A290447 -id:A290447 - cp's OEIS frontend

A290865 a(n) = number of regions in the configuration A290447(n).

0, 1, 3, 7, 15, 30, 56, 98, 161, 250, 370, 536, 748, 1027, 1379, 1807, 2320, 2954, 3702, 4604, 5652, 6852, 8239, 9858, 11683, 13748, 16086, 18700, 21604, 24887, 28471, 32491, 36907, 41751, 47080, 52876, 59105, 65965, 73440, 81521, 90176

Offset: 1

David Applegate, Aug 12 2017

nonn

			With 3 points, there are 3 semicircles above the baseline, which bound a(3) = 3 regions. With 4 points, there are 6 semicircles, defining 7 regions (use the Halser webpage with n = 3 and 4). - _N. J. A. Sloane_, Aug 12 2017

David Applegate, Table of n, a(n) for n = 1..100
M. F. Hasler, Interactive web page for drawing the illustration for a(n).
N. J. A. Sloane (in collaboration with Scott R. Shannon), Art and Sequences, Slides of guest lecture in Math 640, Rutgers Univ., Feb 8, 2020. Mentions this sequence.

Cf. A290447, A290866, A290867, A332723 (number of regions with k edges).

A290866 a(n) = number of segments (edges) in the configuration A290447(n).

Original entry on oeis.org

0, 1, 3, 8, 20, 45, 91, 168, 285, 450, 670, 981, 1375, 1902, 2568, 3371, 4326, 5522, 6927, 8639, 10624, 12882, 15489, 18559, 22006, 25904, 30321, 35254, 40728, 46959, 53721, 61354, 69734, 78917, 89029, 100018, 111758, 124759, 138943

Offset: 1

David Applegate, Aug 12 2017

nonn

Only edges above the line are counted. Total edges = a(n) + n - 1.

David Applegate, Table of n, a(n) for n = 1..100

Cf. A290447, A290865, A290867.

a(n) = A290447(n) + A290865(n).

A290867 Irregular triangle read by rows: the number of points that are the intersections of k semicircles in the configuration A290447(n).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 5, 0, 15, 0, 35, 0, 70, 0, 123, 1, 0, 195, 5, 0, 285, 15, 0, 420, 25, 0, 586, 39, 2, 0, 818, 53, 4, 0, 1110, 73, 6, 0, 1451, 103, 10, 0, 1846, 142, 18, 0, 2361, 181, 26, 0, 2956, 234, 33, 2, 0, 3704, 287, 40, 4, 0, 4567, 348, 49, 8

Offset: 1

David Applegate, Aug 12 2017

Row lengths are A290726(n).
The first entry of each row is 0, because an intersection requires at least 2 lines.
The first row with 3 entries is for n=9, because that is the first configuration with a nontrivial intersection.
Row sums give A290447.

			Triangle begins:
  0;
  0;
  0;
  0,   1;
  0,   5;
  0,  15;
  0,  35;
  0,  70;
  0, 123,   1;
  0, 195,   5;
  0, 285,  15;
  0, 420,  25;
  0, 586,  39,   2;

David Applegate, Table of n, a(n) for n = 1..800
David Applegate, Triangular table T(n,k) for n = 1..100
N. J. A. Sloane, Three (No, 8) Lovely Problems from the OEIS, Experimental Mathematics Seminar, Rutgers University, Oct 05 2017, Part I, Part 2, Slides. (Mentions this sequence)
N. J. A. Sloane (in collaboration with Scott R. Shannon), Art and Sequences, Slides of guest lecture in Math 640, Rutgers Univ., Feb 8, 2020. Mentions this sequence.

Cf. A290447, A290726, A290865, A290866.

Sum_{k} T(n,k) * binomial(k,2) = binomial(n,4), because there are binomial(n,4) total pairs of semicircles, and an intersection of k consists of binomial(k,2) of those pairs.

