A008315 Catalan triangle read by rows. Also triangle of expansions of powers of x in terms of Chebyshev polynomials U_n(x).

1, 1, 1, 1, 1, 2, 1, 3, 2, 1, 4, 5, 1, 5, 9, 5, 1, 6, 14, 14, 1, 7, 20, 28, 14, 1, 8, 27, 48, 42, 1, 9, 35, 75, 90, 42, 1, 10, 44, 110, 165, 132, 1, 11, 54, 154, 275, 297, 132, 1, 12, 65, 208, 429, 572, 429, 1, 13, 77, 273, 637, 1001, 1001, 429, 1, 14, 90, 350, 910, 1638, 2002, 1430, 1, 15, 104

Offset

0,6

Comments

There are several versions of a Catalan triangle: see A009766, A008315, A028364, A053121.

Number of standard tableaux of shape (n-k,k) (0<=k<=floor(n/2)). Example: T(4,1)=3 because in th top row we can have 124, 134, or 123 (but not 234). - Emeric Deutsch, May 23 2004

T(n,k) is the number of n-digit binary words (length n sequences on {0,1}) containing k 1's such that no initial segment of the sequence has more 1's than 0's. - Geoffrey Critzer, Jul 31 2009

T(n,k) is the number of dispersed Dyck paths (i.e. Motzkin paths with no (1,0) steps at positive heights) of length n and having k (1,1)-steps. Example: T(5,1)=4 because, denoting U=(1,1), D=(1,-1), H=1,0), we have HHHUD, HHUDH, HUDHH, and UDHHH. - Emeric Deutsch, May 30 2011

T(n,k) is the number of length n left factors of Dyck paths having k (1,-1)-steps. Example: T(5,1)=4 because, denoting U=(1,1), D=(1,-1), we have UUUUD, UUUDU, UUDUU, and UDUUU. There is a simple bijection between length n left factors of Dyck paths and dispersed Dyck paths of length n, that takes D steps into D steps. - Emeric Deutsch, Jun 19 2011

Triangle, with zeros omitted, given by (1, 0, 0, -1, 1, 0, 0, -1, 1, 0, 0, -1, 1, ...) DELTA (0, 1, -1, 0, 0, 1, -1, 0, 0, 1, -1, 0, 0, 1, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 12 2011

T(n,k) are rational multiples of A055151(n,k). - Peter Luschny, Oct 16 2015

T(2*n,n) = Sum_{k>=0} T(n,k)^2 = A000108(n), T(2*n+1,n) = A000108(n+1). - Michael Somos, Jun 08 2020

References

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 796.

P. J. Larcombe, A question of proof..., Bull. Inst. Math. Applic. (IMA), 30, Nos. 3/4, 1994, 52-54.

Links

T. D. Noe, Rows n=0..100 of triangle, flattened

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Tewodros Amdeberhan, Moa Apagodu, and Doron Zeilberger, Wilf's "Snake Oil" Method Proves an Identity in The Motzkin Triangle, arXiv:1507.07660 [math.CO], 2015.

Nantel Bergeron, Kelvin Chan, Yohana Solomon, Farhad Soltani, and Mike Zabrocki, Quasisymmetric harmonics of the exterior algebra, arXiv:2206.02065 [math.CO], 2022.

Suyoung Choi and Hanchul Park, A new graph invariant arises in toric topology, arXiv preprint arXiv:1210.3776 [math.AT], 2012.

C. Kenneth Fan, Structure of a Hecke algebra quotient, J. Amer. Math. Soc. 10 (1997), no. 1, 139-167.

R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.

L. Jiu, V. H. Moll, and C. Vignat, Identities for generalized Euler polynomials, arXiv:1401.8037 [math.PR], 2014.

N. Lygeros and O. Rozier, A new solution to the equation tau(rho) == 0 (mod p), J. Int. Seq. 13 (2010) # 10.7.4.

M. A. A. Obaid, S. K. Nauman, W. M. Fakieh, and C. M. Ringel, The numbers of support-tilting modules for a Dynkin algebra, 2014 and J. Int. Seq. 18 (2015) 15.10.6.

Alon Regev, The central component of a triangulation, Journal of Integer Sequences, Vol. 16 (2013), Article 13.4.1, p. 7.

J. Riordan, The distribution of crossings of chords joining pairs of 2n points on a circle, Math. Comp., 29 (1975), 215-222.

