Eric Jovinelly - cp's OEIS frontend

A317027 Perfect powers that appear more than once in the Catalan triangle.

9, 27, 324, 8000, 11025, 374544, 12723489, 432224100, 14682895929, 498786237504, 16944049179225, 575598885856164, 19553418069930369, 664240615491776400, 22564627508650467249, 766533094678624110084

Offset: 1

Eric Jovinelly, Eugene Fiorini, Jul 19 2018

Since every integer appears exactly once in the first column of the Catalan triangle, this sequence lists perfect powers that appear in any other column of the triangle. Pell's equation can be used to prove that infinitely many perfect squares appear in this sequence.

Nathaniel Benjamin, Grant Fickes, Eugene Fiorini, Edgar Jaramillo Rodriguez, Eric Jovinelly, Tony W. H. Wong, Primes and Perfect Powers in the Catalan Triangle, J. Int. Seq., Vol. 22 (2019), Article 19.7.6.
Eric Jovinelly, Python program

Cf. A009766.

Subsequence of A275586.

a(9)-a(16) from Giovanni Resta, Jul 29 2018

A290322 Sum modulo n of all units u in Z/nZ such that Phi(5,u) is a unit, where Phi is the cyclotomic polynomial.

Original entry on oeis.org

1, 0, 0, 4, 0, 0, 0, 0, 9, 1, 0, 0, 0, 3, 0, 0, 0, 0, 8, 0, 12, 0, 0, 20, 0, 0, 0, 0, 18, 1, 0, 24, 0, 14, 0, 0, 0, 0, 16, 1, 0, 0, 24, 9, 0, 0, 0, 0, 45, 0, 0, 0, 0, 14, 0, 0, 0, 0, 36, 1, 32, 0, 0, 13, 24, 0, 0, 0, 14, 1, 0, 0, 0, 15, 0, 28, 0, 0, 32, 0, 42, 0

Offset: 2

Joshua Harrington, Michael Mueller, Jordan Lenchitz, Tristan Phillips, Madison Wellen, Eric Jovinelly, Jul 27 2017

Conjecture: If n is divisible by 5 then a(n) > 0. - Robert Israel, Jan 23 2024

Robert Israel, Table of n, a(n) for n = 2..10000

Cf. A058026, A289460.

with(numtheory): m:=5: for n from 2 to 100 do S:={}: for a from 1 to n-1 do if gcd(a,n)=1 and gcd(cyclotomic(m,a),n)=1 then S:={op(S),a}: fi: od: print(sum(op(i,S),i=1..nops(S)) mod n): od:

Table[Mod[Total@ Select[Range[n - 1], CoprimeQ[#, n] && CoprimeQ[Cyclotomic[5, #], n] &], n], {n, 83}] (* Michael De Vlieger, Jul 29 2017 *)

a(n) = sum(k=0, n-1, k*((gcd(n, k)==1) && (gcd(n, polcyclo(5, k))==1))) % n; \\ Michel Marcus, Jul 29 2017

A290321 Sum modulo n of all units u in Z/nZ such that Phi(3,u) is a unit, where Phi is the cyclotomic polynomial.

Original entry on oeis.org

1, 2, 0, 0, 5, 1, 0, 6, 0, 0, 4, 1, 8, 5, 0, 0, 15, 1, 0, 8, 0, 0, 8, 0, 14, 18, 16, 0, 20, 1, 0, 11, 0, 25, 12, 1, 20, 14, 0, 0, 8, 1, 0, 15, 0, 0, 16, 7, 0, 17, 28, 0, 45, 0, 32, 20, 0, 0, 40, 1, 32, 24, 0, 30, 44, 1, 0, 23, 60, 0, 24, 1, 38, 25, 40, 66, 14, 1

Offset: 2

Michael Mueller, Jordan Lenchitz, Tristan Phillips, Madison Wellen, Eric Jovinelly, Joshua Harrington, Jul 27 2017

nonn

Cf. A058026, A289460.

with(numtheory): m:=3: for n from 2 to 100 do S:={}: for a from 1 to n-1 do if gcd(a,n)=1 and gcd(cyclotomic(m,a),n)=1 then S:={op(S),a}: fi: od: print(sum(op(i,S),i=1..nops(S)) mod n): od:

Table[Mod[Total@ Select[Range[n - 1], CoprimeQ[#, n] && CoprimeQ[Cyclotomic[3, #], n] &], n], {n, 79}] (* Michael De Vlieger, Jul 29 2017 *)

a(n) = sum(k=0, n-1, k*((gcd(n, k)==1) && (gcd(n, polcyclo(3, k))==1))) % n; \\ Michel Marcus, Jul 29 2017

A289835 Number of units u in Z/(2n-1)Z such that Phi(4,u) is a unit, where Phi is the cyclotomic polynomial.