A290865(n) = binomial(n,2) + Sum_{k} T(n,k) * (k-1).

A332723 Irregular table read by rows: Take a line with n equally spaced points with semicircles drawn between them, as in A290447. Then T(n,k) = number of k-sided regions in that figure, where k>=2.

Original entry on oeis.org

1, 2, 1, 3, 4, 4, 10, 0, 1, 5, 19, 3, 3, 6, 31, 13, 6, 7, 46, 35, 10, 8, 65, 74, 14, 9, 92, 131, 18, 10, 140, 192, 27, 1, 11, 202, 274, 46, 3, 12, 275, 396, 62, 3, 13, 363, 563, 79, 9, 14, 467, 784, 100, 14, 15, 598, 1054, 126, 12, 2

Offset: 2

Scott R. Shannon and N. J. A. Sloane, Feb 21 2020

			The first 25 rows are:
1;
2,1;
3,4;
4,10,0,1;
5,19,3,3;
6,31,13,6;
7,46,35,10;
8,65,74,14;
9,92,131,18;
10,140,192,27,1;
11,202,274,46,3;
12,275,396,62,3;
13,363,563,79,9;
14,467,784,100,14;
15,598,1054,126,12,2;
16,772,1358,159,13,2;
17,996,1698,216,24,2,1;
18,1255,2120,266,41,2;
19,1551,2629,346,54,5;
20,1892,3236,425,71,8;
21,2304,3909,525,83,9,1;
22,2793,4676,629,108,9,2;
23,3342,5559,792,125,14,3;
24,3982,6546,948,166,15,2;
25,4705,7658,1145,198,14,3;
The row sums are A290865.

Scott R. Shannon, Table of n, a(n) for n = 2..345. The terms for 2 to 55 points.
Scott R. Shannon, Illustration of regions for n = 5.
Scott R. Shannon, Illustration of regions for n = 6.
Scott R. Shannon, Illustration of regions for n = 10.
Scott R. Shannon, Illustration of regions for n = 11.
Scott R. Shannon, Illustration of regions for n = 15.
Scott R. Shannon, Illustration of regions for n = 16.
Scott R. Shannon, Illustration of regions for n = 20.
Scott R. Shannon, Illustration of regions for n = 21.
Scott R. Shannon, Illustration of regions for n = 25.
Scott R. Shannon, Illustration of regions for n = 26.
Scott R. Shannon, Illustration of regions with random distance-based coloring for n = 10.
Scott R. Shannon, Illustration of regions with random distance-based coloring for n = 11.
Scott R. Shannon, Illustration of regions with random distance-based coloring for n = 15.
Scott R. Shannon, Illustration of regions with random distance-based coloring for n = 16.
Scott R. Shannon, Illustration of regions with random distance-based coloring for n = 20.
Scott R. Shannon, Illustration of regions with random distance-based coloring for n = 21.
Scott R. Shannon, Illustration of regions with random distance-based coloring for n = 25.
Scott R. Shannon, Illustration of regions with random distance-based coloring for n = 26.

Cf. A290447, A290865, A290866, A290867, A290726.

A290726 a(n) = maximal number of semicircles that pass through an intersection point in the configuration A290447(n).

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12

Offset: 1

N. J. A. Sloane, Aug 10 2017

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..190
M. F. Hasler, Illustration for A290447(9) = 124. (First instance where a triple intersection occurs, whence a(9) = 3.)

Cf. A290447.

More terms from Chai Wah Wu, Aug 10 2017

A290461 Binomial(n,4) - A290447(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 2, 10, 30, 50, 88, 126, 176, 256, 374, 492, 651, 810, 1013, 1285, 1605, 1925, 2327, 2794, 3315, 3921, 4627, 5333, 6215, 7097, 8093, 9210, 10411, 11763, 13392, 15021, 16748, 18649, 20833, 23017, 25480, 27943, 30634, 33656, 36806, 39956, 43478, 47215, 51266, 55565, 60168

Offset: 1

N. J. A. Sloane, Aug 07 2017

nonn

Binomial(n,4) is an upper bound on A290447. No formula or recurrence is known for the latter, but perhaps the differences will turn out to be simpler to analyze.