J. Riordan, The distribution of crossings of chords joining pairs of 2n points on a circle, Math. Comp., 29 (1975), 215-222. [Annotated scanned copy]

L. W. Shapiro, A Catalan triangle, Discrete Math. 14 (1976), no. 1, 83-90. [Annotated scanned copy]

Zheng Shi, Impurity entropy of junctions of multiple quantum wires, arXiv preprint arXiv:1602.00068 [cond-mat.str-el], 2016 (See Appendix A).

Index entries for sequences related to Chebyshev polynomials.

Formula

T(n, 0) = 1 if n >= 0; T(2*k, k) = T(2*k-1, k-1) if k>0; T(n, k) = T(n-1, k-1) + T(n-1, k) if k=1, 2, ..., floor(n/2). - Michael Somos, Aug 17 1999

T(n, k) = binomial(n, k) - binomial(n, k-1). - Michael Somos, Aug 17 1999

Rows of Catalan triangle A008313 read backwards. Sum_{k>=0} T(n, k)^2 = A000108(n); A000108 : Catalan numbers. - Philippe Deléham, Feb 15 2004

T(n,k) = C(n,k)*(n-2*k+1)/(n-k+1). - Geoffrey Critzer, Jul 31 2009

Sum_{k=0..n} T(n,k)*x^k = A000012(n), A001405(n), A126087(n), A128386(n), A121724(n), A128387(n), A132373(n), A132374(n), A132375(n), A121725(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 respectively. - Philippe Deléham, Dec 12 2011

Example

Triangle begins:
  1;
  1;
  1, 1;
  1, 2;
  1, 3,  2;
  1, 4,  5;
  1, 5,  9,  5;
  1, 6, 14, 14;
  1, 7, 20, 28, 14;
  ...
T(5,2) = 5 because there are 5 such sequences: {0, 0, 0, 1, 1}, {0, 0, 1, 0, 1}, {0, 0, 1, 1, 0}, {0, 1, 0, 0, 1}, {0, 1, 0, 1, 0}. - Geoffrey Critzer, Jul 31 2009

Maple

b:= proc(x, y) option remember; `if`(y<0 or y>x, 0,
     `if`(x=0, 1, add(b(x-1, y+j), j=[-1, 1])))
    end:
T:= (n, k)-> b(n, n-2*k):
seq(seq(T(n, k), k=0..n/2), n=0..16);  # Alois P. Heinz, Oct 14 2022

Mathematica

Table[Binomial[k, i]*(k - 2 i + 1)/(k - i + 1), {k, 0, 20}, {i, 0, Floor[k/2]}] // Grid (* Geoffrey Critzer, Jul 31 2009 *)

Prog

(PARI) {T(n, k) = if( k<0 || k>n\2, 0, if( n==0, 1, T(n-1, k-1) + T(n-1, k)))}; /* Michael Somos, Aug 17 1999 */
(Haskell)
a008315 n k = a008315_tabf !! n !! k
a008315_row n = a008315_tabf !! n
a008315_tabf = map reverse a008313_tabf
-- Reinhard Zumkeller, Nov 14 2013

Crossrefs

T(2n, n) = A000108 (Catalan numbers), row sums = A001405 (central binomial coefficients).

This is also the positive half of the triangle in A008482. - Michael Somos

This is another reading (by shallow diagonals) of the triangle A009766, i.e. A008315[n] = A009766[A056536[n]].

Cf. A120730, A055151.

Keyword

nonn,tabf,nice,easy

Author

N. J. A. Sloane

Extensions

Expanded description from Clark Kimberling, Jun 15 1997

Status

approved

A373672 Length of the n-th maximal antirun of non-prime-powers.

5, 3, 1, 6, 1, 1, 2, 1, 3, 1, 3, 1, 2, 1, 1, 1, 3, 2, 2, 1, 3, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 3, 1, 3, 1, 2, 1, 1, 1, 1, 1, 2, 1, 3, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1

Offset

1,1

Comments

An antirun of a sequence (in this case A361102 or A024619 with 1) is an interval of positions at which consecutive terms differ by more than one.

Links

Table of n, a(n) for n=1..87.

Gus Wiseman, Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers.

Formula

Partial sums are A356068(A255346(n)).

Example

The maximal antiruns of non-prime-powers begin:
   1   6  10  12  14
  15  18  20
  21
  22  24  26  28  30  33
  34
  35
  36  38
  39
  40  42  44
  45
  46  48  50

Mathematica

Length/@Split[Select[Range[100], !PrimePowerQ[#]&], #1+1!=#2&]//Most

Crossrefs

For prime antiruns we have A027833.