Original entry on oeis.org

1, 2, 2, 6, 6, 10, 10, 4, 14, 18, 12, 22, 10, 18, 26, 30, 20, 12, 34, 20, 38, 42, 12, 46, 42, 28, 50, 20, 36, 58, 58, 36, 20, 66, 44, 70, 70, 20, 60, 78, 54, 82, 28, 52, 86, 60, 60, 36, 94, 60, 98, 102, 24, 106, 106, 68, 110, 44, 60, 84, 110, 76, 50, 126, 84

Offset: 1

Jordan Lenchitz, Michael Mueller, Tristan Phillips, Madison Wellen, Eric Jovinelly, Joshua Harrington, Jul 25 2017

If k is even, the number of units u in Z/kZ such that Phi(4,u) is a unit is zero.

Jordan Lenchitz, Table of n, a(n) for n = 1..1000

Cf. A058026, A289460.

m:=4 do for t from 1 to 1000 do n:=2*t-1: S:={}: for a from 0 to n-1 do if gcd(a,n)=1 and gcd(cyclotomic(m,a),n)=1 then S:={op(S),a}: fi: od: print(t,nops(S)): od: od:

a(n) = sum(k=0, 2*n-2, (gcd(2*n-1, k)==1) && (gcd(2*n-1, polcyclo(4, k))==1)); \\ Michel Marcus, Jul 29 2017

A290013 Length of the period of the continued fraction expansion of phi/n where phi is the golden ratio.

Original entry on oeis.org

1, 1, 2, 2, 1, 6, 2, 2, 6, 5, 4, 4, 1, 10, 8, 4, 3, 2, 8, 14, 2, 12, 10, 4, 11, 5, 14, 10, 4, 28, 8, 8, 8, 1, 20, 2, 7, 4, 8, 14, 6, 6, 18, 8, 24, 6, 2, 4, 22, 31, 12, 14, 9, 10, 2, 12, 16, 12, 20, 20, 5, 8, 8, 20, 13, 20, 22, 2, 10, 52, 28, 2, 15, 19, 36, 4

Offset: 1

Vanessa Gomez, Eric Jovinelly, Jacob A. McCann, Bryce Orloski, Catherine Rea, Shannon Talbott, Jul 17 2017

We calculated the continued fraction expansion of phi/n and observed that the expansion is periodic after the first nonzero term. We tracked the periodicity of the expansions and present them here. The authors acknowledge the National Science Foundation (DMS-1560019) and Muhlenberg College for supporting the REU (Research Experiences for Undergraduates) on which this sequence is based.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A001622 (phi), A019863 (phi/2), A134943 (phi/3), A134944 (phi/4), A134946 (phi/6).

a[n_] := ContinuedFraction[GoldenRatio/n] // Last // Length; Array[a, 80] (* Jean-François Alcover, Jul 28 2017 *)

A290309 Number of units u in Z/nZ such that Phi(5,u) is a unit, where Phi is the cyclotomic polynomial.

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 6, 4, 6, 3, 6, 4, 12, 6, 6, 8, 16, 6, 18, 6, 12, 6, 22, 8, 15, 12, 18, 12, 28, 6, 26, 16, 12, 16, 18, 12, 36, 18, 24, 12, 36, 12, 42, 12, 18, 22, 46, 16, 42, 15, 32, 24, 52, 18, 18, 24, 36, 28, 58, 12, 56, 26, 36, 32, 36, 12, 66, 32, 44, 18

Offset: 1

Tristan Phillips, Jordan Lenchitz, Michael Mueller, Madison Wellen, Eric Jovinelly, Joshua Harrington, Jul 27 2017

If n is a prime other than 5, then a(n) = n - 5 if n == 1 (mod 10), otherwise a(n) = n - 1. - Robert Israel, Jul 31 2017

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A058026 (with Phi(1,u) or Phi(2,u)), A289460 (with Phi(3,u)).