N. J. A. Sloane, Table of n, a(n) for n = 1..500 [Based on David Applegate's b-file for A290447]

Cf. A000332, A290447.

A290465 a(n) = A006561(n) - A290447(n).

Original entry on oeis.org

0, 0, 0, 0, 0, -2, 0, -21, 2, -39, 30, -144, 88, -118, 176, -187, 374, -731, 651, -194, 1013, -89, 1605, -1404, 2327, 325, 3315, 695, 4627, -5271, 6215, 2050, 8093, 3091, 10411, -161, 13392, 6187, 16748, 8140, 20833, -4072, 25480, 13633, 30634, 17212, 36806, 9537, 43478, 25716, 51266, 31113, 60168

Offset: 1

N. J. A. Sloane, Aug 09 2017

sign

A006561 and A290447 are closely related problems. The former is well-understood, while the latter is at present a mystery.

Cf. A006561, A290447, A290461.

A292103 Number of points that are the intersections of exactly two semicircles in the configuration A290447(n).

Original entry on oeis.org

0, 0, 0, 1, 5, 15, 35, 70, 123, 195, 285, 420, 586, 818, 1110, 1451, 1846, 2361, 2956, 3704, 4567, 5530, 6631, 7963, 9443, 11113, 13005, 15111, 17450, 20167, 23064, 26396, 30053, 34046, 38447, 43230, 48245, 53890, 60061, 66703, 73713, 81503, 89746

Offset: 1

N. J. A. Sloane, Sep 14 2017

nonn

No formula or recurrence is known.

Needs a b-file (A290867 gives first 100 terms).

Cf. A290447, A292104.

Column k=2 of triangle in A290867.

A037274 Home primes: for n >= 2, a(n) = the prime that is finally reached when you start with n, concatenate its prime factors (A037276) and repeat until a prime is reached (a(n) = -1 if no prime is ever reached).

Original entry on oeis.org

1, 2, 3, 211, 5, 23, 7, 3331113965338635107, 311, 773, 11, 223, 13, 13367, 1129, 31636373, 17, 233, 19, 3318308475676071413, 37, 211, 23, 331319, 773, 3251, 13367, 227, 29, 547, 31, 241271, 311, 31397, 1129, 71129, 37, 373, 313, 3314192745739, 41, 379, 43, 22815088913, 3411949, 223, 47, 6161791591356884791277

Offset: 1

N. J. A. Sloane, Jeff Burch

The initial 1 could have been omitted.
Probabilistic arguments give exactly zero for the chance that the sequence of integers starting at n contains no prime, the expected number of primes being given by a divergent sequence. - J. H. Conway
After over 100 iterations, a(49) is still composite - see A056938 for the latest information.
More terms:
a(50) to a(60) are 3517, 317, 2213, 53, 2333, 773, 37463, 1129, 229, 59, 35149;
a(61) to a(65) are 61, 31237, 337, 1272505013723, 1381321118321175157763339900357651;
a(66) to a(76) are 2311, 67, 3739, 33191, 257, 71, 1119179, 73, 379, 571, 333271.
This is different from A195264. Here 8 = 2^3 -> 222 -> ... -> 3331113965338635107 (a prime), whereas in A195264 8 = 2^3 -> 23 (a prime). - N. J. A. Sloane, Oct 12 2014

			9 = 3*3 -> 33 = 3*11 -> 311, prime, so a(9) = 311.
The trajectory of 8 is more interesting:
8 ->
2 * 2 * 2 ->
2 * 3 * 37 ->
3 * 19 * 41 ->
3 * 3 * 3 * 7 * 13 * 13 ->
3 * 11123771 ->
7 * 149 * 317 * 941 ->
229 * 31219729 ->
11 * 2084656339 ->
3 * 347 * 911 * 118189 ->
11 * 613 * 496501723 ->
97 * 130517 * 917327 ->
53 * 1832651281459 ->
3 * 3 * 3 * 11 * 139 * 653 * 3863 * 5107
and 3331113965338635107 is prime, so a(8) = 3331113965338635107.

Jeffrey Heleen, Family Numbers: Mathemagical Black Holes, Recreational and Educational Computing, 5:5, pp. 6, 1990.
Jeffrey Heleen, Family numbers: Constructing Primes by Prime Factor Splicing, J. Recreational Math., Vol. 28 #2, 1996-97, pp. 116-119.

Christian N. K. Anderson, Table of known values of n, # of steps to reach a(n), and a(n) or NA if a(n) has 30 digits or more. Also, the trajectory, with factors separated by a |, terminated by either "(end)" or "-> ?" if a(n) has 30 digits or more.
Patrick De Geest, Home Primes < 100 and Beyond
M. Herman and J. Schiffman, Investigating home primes and their families, Math. Teacher, 107 (No. 8, 2014), 606-614.
N. J. A. Sloane, Confessions of a Sequence Addict (AofA2017), slides of invited talk given at AofA 2017, Jun 19 2017, Princeton. Mentions this sequence.
N. J. A. Sloane, Three (No, 8) Lovely Problems from the OEIS, Experimental Mathematics Seminar, Rutgers University, Oct 05 2017, Part I, Part 2, Slides. (Mentions this sequence)
Eric Weisstein's World of Mathematics, Home Prime.
Wikipedia, Home prime

Cf. A006919, A037271, A037272, A037273, A037275, A037276, A037919-A037941, A048986, A056938.
Cf. A195264 (use exponents instead of repeating primes).
Cf. A084318 (use only one copy of each prime), A248713 (Fermi-Dirac analog: use unique representation of n>1 as a product of distinct terms of A050376).
Cf. also A120716 and related sequences.

b:= n-> parse(cat(sort(map(i-> i[1]$i[2], ifactors(n)[2]))[])):
a:= n-> `if`(isprime(n) or n=1, n, a(b(n))):
seq(a(n), n=1..48);  # Alois P. Heinz, Jan 09 2021

f[n_] := FromDigits@ Flatten[ IntegerDigits@ Table[ #[[1]], { #[[2]] }] & /@ FactorInteger@n, 2]; g[n_] := NestWhile[ f@# &, n, !PrimeQ@# &]; g[1] = 1; Array[g, 41] (* Robert G. Wilson v, Sep 22 2007 *)

step(n)=my(f=factor(n),s="");for(i=1,#f~,for(j=1,f[i,2],s=Str(s,f[i,1]))); eval(s)
a(n)=if(n<4,return(n)); while(!isprime(n), n=step(n)); n \\ Charles R Greathouse IV, May 14 2015

from sympy import factorint, isprime
def f(n): return int("".join(str(p)*e for p, e in factorint(n).items()))
def a(n):
    if n == 1: return 1
    fn = n
    while not isprime(fn): fn = f(fn)
    return fn
print([a(n) for n in range(1, 40)]) # Michael S. Branicky, Jul 11 2022

def digitLen(x,n):
    r=0
    while(x>0):
        x//=n
        r+=1
    return r
def concatPf(x,n):
    r=0
    f=list(factor(x))
    for c in range(len(f)):
        for d in range(f[c][1]):
            r*=(n**digitLen(f[c][0],n))
            r+=f[c][0]
    return r
def hp(x,n):
    x1=concatPf(x,n)
    while(x1!=x):
        x=x1
        x1=concatPf(x1,n)
    return x
#example: prints the home prime of 8 in base 10
print(hp(8,10))

Corrected and extended by Karl W. Heuer, Sep 30 2003

A006561 Number of intersections of diagonals in the interior of a regular n-gon.

