Jörg Zurkirchen - cp's OEIS frontend

User: Jörg Zurkirchen

Jörg Zurkirchen has authored 4 sequences.

A346630 Smallest number which reaches the narcissistic number 153 after n steps when repeatedly summing the cubes of its digits.

153, 135, 18, 3, 9, 12, 33, 114, 78, 126, 6, 117, 669, 177, 12558, 44499999999999999999

Offset: 0

Jörg Zurkirchen, Jul 25 2021

All the terms a(n) as well as the intermediate results will be multiples of 3:
x^3 mod 3 = x mod 3 [0^3 = 0; 1^3 = 1; (-1)^3 = -1].
Therefore (sum of cubes of digits) mod 3 = (sum of digits) mod 3.
Because the only multiple of 3 in A046197 is 153, every number which is a multiple of 3 will end up at 153.
Some other terms (not dealt with here) may reach a cycle of length > 1:
Elizabeth Todd has shown that only numbers (1 mod 3) and (2 mod 3) may reach a cycle, and the only possible cycles are {55, 230, 130}, {136, 244}, {160, 217, 352}, {919, 1459}. That means that numbers (0 mod 3) never reach a cycle but just a single number, namely 153.
Shyam Sunder Gupta tested all the multiples of 3 less than 10^5. He found that they all reach 153, in accordance with the above statements.
The values a(n) for n>15 are really too big to be fully written out (and so are missing in the list), as Jon E. Schoenfield calculated for n=16 and n=17:
a(16) = 3.777999...999*10^61042524005486970; it has one 3, three 7's, and 61042524005486967 9's, so the sum of the cubes of its digits is 1*3^3 + 3*7^3 + 61042524005486967*9^3 = 44499999999999999999 = a(15).
a(17) consists of the digit string 45888 followed by a very, very long string of 9's. The number of 9's in that string is (a(16) - 1725)/729, which is a 61042524005486968-digit number consisting of the digit 5 followed by 753611407475147 copies of the 81-digit string 182441700960219478737997256515775034293552812071330589849108367626886145404663923 followed by a single instance of the 60-digit string 182441700960219478737997256515775034293552812071330589849106.

			a(3) = 3, for 3^3 = 27, 2^3 + 7^3 = 351, 3^3 + 5^3 + 1^3 = 153.
a(13) = 177, for 177 -> 687 -> 1071 -> 345 -> 216 -> 225 -> 141 -> 66 -> 432 -> 99 -> 1458 -> 702 -> 351 -> 153 (13 = longest chain for numbers up to 10^4).
The process ends because 153 = 1^3 + 5^3 + 3^3.

Table of n, a(n) for n = 0..15.
Shyam Sunder Gupta, Curious Properties of 153.
Shyam Sunder Gupta, Beauty of Number 153, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 15, 399-410.
Elizabeth Todd, Happy numbers, p. 11.

Cf. A055012 (sum of cubes of digits), A182111 (number of steps to a cycle), A165330 (cycle end), A046156.
Cf. A046197 (proving that {0, 1, 153, 370, 371, 407} are the only possible fixed points for all numbers of any size when repeatedly summing the cubes of its digits).
Cf. A346789 (concluding the number from the sum of the cubes of its digits).

Table[k=0;While[Last[s=NestList[Total[IntegerDigits@#^3]&,k,n]]!=153||Count[s,153]!=1,k=k+3];k,{n,0,14}] (* Giorgos Kalogeropoulos, Jul 30 2021 *)

A280944 Maximum sum of a set of different positive integers less than or equal to n whose pairwise sums are all different.

Original entry on oeis.org

3, 6, 9, 13, 17, 21, 26, 31, 36, 41, 46, 52, 58, 64, 70, 76, 82, 88, 94, 101, 108, 115, 122, 129, 136, 144, 152, 160, 168, 176, 184, 192, 200, 208, 216, 224, 232, 240, 248, 257, 266, 275, 284, 293, 302, 311, 320, 329, 339, 349, 359, 369, 379, 389, 399, 409, 419, 429, 439, 449, 459, 469, 479, 489, 499, 510, 521

Offset: 2

Jörg Zurkirchen, Jan 11 2017

nonn

Only the terms a(2)..a(26) are identical to sequence A256966.