For nonsquarefree runs we have A053797, firsts A373199.

For non-prime-powers runs we have A110969, firsts A373669, sorted A373670.

For squarefree runs we have A120992.

For prime-power runs we have A174965.

For prime runs we have A175632.

For composite runs we have A176246, firsts A073051, sorted A373400.

For squarefree antiruns we have A373127, firsts A373128.

For composite antiruns we have A373403.

For antiruns of prime-powers:

- length A373671

- min A120430

- max A006549

For antiruns of non-prime-powers:

- length A373672 (this sequence), firsts (3,7,2,25,1,4)

- min A373575

- max A255346

A000961 lists all powers of primes. A246655 lists just prime-powers.

A057820 gives first differences of consecutive prime-powers, gaps A093555.

A356068 counts non-prime-powers up to n.

A361102 lists all non-prime-powers (A024619 if not including 1).

Cf. A001359, A008864, A014963, A038664, A054265, A067774, A251092, A373401.

Keyword

nonn,new

Author

Gus Wiseman, Jun 14 2024

Status

approved

A064532 Total number of holes in decimal expansion of the number n, assuming 4 has no hole.

1, 0, 0, 0, 0, 0, 1, 0, 2, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 1, 1, 0, 0, 0, 0, 0, 1, 0, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 3, 2, 1, 0, 0, 0, 0, 0, 1, 0, 2, 1, 3, 2, 2, 2, 2, 2, 3, 2, 4, 3, 2, 1, 1, 1, 1, 1, 2, 1, 3, 2, 2, 1, 1, 1, 1

Offset

0,9

Comments

Assumes that 4 is represented without a hole.

Links

Indranil Ghosh, Table of n, a(n) for n = 0..50000

Formula

a(10i+j) = a(i) + a(j), etc.

Example

8 has two holes so a(8) = 2.

Mathematica

a[n_ /; 0 <= n <= 9] := a[n] = {1, 0, 0, 0, 0, 0, 1, 0, 2, 1}[[n + 1]]; a[n_] := Total[a[#] + 1 &  /@ (id = IntegerDigits[n])] - Length[id];  Table[a[n], {n, 0, 104}] (* Jean-François Alcover, Nov 22 2013 *)
Table[DigitCount[x].{0, 0, 0, 0, 0, 1, 0, 2, 1, 1}, {x, 0, 104}] (* Michael De Vlieger, Feb 02 2017, after Zak Seidov at A064692 *)

Prog

(Python)
def A064532(n):
    x=str(n)
    return x.count("0")+x.count("6")+x.count("8")*2+x.count("9") # Indranil Ghosh, Feb 02 2017
(PARI) h(n) = [1, 0, 0, 0, 0, 0, 1, 0, 2, 1][n];
a(n) = if (n, my(d=digits(n)); sum(i=1, #d, h(d[i]+1)), 1); \\ Michel Marcus, Nov 11 2022

Crossrefs

Cf. A064529, A064530. Equals A064531 - 1.

Cf. A358439 (sum by number of digits).

Keyword

nonn,easy,base

Author

N. J. A. Sloane, Oct 07 2001

Extensions

More terms from Matthew Conroy, Oct 09 2001

Status

approved

A226247 Let S be the set of numbers defined by these rules: 0 is in S; if x is in S, then x+1 is in S, and if nonzero x is in S, then -1/x are in S. (See Comments.)

1, 1, 1, 1, 1, 2, 1, 3, 2, 1, 4, 3, 2, 1, 1, 5, 4, 3, 2, 2, 3, 1, 6, 5, 4, 3, 3, 5, 2, 5, 3, 1, 7, 6, 5, 4, 4, 7, 3, 8, 5, 2, 7, 5, 3, 1, 1, 8, 7, 6, 5, 5, 9, 4, 11, 7, 3, 11, 8, 5, 2, 2, 9, 7, 5, 3, 3, 4, 1, 9, 8, 7, 6, 6, 11, 5, 14, 9, 4, 15, 11, 7, 3, 3

Offset

1,6

Comments

Let S be the set of numbers defined by these rules: 0 is in S; if x is in S, then x+1 is in S, and if nonzero x is in S, then -1/x are in S. Then S is the set of all rational numbers, produced in generations as follows:

g(1) = (0), g(2) = (1), g(3) = (2, -1), g(4) = (3, -1/2), g(5) = (4, -1/3, 1/2), ... For n > 2, once g(n-1) = (c(1), ..., c(z)) is defined, g(n) is formed from the vector (c(1)+1, -1/c(1), c(2)+1, -1/c(2), ..., c(z)+1, -1/c(z)) by deleting previously generated elements. Let S'' denote the sequence formed by concatenating the generations.