Original entry on oeis.org

0, 0, 0, 1, 5, 13, 35, 49, 126, 161, 330, 301, 715, 757, 1365, 1377, 2380, 1837, 3876, 3841, 5985, 5941, 8855, 7297, 12650, 12481, 17550, 17249, 23751, 16801, 31465, 30913, 40920, 40257, 52360, 46981, 66045, 64981, 82251, 80881, 101270, 84841, 123410, 121441

Offset: 1

N. J. A. Sloane, Bjorn Poonen (poonen(AT)math.princeton.edu)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, Table of n, a(n) for n = 1..20000 (first 1000 terms from T. D. Noe)
Johan Gielis and Ilia Tavkhelidze, The general case of cutting of GML surfaces and bodies, arXiv:1904.01414 [math.GM], 2019.
Jessica Gonzalez, Illustration of a(4) through a(9).
M. Griffiths, Counting the regions in a regular drawing of K_{n,n}, J. Int. Seq. 13 (2010) # 10.8.5.
M. F. Hasler, Interactive illustration of A006561(n), Sep 01 2017. (For colored versions see A006533.)
Sascha Kurz, m-gons in regular n-gons.
Roger Mansuy, Des croisements pas si faciles à compter, La Recherche, 547, Mai 2019 (in French).
B. Poonen and M. Rubinstein, The Number of Intersection Points Made by the Diagonals of a Regular Polygon, SIAM J. Discrete Mathematics, Vol. 11, No.1 (1998) pp. 135-156; DOI:10.1137/S0895480195281246. [Copy on B. Poonen's web site.]
B. Poonen and M. Rubinstein, The number of intersection points made by the diagonals of a regular polygon, arXiv:math/9508209 [math.MG]: revision from 2006 has a few typos from the published version corrected.
B. Poonen and M. Rubinstein, Mathematica programs for A006561 and related sequences.
M. Rubinstein, Drawings for n=4,5,6,....
N. J. A. Sloane, Illustrations of a(8) and a(9).
N. J. A. Sloane, Three (No, 8) Lovely Problems from the OEIS, Experimental Mathematics Seminar, Rutgers University, Oct 05 2017, Part I, Part 2, Slides. (Mentions this sequence)
N. J. A. Sloane (in collaboration with Scott R. Shannon), Art and Sequences, Slides of guest lecture in Math 640, Rutgers Univ., Feb 8, 2020. Mentions this sequence.
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 18.
Robert G. Wilson v, Illustration of a(10)
Index entry for Sequences formed by drawing all diagonals in regular polygon

Sequences related to chords in a circle: A001006, A054726, A006533, A006561, A006600, A007569, A007678. See also entries for chord diagrams in Index file.
See also A101363, A292104, A292105.
See A290447 for an analogous problem on a line.

delta:=(m,n) -> if (n mod m) = 0 then 1 else 0; fi;
f:=proc(n) global delta;
if n <= 2 then 0 else \
binomial(n,4)  \
+ (-5*n^3 + 45*n^2 - 70*n + 24)*delta(2,n)/24 \
- (3*n/2)*delta(4,n) \
+ (-45*n^2 + 262*n)*delta(6,n)/6  \
+ 42*n*delta(12,n) \
+ 60*n*delta(18,n) \
+ 35*n*delta(24,n) \
- 38*n*delta(30,n) \
- 82*n*delta(42,n) \
- 330*n*delta(60,n) \
- 144*n*delta(84,n) \
- 96*n*delta(90,n) \
- 144*n*delta(120,n) \
- 96*n*delta(210,n); fi; end;
[seq(f(n),n=1..100)]; # N. J. A. Sloane, Aug 09 2017

del[m_,n_]:=If[Mod[n,m]==0,1,0]; Int[n_]:=If[n<4, 0, Binomial[n,4] + del[2,n](-5n^3+45n^2-70n+24)/24 - del[4,n](3n/2) + del[6,n](-45n^2+262n)/6 + del[12,n]*42n + del[18,n]*60n + del[24,n]*35n - del[30,n]*38n - del[42,n]*82n - del[60,n]*330n - del[84,n]*144n - del[90,n]*96n - del[120,n]*144n - del[210,n]*96n]; Table[Int[n], {n,1,1000}] (* T. D. Noe, Dec 21 2006 *)

apply( {A006561(n)=binomial(n,4)+if(n%2==0, (n>2) + (-5*n^2+45*n-70)*n/24 + vecsum([t[2] | t<-[4,6,12,18,24,30,42,60,84,90,120,210;-3/2,(262-45*n)/6,42,60,35,-38,-82,-330,-144,-96,-144,-96], n%t[1]==0])*n)}, [1..44]) \\ M. F. Hasler, Aug 23 2017, edited Aug 06 2021

def d(n,m): return not n % m
def A006561(n): return 0 if n == 2 else n*(42*d(n,12) - 144*d(n,120) + 60*d(n,18) - 96*d(n,210) + 35*d(n,24)- 38*d(n,30) - 82*d(n,42) - 330*d(n,60) - 144*d(n,84) - 96*d(n,90)) + (n**4 - 6*n**3 + 11*n**2 - 6*n -d(n,2)*(5*n**3 - 45*n**2 + 70*n - 24) - 36*d(n,4)*n - 4*d(n,6)*n*(45*n - 262))//24 # Chai Wah Wu, Mar 08 2021

Let delta(m,n) = 1 if m divides n, otherwise 0.
For n >= 3, a(n) = binomial(n,4) + (-5*n^3 + 45*n^2 - 70*n + 24)*delta(2,n)/24
- (3*n/2)*delta(4,n) + (-45*n^2 + 262*n)*delta(6,n)/6 + 42*n*delta(12,n)
+ 60*n*delta(18,n) + 35*n*delta(24,n) - 38*n*delta(30,n)
- 82*n*delta(42,n) - 330*n*delta(60,n) - 144*n*delta(84,n)
- 96*n*delta(90,n) - 144*n*delta(120,n) - 96*n*delta(210,n). [Poonen and Rubinstein, Theorem 1] - N. J. A. Sloane, Aug 09 2017
For odd n, a(n) = binomial(n,4) = n*(n-1)*(n-2)*(n-3)/24, see A053126. For even n, use this formula, but then subtract 2 for every 3-crossing, subtract 5 for every 4-crossing, subtract 9 for every 5-crossing, etc. The number to be subtracted for a d-crossing is (d-1)*(d-2)/2. - Graeme McRae, Dec 26 2004
a(n) = A007569(n) - n. - T. D. Noe, Dec 23 2006
a(2n+5) = A053126(n+4). - Philippe Deléham, Jun 07 2013

cp's OEIS Frontend

A290865 a(n) = number of regions in the configuration A290447(n).

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A290866 a(n) = number of segments (edges) in the configuration A290447(n).

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A290867 Irregular triangle read by rows: the number of points that are the intersections of k semicircles in the configuration A290447(n).

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A332723 Irregular table read by rows: Take a line with n equally spaced points with semicircles drawn between them, as in A290447. Then T(n,k) = number of k-sided regions in that figure, where k>=2.

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A290726 a(n) = maximal number of semicircles that pass through an intersection point in the configuration A290447(n).

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A290461 Binomial(n,4) - A290447(n).

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A290465 a(n) = A006561(n) - A290447(n).

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A292103 Number of points that are the intersections of exactly two semicircles in the configuration A290447(n).

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A037274 Home primes: for n >= 2, a(n) = the prime that is finally reached when you start with n, concatenate its prime factors (A037276) and repeat until a prime is reached (a(n) = -1 if no prime is ever reached).

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A006561 Number of intersections of diagonals in the interior of a regular n-gon.

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