Values a(n) found by exhaustive search by a Rexx program.

			For n = 10, the optimum set of integers is {10, 9, 8, 6, 3} and thus a(10) = 36. The 10 pair sums {19, 18, 17, 16, 15, 14, 13, 12, 11, 9} are all different.

Jörg Zurkirchen, Table of n, a(n) for n = 2..240
PuzzleUp, Problem No 06, Different sums, August 31, 2016.
Jörg Zurkirchen, Table of optimum solutions for n = 2..240

Cf. A256966.

Array[Max[Total /@ Select[Rest@ Subsets@ Range@ #, Length@ Union[Total /@ #] == Length[#] &@ Select[Union[Sort /@ Tuples[#, 2]], UnsameQ @@ # &] &]] &, 15, 2] (* Michael De Vlieger, Jan 21 2018 *)

A240438 Greatest minimal difference between numbers of adjacent cells in a regular hexagonal honeycomb of order n with cells numbered from 1 through the total number of cells, the order n corresponding to the number of cells on one side of the honeycomb.

Original entry on oeis.org

0, 1, 5, 11, 18, 28, 40, 53, 69, 87, 106, 128, 152, 177, 205, 235, 266, 300, 336, 373, 413, 455, 498, 544, 592, 641, 693, 747, 802, 860, 920, 981, 1045, 1111, 1178, 1248, 1320, 1393, 1469, 1547, 1626, 1708, 1792, 1877, 1965, 2055, 2146, 2240, 2336, 2433, 2533, 2635

Offset: 1

Jörg Zurkirchen, Apr 05 2014

Difference table of a(n), with a(0)=0 and offset=0:
0,   0,  1,  5, 11, 18, 28, 40, 53, 69, ...
0,   1,  4,  6,  7, 10, 12, 13, 16, 18, ...   =  A047234(n+1)
1,   3,  2,  1,  3,  2,  1,  3,  2,  1, ...   =  A130784
2,  -1, -1,  2, -1, -1,  2, -1, -1,  2, ...   = -A131713(n+1)
-3,  0,  3, -3,  0,  3, -3,  0,  3, -3; ...   =  A099838(n+4)
3,   3, -6,  3,  3, -6,  3,  3, -6,  3, ...
0,  -9,  9,  0, -9,  9,  0, -9,  9,  0, ...
-9, 18, -9, -9, 18, -9, -9, 18, -9, -9, ...
First column: see A057682. - Paul Curtz, Nov 11 2014
Diameter of the chamber graph Γ(Alt(2n+1)). Definition of this graph:
Each vertex v is a sequence (v[1],v[2],...,v[n]) of length n, where each v[i] is a 2-subset of {1,2,...,2n+1} and v[i] and v[j] are disjoint unless i=j.
Vertices u and v are connected iff either:
u and v are identical except for their first elements u[1] and v[1], or
u and v are identical except for some i for which u[i]=v[i+1] and v[i]=u[i+1] - Tim Crinion, 17 Feb 2019

			For n = 3 an example of a honeycomb with the greatest minimal difference of a(3) = 5 is:
.         __
.      __/ 7\__
.   __/15\__/13\__
.  / 4\__/ 2\__/ 1\
.  \__/10\__/ 8\__/
.  /18\__/16\__/14\
.  \__/ 5\__/ 3\__/
.  /12\__/11\__/ 9\
.  \__/19\__/17\__/
.     \__/ 6\__/
.        \__/
.

22ème Championnat des jeux mathématiques et logiques - 1/4 de finale individuels 2008, problème 18, «Les ruches d’Abella»

Vincenzo Librandi, Table of n, a(n) for n = 1..1000 (first 100 terms from Jörg Zurkirchen)
Tim Crinion, Chamber Graphs of some geometries related to the Petersen Graph, 2013.
Fédération Suisse des Jeux Mathématiques, 22nd Championship of Mathematical and Logical Games - Quarter Final 2008, 18 problems in French; problem number 18 was decisive to creating this sequence. See following pdf for an English version of problem 18.
Jörg Zurkirchen, Honeycomb.pdf
Index entries for linear recurrences with constant coefficients, signature (2, -1, 1, -2, 1).