m:=5; T:=[]: for n from 1 to 100 do S:={}: for a from 0 to n-1 do if gcd(a,n)=1 and gcd(cyclotomic(m,a),n)=1 then S:={op(S),a}: fi: od: T:=[op(T),nops(S)]: od: print(T):

Table[Count[Range[n - 1], k_ /; And[CoprimeQ[k, n], CoprimeQ[Cyclotomic[5, k], n]]], {n, 70}] (* Michael De Vlieger, Jul 30 2017 *)

a(n) = sum(k=0, n-1, (gcd(n, k)==1) && (gcd(n, polcyclo(5, k))==1)); \\ Michel Marcus, Jul 29 2017

A289460 Number of units u in Z/nZ such that Phi(3,u) is a unit, where Phi is the cyclotomic polynomial.

Original entry on oeis.org

1, 1, 1, 2, 4, 1, 4, 4, 3, 4, 10, 2, 10, 4, 4, 8, 16, 3, 16, 8, 4, 10, 22, 4, 20, 10, 9, 8, 28, 4, 28, 16, 10, 16, 16, 6, 34, 16, 10, 16, 40, 4, 40, 20, 12, 22, 46, 8, 28, 20, 16, 20, 52, 9, 40, 16, 16, 28, 58, 8, 58, 28, 12, 32, 40, 10, 64, 32, 22, 16, 70, 12

Offset: 1

Jordan Lenchitz, Michael Mueller, Tristan Phillips, Madison Wellen, Eric Jovinelly, Joshua Harrington, Jul 06 2017

The number of units u in Z/nZ such that Phi(1,u) or Phi(2,u) is a unit is given by A058026.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (terms to 1000 from Jordan Lenchitz)

Cf. A058026.

m:=3; T:=[]: for n from 2 to 1000 do S:={}: for a from 0 to n-1 do if gcd(a,n)=1 and gcd(cyclotomic(m,a),n)=1 then S:={op(S),a}: fi: od: T:=[op(T),nops(S)]: od: print(m,T):

Table[Count[Map[Cyclotomic[3, #] &, Select[Range@ n, CoprimeQ[#, n] &]], u_ /; CoprimeQ[u, n]], {n, 72}] (* Michael De Vlieger, Jul 11 2017 *)

g(n)=sum(k=0,n-1, gcd(k,n)==1 && gcd(polcyclo(3,k),n)==1)
a(n)=my(f=factor(n)); prod(i=1,#f~, g(f[i,1]^f[i,2])) \\ Charles R Greathouse IV, Jul 06 2017

A275586 Numbers k that appear more than once in c_{m,n} for integers m >= n >= 1 where c_{m,n} = ((m+n)!(m-n+1))/((n)!(m+1)!).

Original entry on oeis.org

1, 2, 5, 9, 14, 20, 27, 28, 35, 42, 44, 48, 54, 65, 75, 77, 90, 104, 110, 119, 132, 135, 152, 154, 165, 170, 189, 208, 209, 230, 252, 273, 275, 297, 299, 324, 350, 377, 405, 429, 434, 440, 464, 495, 527, 544, 560, 572, 594, 629, 637, 663, 665, 702, 740, 779, 798, 819, 860, 902, 910, 945, 950, 989

Offset: 1

Edgar Jaramillo Rodriguez, Nathaniel Benjamin, Eric Jovinelly, Eugene Fiorini, Aug 02 2016

nonn

Integers that do not appear uniquely in the Catalan triangle A009766.

			The Catalan triangle (A009766) starts:
1
1, 1
1, 2, 2
1, 3, 5,  5
1, 4, 9, 14, 14
Each entry is the sum of elements in the previous row except for those which are further right. The columns are nondecreasing, and all positive integers appear in the second column.
Since 2 appears twice in the triangle, it is in the sequence. Since 6 appears only once in the triangle, it is not in the sequence. - _Michael B. Porter_, Aug 05 2016

Giovanni Resta, Table of n, a(n) for n = 1..10000
D. F. Bailey, Counting arrangements of 1's and -1's, Mathematics Magazine, 69 (1996): 128-131.
Nathaniel Benjamin, Grant Fickes, Eugene Fiorini, Edgar Jaramillo Rodriguez, Eric Jovinelly, Tony W. H. Wong, Primes and Perfect Powers in the Catalan Triangle, J. Int. Seq., Vol. 22 (2019), Article 19.7.6.
Eric W. Weisstein, Catalan's Triangle

Cf. A009766, A275481 (complement).

def remove_duplicates(values):
    output = []
    seen = set()
    for value in values:
        if value not in seen:
            output.append(value)
            seen.add(value)
    return output
def Non_Unique_Catalan_Triangle(k):
    t = []
    t.append([])
    t[0].append(1)
    for h in range(1, k):
        t.append([])
        t[0].append(1)
    for i in range(1, k):
        for j in range(0, k):
            if i>j:
                t[i].append(0)
            else:
                t[i].append(t[i-1][j] + t[i][j-1])
    l = []
    for r in range(0, k):
        for s in range(0, k):
            l.append(t[r][s])
    non_unique = []
    for n in l:
        if  n <= k and n>1 and l.count(n) > 1:
            non_unique.append(n)
    non_unique = remove_duplicates(non_unique)
    print (non_unique)

A275481 Integers that appear uniquely in the Catalan triangle, A009766.

Original entry on oeis.org

3, 4, 6, 7, 8, 10, 11, 12, 13, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 41, 43, 45, 46, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 66, 67, 68, 69, 70, 71, 72, 73, 74, 76, 78, 79, 80, 81, 82, 83, 84, 85

Offset: 1

Edgar Jaramillo Rodriguez, Nathaniel Benjamin, Eric Jovinelly, and Eugene Fiorini, Jul 29 2016

n appears once in c_{m,k} for integers m >= k >= 1 where c_{m,k} = ((n+k)!(n-k+1))/((k)!(n+1)!).

D. F. Bailey, Counting arrangements of 1's and-1's, Mathematics Magazine, 69 (1996): 128-131.
Nathaniel Benjamin, Grant Fickes, Eugene Fiorini, Edgar Jaramillo Rodriguez, Eric Jovinelly, and Tony W. H. Wong, Primes and Perfect Powers in the Catalan Triangle, J. Int. Seq., Vol. 22 (2019), Article 19.7.6.
Eric W. Weisstein, Catalan's Triangle

Subsequence of A007401, which is the complement of A000096.

Cf. A009766, A275586 (complement).

Block[{T, nn = 85}, T[n_, k_] := T[n, k] = Which[k == 0, 1, k > n, 0, True, T[n - 1, k] + T[n, k - 1]]; Rest@ Complement[Range@ nn, Union@ Flatten@ Table[T[n, k], {n, 2, nn}, {k, 2, n}]]] (* Michael De Vlieger, Feb 04 2020, after Jean-François Alcover at A009766 *)

#prints the unique integers less than k
def Unique_Catalan_Triangle(k):
    t = []
    t.append([])
    t[0].append(1)
    for h in range(1, k):
        t.append([])
        t[0].append(1)
    for i in range(1, k):
        for j in range(0, k):
            if i>j:
                t[i].append(0)
            else:
                t[i].append(t[i-1][j] + t[i][j-1])
    l = []
    for r in range(0, k):
        for s in range(0, k):
            l.append(t[r][s])
    unique = []
    for n in l:
        if n <= k and l.count(n) == 1 :
            unique.append(n)
    print(sorted(unique))

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User: Eric Jovinelly

A317027 Perfect powers that appear more than once in the Catalan triangle.

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A290322 Sum modulo n of all units u in Z/nZ such that Phi(5,u) is a unit, where Phi is the cyclotomic polynomial.

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A290321 Sum modulo n of all units u in Z/nZ such that Phi(3,u) is a unit, where Phi is the cyclotomic polynomial.

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A289835 Number of units u in Z/(2n-1)Z such that Phi(4,u) is a unit, where Phi is the cyclotomic polynomial.

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A290013 Length of the period of the continued fraction expansion of phi/n where phi is the golden ratio.

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A290309 Number of units u in Z/nZ such that Phi(5,u) is a unit, where Phi is the cyclotomic polynomial.

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Maple

Mathematica

PARI

A289460 Number of units u in Z/nZ such that Phi(3,u) is a unit, where Phi is the cyclotomic polynomial.

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Maple

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A275586 Numbers k that appear more than once in c_{m,n} for integers m >= n >= 1 where c_{m,n} = ((m+n)!(m-n+1))/((n)!(m+1)!).

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