A226247: Denominators of terms of S''

A226248: Numerators of terms of S''

A226249: Positions of nonnegative numbers in S''

A226250: Positions of positive numbers in S''

A closely related sequence S' (for which the rules of generation are shorter but the resulting sequence is slightly less natural) is discussed at A226130. For both S' and S'', the number of numbers in g(n) is given by A097333.

Links

Clark Kimberling, Table of n, a(n) for n = 1..1000

Example

The denominators and numerators are read from S'':
  0/1, 1/1, 2/1, -1/1, 3, -1/2, 4/1, -1/3, 1/2, 5, -1/4, 2/3, 3/2, -2, ...
Table begins:
  n |
  --+-----------------------------------------------
  1 | 1;
  2 | 1, 1;
  3 | 1, 2;
  4 | 1, 3, 2;
  5 | 1, 4, 3, 2, 1;
  6 | 1, 5, 4, 3, 2, 2, 3;
  7 | 1, 6, 5, 4, 3, 3, 5, 2, 5, 3;
  8 | 1, 7, 6, 5, 4, 4, 7, 3, 8, 5, 2, 7, 5, 3, 1;

Mathematica

Clear[g]; z = 12; g[1] := {0}; g[2] := {1}; g[n_] :=  g[n] = DeleteCases[Flatten[Transpose[{# + 1, -1/#}]] &[g[n - 1]], Apply[Alternatives, Flatten[Map[g, Range[n - 1]]]]]; f = Flatten[Map[g, Range[z]]]; Take[Denominator[f], 100]  (*A226247*)
t = Take[Numerator[f], 100]  (*A226248*)
s[n_] := If[t[[n]] > 0, 1, 0]; u = Table[s[n], {n, 1, Length[t]}]
Flatten[Position[u, 1]] (*A226249*)
p = Flatten[Position[u, 0]] (*A226250*) (* Peter J. C. Moses, May 30 2013 *)

Prog

(Python)
from fractions import Fraction
from itertools import count, islice
def agen():
    rats = [Fraction(0, 1)]
    seen = {Fraction(0, 1)}
    for n in count(1):
        yield from [r.denominator for r in rats]
        newrats = []
        for r in rats:
            f = 1+r
            if f not in seen:
                newrats.append(1+r)
                seen.add(f)
            if r != 0:
                g = -1/r
                if g not in seen:
                    newrats.append(-1/r)
                    seen.add(g)
        rats = newrats
print(list(islice(agen(), 84))) # Michael S. Branicky, Jan 17 2022

Crossrefs

Cf. A226080 (rabbit ordering of positive rationals), A226130.

Keyword

nonn,easy,tabf

Author

Clark Kimberling, Jun 01 2013

Status

approved

A327981 Distances between successive ones in A051023, the middle column of rule-30 1-D cellular automaton, when started from a lone 1 cell.

1, 2, 1, 1, 3, 1, 4, 2, 1, 3, 3, 1, 1, 2, 2, 1, 1, 3, 1, 1, 2, 2, 2, 1, 5, 1, 3, 2, 2, 1, 2, 2, 2, 1, 1, 1, 1, 1, 5, 1, 1, 1, 4, 2, 2, 1, 1, 6, 3, 2, 1, 4, 1, 1, 4, 1, 2, 1, 2, 1, 2, 8, 4, 1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 4, 1, 1, 2, 2, 1, 1, 6, 1, 3, 4, 1, 3, 1, 1, 1, 3, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 2, 1, 3, 2, 1, 2, 1, 1

Offset

1,2

Comments

First differences of A327984, which gives indices of ones in A051023.

Links

Antti Karttunen, Table of n, a(n) for n = 1..100000

Formula

a(n) = A327984(1+n) - A327984(n).

Example

The evolution of one-dimensional cellular automaton rule 30 proceeds as follows, when started from a single alive (1) cell:
   0:              (1)
   1:             1(1)1
   2:            11(0)01
   3:           110(1)111
   4:          1100(1)0001
   5:         11011(1)10111
   6:        110010(0)001001
   7:       1101111(0)0111111
   8:      11001000(1)11000001
   9:     110111101(1)001000111
  10:    1100100001(0)1111011001
  11:   11011110011(0)10000101111
  12:  110010001110(0)110011010001
  13: 1101111011001(1)1011100110111
The distances between successive 1's in its central column (indicated here with parentheses) are 1-0 (as the first 1 is on row 0, and the second is on row 1), 3-1, 4-3, 5-4, 8-5, 9-8, 13-9, ..., that is, the first terms of this sequence: 1, 2, 1, 1, 3, 1, 4, ...

Mathematica

A327981list[upto_]:=Differences[Flatten[Position[CellularAutomaton[30, {{1}, 0}, {upto, {{0}}}], 1]]]; A327981list[300] (* Paolo Xausa, Jun 27 2023 *)

Prog

(PARI)
up_to = 105;
A269160(n) = bitxor(n, bitor(2*n, 4*n));
A327981list(up_to) = { my(v=vector(up_to), s=1, n=0, on=n, k=0); while(k<up_to, n++; s = A269160(s); if((s>>n)%2, k++; v[k] = (n-on); on=n)); (v); }
v327981 = A327981list(up_to);
A327981(n) = v327981[n];

Crossrefs

Cf. A051023, A110240, A269160, A327980, A327982, A327983, A327984.

Keyword

nonn

Author

Antti Karttunen, Oct 03 2019

Status

approved

A213883 Least number k such that (10^k-j)*10^n-1 is prime for some single-digit j or 0 if no such prime with 1<=k, 0<=j<=9 exists.

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 2, 1, 1, 3, 5, 5, 3, 1, 3, 3, 1, 1, 9, 1, 1, 1, 1, 1, 7, 3, 6, 4, 1, 4, 4, 1, 15, 10, 1, 7, 3, 1, 3, 2, 2, 4, 6, 1, 3, 5, 20, 1, 1, 1, 8, 10, 7, 15, 10, 1, 4, 2, 5, 8, 3, 23, 11, 2, 2, 9, 3, 1, 5, 4, 1, 6, 3, 18, 2

Offset

1,9

Comments

j cannot be 0, 3, 6 or 9 because we are searching for repdigit primes with k-1 times the digit 9, one digit (9-j), and n least-significant digits 9 (so n+k-1 times the digit 9 in total). If j is a multiple of 3, that number is also a multiple of 3 and not prime.

Conjecture: there is always at least one (k,j) solution for each n.

Links

Pierre CAMI, Table of n, a(n) for n = 1..2200

Example

Refers to the primes 89, 599, 8999, 79999, 799999, 4999999, 89999999,...

Maple

A213883 := proc(n)
    for k from 1 to 2*n-1 do
        for j from 0 to 9 do
            if isprime( (10^k-j)*10^n-1) then
                return k;
            end if;
        end do:
    end do:
    return 0 ;
end proc: # R. J. Mathar, Jul 20 2012

Prog

SCRIPT
DIM nn, 0
DIM jj
DIM kk
DIMS tt
OPENFILEOUT myfile, a(n).txt
LABEL loopn
SET nn, nn+1
IF nn>2200 THEN END
SET kk, 0
LABEL loopk
SET kk, kk+1
IF kk>2*nn THEN GOTO loopn
SET jj, 0
LABEL loopj
SET jj, jj+1
IF jj%3==0 THEN SET jj, jj+1
IF jj>9 THEN GOTO loopk
SETS tt, %d, %d, %d\,; nn; kk; jj
PRP (10^kk-jj)*10^nn-1, tt
IF ISPRP THEN GOTO a
IF ISPRIME THEN GOTO a
GOTO loopj
LABEL a
WRITE myfile, tt
GOTO loopn

Crossrefs

Cf. A213790, A213884 (corresponding j).

Keyword

nonn

Author

Pierre CAMI, Jun 26 2012

Status

approved

A220901 Triangle read by rows: k-th "a-number" of star graph K_{1,n-1}.

1, 1, 1, 1, 1, 2, 1, 3, 2, 1, 4, 8, 1, 5, 20, 16, 1, 6, 40, 96, 1, 7, 70, 336, 272, 1, 8, 112, 896, 2176, 1, 9, 168, 2016, 9792, 7936, 1, 10, 240, 4032, 32640, 79360, 1, 11, 330, 7392, 89760, 436480, 353792, 1, 12, 440, 12672, 215424, 1745920, 4245504

Offset

0,6

Links

Alois P. Heinz, Rows n = 0..200, flattened

Suyoung Choi and Hanchul Park, A new graph invariant arises in toric topology, arXiv preprint arXiv:1210.3776 [math.AT], 2012-2013. See Table 4.

Formula

T(n,k) = binomial(n-1, 2*k-1)*A000111(2*k-1) (see Theorem 2.9 in paper). - Michel Marcus, Feb 07 2013

Example

Triangle begins:
  1;
  1;
  1, 1;
  1, 2;
  1, 3,  2;
  1, 4,  8;
  1, 5, 20,  16;
  1, 6, 40,  96;
  1, 7, 70, 336, 272;
  ...

Mathematica

t[n_, k_] := t[n, k] = If[k == 0, Boole[n == 0], t[n, k-1] + t[n-1, n-k]];
T[n_, k_] := If[k == 0, 1, Binomial[n-1, 2k-1] t[2k-1, 2k-1]];
Table[T[n, k], {n, 0, 13}, {k, 0, Floor[n/2]}] // Flatten (* Jean-François Alcover, Oct 06 2018 *)

Prog

(PARI) T(n, k) = {if (k == 0, return(1)); binomial(n - 1, 2*k - 1)*(2*k - 1)!*polcoeff(tan(x + O(x^(2*n + 2))), 2*k - 1); } \\ Michel Marcus, Feb 07 2013

Keyword

nonn,tabf

Author

N. J. A. Sloane, Jan 01 2013

Extensions

More terms from Michel Marcus, Feb 07 2013

Status

approved

A249622 a(n) = number of ways to express A117048(n) as the sum of two positive triangular numbers.

1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 2, 1, 1, 3, 1, 2, 1, 1, 2, 1, 3, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 4, 1, 3, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 4, 1, 1, 1, 3, 1, 1, 2, 1, 2, 2, 1, 1, 2, 1, 1, 3, 1, 2, 1, 2, 1, 1, 6, 2, 1, 1, 1, 1

Offset

1,6

Links

Zak Seidov, Table of n, a(n) for n = 1..10000

Example

a(6) = 2 because A117048(6) = 31 and 31 = 3 + 28 = 10 + 21 (first case of two-way expression).
a(22) = 3 because A117048(22) = 181 and 181 = A000217(i) + A000217(k), for {i,k} = {{4, 18}, {7, 17}, {9, 16}} (first case of three-way expression): 181 = 10 + 171 = 28 + 153 = 45 + 136.

Crossrefs

Cf. A000040, A000217, A002383.

Keyword

nonn

Author

Zak Seidov, Nov 03 2014

Status

approved

A349497 a(n) is the smallest element in the continued fraction of the harmonic mean of the divisors of n.

1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 2, 1, 3, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 3, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1

Offset

1,6

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

a(p) = 1 for a prime p.

a(p^2) = 1 for a prime p != 3.

a(A129521(n)) = 1 for n > 3.

For a harmonic number m = A001599(k), a(m) = A099377(m) = A001600(k).

Example

a(2) = 1 since the continued fraction of the harmonic mean of the divisors of 2, 4/3 = 1 + 1/3, has 2 elements, {1, 3}, and the smallest of them is 1.

Mathematica

a[n_] := Min[ContinuedFraction[DivisorSigma[0, n] / DivisorSigma[-1, n]]]; Array[a, 100]

Crossrefs

Cf. A001248, A001599, A001600, A099377, A099378, A129521, A349473.

Keyword

nonn

Author

Amiram Eldar, Nov 20 2021

Status

approved

A063059 a(n) = number of 'Reverse and Add!' operations that have to be applied to the n-th term of A063058 in order to obtain a term in the trajectory of 7059.

1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 0, 2, 1, 3, 2, 1, 3, 2, 1, 1, 1, 1, 1, 2, 1, 3, 2, 1, 3, 2, 1, 1, 1, 1, 1, 2, 1, 3, 2, 1, 3, 2, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 2, 1, 3, 2, 1

Offset

0,6

Links

Table of n, a(n) for n=0..104.

Index entries for sequences related to Reverse and Add!

Example

7239 is a term of A063058. One 'Reverse and Add!' operation applied to 7239 leads to a term (16566) in the trajectory of 7059, so the corresponding term of the present sequence is 1.

Crossrefs

A023108, A033865, A063057, A063058.

Keyword

base,nonn

Author

Klaus Brockhaus, Jul 07 2001

Status

approved