Cf. A042965, A047234, A057682, A099838, A130784, A131713.

[n*(n-1)-Floor((n+1)/3): n in [1..60]]; // Vincenzo Librandi, Nov 12 2014

A240438:=n->n*(n-1)-floor((n+1)/3); seq(A240438(n), n=1..50); # Wesley Ivan Hurt, Apr 08 2014

Table[n (n - 1) - Floor[(n + 1)/3], {n, 50}] (* Wesley Ivan Hurt, Apr 08 2014 *)
CoefficientList[Series[x (x + 1) (2 x + 1) / ((1 - x)^3 (x^2 + x + 1)), {x, 0, 60}], x] (* Vincenzo Librandi, Nov 12 2014 *)
LinearRecurrence[{2, -1, 1, -2, 1},{0, 1, 5, 11, 18},52] (* Ray Chandler, Sep 24 2015 *)

a(n) = n*(n-1)-floor((n+1)/3).
G.f.: -x^2*(x+1)*(2*x+1) / ((x-1)^3*(x^2+x+1)). - Colin Barker, Apr 08 2014
a(n+3) = a(n) + 6*n+5. - Paul Curtz, Nov 11 2014
a(n) = n^2 - (A042965(n+1)=0, 1, 3, 4, ...). - Paul Curtz, Nov 11 2014
a(n+1) = a(n) + A047234(n+1). - Paul Curtz, Nov 11 2014

A035331 Base-1000 expansion of Pi.

Original entry on oeis.org

3, 141, 592, 653, 589, 793, 238, 462, 643, 383, 279, 502, 884, 197, 169, 399, 375, 105, 820, 974, 944, 592, 307, 816, 406, 286, 208, 998, 628, 34, 825, 342, 117, 67, 982, 148, 86, 513, 282, 306, 647, 93, 844, 609, 550, 582, 231, 725, 359, 408, 128, 481, 117, 450

Offset: 0

Jörg Zurkirchen

Start with a(0)=3; other terms are formed from triples of successive digits in the decimal expansion of Pi.

This sequence can be considered as a (pseudo)random generator with range 0..999. Its scatterplot graph is very similar to that of other random generators, e.g., A096558. - M. F. Hasler, May 14 2015

			Pi = 3.141 592 653 589 793 238 462 643 383 279 502 884 197 169 399 375 105 ...

T. D. Noe, Table of n, a(n) for n = 0..10000

Cf. A000796, A016062, A037244, A050819, A032445, A064467.

RealDigits[Pi,1000,60][[1]] (* Harvey P. Dale, Nov 22 2015 *)

default(realprecision,3*N=100);vector(N,i,Pi\1000^(1-i)%1000) \\ or: {P=Pi;vector(N,i,P\1+0*P=frac(P)*1000)} or {P=Pi/1000;vector(N,i,floor(P=frac(P)*1000))}. \\ M. F. Hasler, May 11 2015

a(n) = floor(Pi*10^(3n)) mod 1000. - M. F. Hasler, May 14 2015

More terms from Larry Reeves (larryr(AT)acm.org), Oct 04 2001

Better definition from Franklin T. Adams-Watters, Apr 10 2006

cp's OEIS Frontend

User: Jörg Zurkirchen

A346630 Smallest number which reaches the narcissistic number 153 after n steps when repeatedly summing the cubes of its digits.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A280944 Maximum sum of a set of different positive integers less than or equal to n whose pairwise sums are all different.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A240438 Greatest minimal difference between numbers of adjacent cells in a regular hexagonal honeycomb of order n with cells numbered from 1 through the total number of cells, the order n corresponding to the number of cells on one side of the honeycomb.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Formula

A035331 Base-1000 expansion of Pi.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions