13) from Eric W. Weisstein, Mar 13 2009
a(14)-a(21) from Seiichi Manyama, Apr 25 2016
a(22)-a(28) from Andrew Howroyd, Jul 01 2017

A117431 String n is at position n in decimal digits of golden ratio (phi).

Original entry on oeis.org

1, 20, 62, 9956

Offset: 1

Colin Rose, Mar 14 2006

The next such number is greater than 10^7. Not only does number 20 occur at the 20th digit, but it occurs again as the 20th pair of digits (cf. A117432).

			1 is a term because the first digit in the golden ratio phi is 1. (phi = 1.6180339887498948482045 ...)

Eric Weisstein's World of Mathematics, The Golden Ratio

Cf. A001622, A057679, A117432.

StringFinder[m_] := Module[{cc = 10^m + m, sol, aa}, sol = Partition[RealDigits[(1+Sqrt[5])/2, 10, cc] // First, m, 1]; Do[aa = FromDigits[sol[[i]]]; If[aa==i, Print[{i, aa}]], {i,Length[sol]}];] (* Example: StringFinder[2] produces all 2-digit members of the sequence. *)

from sympy import S
def aupto(nn):
  phistr = str(S.GoldenRatio.n(nn+len(str(nn))+1)).replace(".", "")[:-1]
  for n in range(1, nn+1):
    nstr = str(n)
    if phistr[n-1:n-1+len(nstr)] == nstr: print(n, end=", ")
aupto(10**5) # Michael S. Branicky, Jan 20 2021

A117432 Let n be an integer consisting of m digits. Then n is a Phithy number if the n-th m-tuple in the decimal digits of golden ratio phi is string n.

Original entry on oeis.org

1, 20, 63, 104, 7499430, 9228401

Offset: 0

Colin Rose, Mar 14 2006

			1 is a term because the first single digit in golden ratio phi is 1.
Number 20 is a term because the 20th pair of digits in phi is 20.
(cf. phi = 1.6180339887498948482045868343656381177203...)

Eric Weisstein's World of Mathematics, The Golden Ratio

Cf. A001622, A109513, A109514, A117431.

PhithyNumbers[m_] := Module[{cc = m(10^m)+m, sol, aa}, sol = Partition[RealDigits[GoldenRatio, 10, cc] // First, m]; Do[aa = FromDigits[sol[[i]]]; If[aa==i, Print[{i, aa}]], {i,Length[sol]}];] Example: PhithyNumbers[3] produces all 3-digit Phithy numbers

from sympy import S
def aupto(nn):
  mm = len(str(nn))
  phistr = str(S.GoldenRatio.n(nn*mm+1)).replace(".", "")[:-1]
  for n in range(1, nn+1):
    nstr = str(n)
    m = len(nstr)
    if phistr[(n-1)*m:n*m] == nstr: print(n, end=", ")
aupto(10**5) # Michael S. Branicky, Jan 20 2021

a(4)-a(5) from Michael S. Branicky, Jan 21 2021

A109513 Let k be an m-digit integer. Then k is a Pithy number if the k-th m-tuple in the decimal digits of Pi (after the decimal point) is the string k.

Original entry on oeis.org

1, 19, 94, 3542, 7295, 318320, 927130, 939240, 688370303, 7682437410, 7996237896, 89594051933

Offset: 0

Colin Rose, Jul 01 2005

			1 is a term because the first digit in Pi (after the decimal point) is 1.
19 is a term because the 19th pair of digits (after the decimal point) in Pi is 19:
      1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19
  3. 14 15 92 65 35 89 79 32 38 46 26 43 38 32 79 50 28 84 19 ...

David G. Andersen, The Pi-Search Page.

Cf. A000796, A109514, A057679, A057680.

PithyNumbers[m_] := Module[{cc = m(10^m)+m, sol, aa}, sol = Partition[RealDigits[Pi, 10, cc] // First // Rest, m]; Do[aa = FromDigits[sol[[i]]]; If[aa==i, Print[{i, aa}]], {i, Length[sol]}];] Example: PithyNumbers[4] produces all 4-digit Pithy numbers

a(8)-a(11) from J. Volkmar Schmidt, Dec 17 2023

A109514 Let k be an integer consisting of m digits. Then k is a Pithy number if the k-th m-tuple in the decimal digits of Pi is k.

Original entry on oeis.org

5, 9696, 19781, 199898, 687784863, 4518673035, 7526556436

Offset: 1

Colin Rose, Jul 01 2005

A near-miss '02805451' occurs at position 2805451. - Vaclav Kotesovec, Feb 19 2020

			5 is a term because the 5th single digit in Pi is 5.
9696 is a term because the 9696th quadruplet in Pi is 9696.

David G. Andersen, The Pi-Search Page.

Cf. A109513, A057679, A057680.

PithyNumbersWith3[m_] := Module[{cc = m(10^m)+m, sol, aa}, sol = Partition[RealDigits[Pi, 10, cc] // First, m]; Do[aa = FromDigits[sol[[i]]]; If[aa==i, Print[{i, aa}]], {i, Length[sol]}];]
(* Example: PithyNumbersWith3[5] produces all 5-digit Pithy numbers *)

a(5)-a(7) from J. Volkmar Schmidt, Dec 17 2023

cp's OEIS Frontend

User: Colin Rose

A193069 Pretty wild narcissistic numbers - numbers that pwn: - an integer n that can be expressed using just the digits of n (each digit used once only and in order from left to right) and the operators + - * / ^ ! and the radical symbol. Concatenation is allowed.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

A118067 Number of (directed) Hamiltonian paths in the 3 X n knight graph.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A117431 String n is at position n in decimal digits of golden ratio (phi).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

A117432 Let n be an integer consisting of m digits. Then n is a Phithy number if the n-th m-tuple in the decimal digits of golden ratio phi is string n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Python

Extensions

A109513 Let k be an m-digit integer. Then k is a Pithy number if the k-th m-tuple in the decimal digits of Pi (after the decimal point) is the string k.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A109514 Let k be an integer consisting of m digits. Then k is a Pithy number if the k-th m-tuple in the decimal digits of Pi is k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions