A003989 -id:A003989 - cp's OEIS frontend

A186230 Triangle T(n,k), n>=1, 1<=k<=n, read by rows: T(n,k) is the number of positive integers j

0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 2, 2, 0, 0, 0, 0, 0, 1, 0, 0, 1, 2, 2, 4, 2, 0, 0, 0, 1, 0, 2, 0, 3, 0, 0, 1, 0, 1, 3, 0, 4, 3, 0, 0, 0, 1, 0, 0, 0, 2, 0, 2, 0, 0, 1, 2, 2, 4, 2, 6, 4, 6, 4, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 3, 0, 0, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4, 0, 0, 0, 1, 0, 2, 0, 0, 0, 2, 0, 4, 0, 5, 0

Offset: 1

Alois P. Heinz, Feb 15 2011

T(n,k) = A000010(k) if n is prime and 1

			T(n,1) = 0 because no positive integer j<1 can be found.
T(n,k) = 0 if GCD(n,k)>1.
T(7,5) = 4 because for j in {1,2,3,4} all conditions are satisfied.
Triangle T(n,k) begins:
  0;
  0, 0;
  0, 1, 0;
  0, 0, 1, 0;
  0, 1, 2, 2, 0;
  0, 0, 0, 0, 1, 0;
  0, 1, 2, 2, 4, 2, 0;

Alois P. Heinz, Rows n = 1..250, flattened

Row sums give: A185953. Column k=2 gives: A000035 for n>1. Lower diagonal gives: A057475(n-1) for n>2. Cf. A000010, A000040, A003989.

with(numtheory):
T:= proc(n,k) local c, i, j, m;
      if k=1 or igcd(n, k)>1 then 0
    elif isprime(n) then phi(k)
    else m:= n*k;
         i:= igcd(m, 2);
         c:= 0;
         for j to k-1 by i do
           if igcd(m, j)=1 then c:= c+1 fi
         od; c
      fi
    end:
seq(seq(T(n, k), k=1..n), n=1..20);

t[n_, k_] := Module[{c, i, j, m}, If[ k == 1 || GCD[n, k] > 1, 0, If[PrimeQ[n], EulerPhi[k], m = n*k; i = GCD[m, 2]; c = 0; For[j = 1, j <= k-1, j = j+i, If[GCD[m, j] == 1, c = c+1]]; c]]]; Table[Table[t[n, k], {k, 1, n}], {n, 1, 20}] // Flatten (* Jean-François Alcover, Dec 19 2013 *)

T(n,k) = |{ j : 1 <= j < k and GCD(n,k) = GCD(n,j) = GCD(k,j) = 1 }|.

A258820 Reversed rows of A178252 presented as diagonals of an irregular triangle.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 2, 1, 1, 5, 2, 1, 1, 3, 10, 1, 1, 7, 5, 5, 1, 1, 4, 7, 5, 1, 1, 9, 28, 35, 3, 1, 1, 5, 12, 14, 7, 1, 1, 11, 15, 21, 14, 7, 1, 1, 6, 55, 30, 126, 28, 1, 1, 13, 22, 165, 42, 21, 4, 1

Offset: 0

Tom Copeland, Jun 18 2015

The diagonals of T are the reversed rows of A178252. E.g., the fifth diagonal of T is (1,2,2,1,1) from the example below, which is the fifth reversed row of A178252.
Factoring out the greatest common divisor (gcd) of the coefficients of the sub-polynomials in the indeterminate q of the polynomials in s presented on p. 12 of the Alexeev et al. link and then evaluating the sub-polynomials at q=1 gives the first nine rows of T given in the example below. E.g., for k=6 (the seventh row), q*s^6 + (6*q + 9*q^2) s^4 + (15*q + 15*q^2) s^2 + 5  = q*s^6 + 3*(2*q + 3*q^2)*s^4 + 15*(q + q^2)*s^2 + 5 generates (1,2+3,1+1,1)=(1,5,2,1).
The row length sequence of this irregular triangle is A008619(n) = 1 + floor(n/2). - Wolfdieter Lang, Aug 25 2015

			The irregular triangle T(n,k) starts
n\k  0 1  2  3 4 5 ...
0:   1
1:   1
2:   1 1
3:   1 1
4:   1 3  1
5:   1 2  1
6:   1 5  2  1
7:   1 3 10  1
8:   1 7  5  5 1
9:   1 4  7  5 1
10:  1 9 28 35 3 1
... reformatted. - _Wolfdieter Lang_, Aug 25 2015

N. Alexeev, J. Andersen, R. Penner, P. Zograf, Enumeration of chord diagrams on many intervals and their non-orientable analogs, arXiv:1307.0967 [math.CO], 2013-2014.

Cf. A050169, A161642, A055151, A088617, A178252, A107711, A165661, A003989, A008619.

max = 15; coes = Table[ PadRight[ CoefficientList[ BernoulliB[n, x], x], max], {n, 0, max-1}]; inv = Inverse[coes] // Numerator; t[n_, k_] := inv[[n, k]]; t[n_, k_] /; k == n+1 = 1; Table[t[n-k+1, k], {n, 2, max+1}, {k, 2, Floor[n/2]+1}] // Flatten (* Jean-François Alcover, Jul 22 2015 *)

T(n,k) = A178252(n-k,n-2k) = A055151(n,k) / A161642(n,k) = A007318(n,2k) * A000108(k) / A161642(n,k) = n! / [(n-2k)! k! (k+1)! A161642(n,k)] = A003989(n-k+1,k+1) * (n-k)! / [ (n-2k)! (k+1)! ], where A003989(j,k) = gcd(j,k).

A285724 Square array read by descending antidiagonals: If n > k, A(n,k) = T(lcm(n,k), gcd(n,k)), otherwise A(n,k) = T(gcd(n,k), lcm(n,k)), where T(n,k) is sequence A000027 considered as a two-dimensional table.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 16, 21, 10, 11, 12, 13, 14, 15, 16, 46, 67, 78, 55, 21, 22, 23, 106, 25, 120, 27, 28, 29, 92, 31, 191, 210, 34, 105, 36, 37, 38, 211, 80, 41, 90, 231, 44, 45, 46, 154, 277, 379, 436, 465, 406, 300, 171, 55, 56, 57, 58, 59, 596, 61, 630, 63, 64, 65, 66, 67, 232, 436, 631, 781, 862, 903, 820, 666, 465, 253, 78, 79, 80, 529, 212, 991, 302, 85, 324, 1035, 230, 561, 90, 91

Offset: 1

Antti Karttunen, May 03 2017

The array is read by descending antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

			The top left 12 X 12 corner of the array:
   1,   2,   4,   7,   11,   16,   22,   29,   37,   46,   56,   67
   3,   5,  16,  12,   46,   23,   92,   38,  154,   57,  232,   80
   6,  21,  13,  67,  106,   31,  211,  277,   58,  436,  529,   94
  10,  14,  78,  25,  191,   80,  379,   59,  631,  212,  947,  109
  15,  55, 120, 210,   41,  436,  596,  781,  991,   96, 1486, 1771
  21,  27,  34,  90,  465,   61,  862,  302,  193,  467, 2146,  142
  28, 105, 231, 406,  630,  903,   85, 1541, 1954, 2416, 2927, 3487
  36,  44, 300,  63,  820,  324, 1596,  113, 2557,  822, 3829,  355
  45, 171,  64, 666, 1035,  208, 2016, 2628,  145, 4006, 4852,  706
  55,  65, 465, 230,  101,  495, 2485,  860, 4095,  181, 5996, 1832
  66, 253, 561, 990, 1540, 2211, 3003, 3916, 4950, 6105,  221, 8647
  78,  90, 103, 117, 1830,  148, 3570,  375,  739, 1890, 8778,  265

Antti Karttunen, Table of n, a(n) for n = 1..10585; the first 145 antidiagonals of array
MathWorld, Pairing Function

Cf. A000124 (row 1), A000217 (column 1), A001844 (main diagonal).

Cf. A000027, A003989, A003990, A003991, A286101, A286102, A286155, A285722.

(define (A285724 n) (A285724bi (A002260 n) (A004736 n)))
(define (A285724bi row col) (if (> row col) (A000027bi (lcm row col) (gcd row col)) (A000027bi (gcd row col) (lcm row col))))
(define (A000027bi row col) (* (/ 1 2) (+ (expt (+ row col) 2) (- row) (- (* 3 col)) 2)))

If n > k, A(n,k) = T(lcm(n,k),gcd(n,k)), otherwise A(n,k) = T(gcd(n,k),lcm(n,k)), where T(n,k) is sequence A000027 considered as a two-dimensional table, that is, as a pairing function from N x N to N.

If n < k, A(n,k) = A286101(n,k), otherwise A(n,k) = A286102(n,k).

A287957 Table read by antidiagonals: T(n, k) = greatest common recursive divisor of n and k; n > 0 and k > 0.

Original entry on oeis.org

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

Offset: 1

Rémy Sigrist, Jun 03 2017

We use the definition of recursive divisor given in A282446.
More informally, the prime tower factorization of T(n, k) is the intersection of the prime tower factorizations of n and k (the prime tower factorization of a number is defined in A182318).
This sequence has connections with the classical GCD (A003989).
For any i > 0, j > 0 and k > 0:
- T(i, j) = 1 iff gcd(i, j) = 1,
- A007947(T(i, j)) = A007947(gcd(i, j)),
- T(i, j) >= 1,
- T(i, j) <= min(i, j),
- T(i, j) <= gcd(i, j),
- T(i, 1) = 1,
- T(i, i) = i,
- T(i, j) = T(j, i) (the sequence is commutative),
- T(i, T(j, k)) = T(T(i, j), k) (the sequence is associative),
- T(i, i*j) <= i,
- if gcd(i, j) = 1 then T(i*j, k) = T(i, k) * T(j, k) (the sequence is multiplicative),
- T(i, 2*i) = A259445(i).
See also A287958 for the LCM equivalent.

			Table starts:
n\k|    1   2   3   4   5   6   7   8   9   10
---+-----------------------------------------------
1  |    1   1   1   1   1   1   1   1   1    1  ...
2  |    1   2   1   2   1   2   1   2   1    2  ...
3  |    1   1   3   1   1   3   1   1   3    1  ...
4  |    1   2   1   4   1   2   1   2   1    2  ...
5  |    1   1   1   1   5   1   1   1   1    5  ...
6  |    1   2   3   2   1   6   1   2   3    2  ...
7  |    1   1   1   1   1   1   7   1   1    1  ...
8  |    1   2   1   2   1   2   1   8   1    2  ...
9  |    1   1   3   1   1   3   1   1   9    1  ...
10 |    1   2   1   2   5   2   1   2   1   10  ...
...
T(4, 8) = T(2^2, 2^3) = 2.

Rémy Sigrist, First 100 antidiagonals of array, flattened
Rémy Sigrist, Illustration of the first terms

Cf. A003989, A007947, A182318, A259445, A282446, A287958.

T(n,k) = my (g=factor(gcd(n,k))); return (prod(i=1, #g~, g[i,1]^T(valuation(n, g[i,1]), valuation(k, g[i,1]))))

A331306 Lexicographically earliest infinite sequence such that a(i) = a(j) => A285732(i) = A285732(j) for all i, j.

Original entry on oeis.org

1, 2, 2, 3, 4, 5, 6, 5, 3, 7, 8, 9, 10, 9, 11, 12, 13, 7, 6, 14, 15, 16, 17, 14, 18, 13, 19, 20, 21, 22, 23, 11, 8, 23, 24, 25, 26, 27, 28, 19, 29, 17, 30, 31, 32, 33, 34, 35, 30, 15, 12, 28, 36, 37, 38, 39, 40, 41, 42, 24, 43, 22, 42, 44, 45, 46, 47, 48, 49, 50, 36, 20, 16, 35, 51, 52, 53, 54, 55, 56, 57, 58, 51, 31, 59, 27, 50, 60, 61, 62, 63, 64, 65, 66, 67, 68, 44, 25, 21, 41

Offset: 1

Antti Karttunen, Jan 19 2020

Restricted growth sequence transform of A285732 (when considered as an one-dimensional sequence).
For all i, j:
  a(i) = a(j) => A003989(i) = A003989(j),
  a(i) = a(j) => A331307(i) = A331307(j) => A072030(i) = A072030(j).

Antti Karttunen, Table of n, a(n) for n = 1..25425

Cf. A003989, A072030, A285732.

Cf. also A331305, A331307.

up_to = 25425; \\ = binomial(225+1,2)
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A000027pairton(a,b) = ((2+((a+b)^2 - a) - (3*b))/2);
A285732sq(n, k) = if(n==k,-n,if(n>k,A000027pairton(n-k,k),A000027pairton(n,k-n)));
A285732list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A285732sq(col,(a-(col-1))))); (v); };
v331306 = rgs_transform(A285732list(up_to));
A331306(n) = v331306[n];

A345418 Table read by upward antidiagonals: Given m, n >= 1, write gcd(prime(m),prime(n)) as d = uprime(m)+vprime(n) where u, v are minimal; T(m,n) = v.

Original entry on oeis.org

1, -1, 1, -2, 1, 1, -3, 2, -1, 1, -5, -2, 1, 1, 1, -6, 4, 3, -2, -1, 1, -8, -4, -2, 1, 1, 1, 1, -9, 6, -5, -3, 2, 2, -1, 1, -11, -6, 7, 2, 1, -1, -2, 1, 1, -14, 8, 4, 5, 6, -5, -2, -1, -1, 1, -15, 10, -9, -8, -3, 1, 2, 3, 2, -1, 1, -18, -10, 6, 10, 7, 4, -3, -4, -3, -1, 1, 1

Offset: 1

N. J. A. Sloane, Jun 19 2021

The gcd is 1 unless m=n when it is m; u is given in A345417. Minimal means minimize u^2+v^2. We follow Maple, PARI, etc., in setting u=0 and v=1 when m=n. If we ignore the diagonal, the v table is the transpose of the u table.

			The u table (A345417) begins:
[0, -1, -2, -3, -5, -6, -8, -9, -11, -14, -15, -18, -20, -21, -23, -26]
[1,  0,  2, -2,  4, -4,  6, -6,   8,  10, -10, -12,  14, -14,  16,  18]
[1, -1,  0,  3, -2, -5,  7,  4,  -9,   6,  -6,  15,  -8, -17,  19, -21]
[1,  1, -2,  0, -3,  2,  5, -8,  10,  -4,   9,  16,   6,  -6, -20, -15]
[1, -1,  1,  2,  0,  6, -3,  7,  -2,   8, -14, -10,  15,   4, -17, -24]
[1,  1,  2, -1, -5,  0,  4,  3,  -7,   9,  12, -17,  19,  10, -18,  -4]
[1, -1, -2, -2,  2, -3,  0,  9,  -4,  12,  11, -13, -12,  -5, -11,  25]
[1,  1, -1,  3, -4, -2, -8,  0,  -6,  -3, -13,   2,  13,  -9,   5,  14]
[1, -1,  2, -3,  1,  4,  3,  5,   0,  -5,  -4,  -8, -16,  15,  -2, -23]
[1, -1, -1,  1, -3, -4, -7,  2,   4,   0,  15, -14,  17,   3,  13,  11]
[1,  1,  1, -2,  5, -5, -6,  8,   3, -14,   0,   6,   4, -18,  -3,  12]
[1,  1, -2, -3,  3,  6,  6, -1,   5,  11,  -5,   0,  10,   7,  14, -10]
...
The v table (this entry) begins:
[  1,   1,  1,  1,   1,   1,   1,   1,  1,   1,   1,  1,   1,  1,   1,  1]
[ -1,   1, -1,  1,  -1,   1,  -1,   1, -1,  -1,   1,  1,  -1,  1,  -1, -1]
[ -2,   2,  1, -2,   1,   2,  -2,  -1,  2,  -1,   1, -2,   1,  2,  -2,  2]
[ -3,  -2,  3,  1,   2,  -1,  -2,   3, -3,   1,  -2, -3,  -1,  1,   3,  2]
[ -5,   4, -2, -3,   1,  -5,   2,  -4,  1,  -3,   5,  3,  -4, -1,   4,  5]
[ -6,  -4, -5,  2,   6,   1,  -3,  -2,  4,  -4,  -5,  6,  -6, -3,   5,  1]
[ -8,   6,  7,  5,  -3,   4,   1,  -8,  3,  -7,  -6,  6,   5,  2,   4, -8]
[ -9,  -6,  4, -8,   7,   3,   9,   1,  5,   2,   8, -1,  -6,  4,  -2, -5]
[-11,   8, -9, 10,  -2,  -7,  -4,  -6,  1,   4,   3,  5,   9, -8,   1, 10]
[-14,  10,  6, -4,   8,   9,  12,  -3, -5,   1, -14, 11, -12, -2,  -8, -6]
[-15, -10, -6,  9, -14,  12,  11, -13, -4,  15,   1, -5,  -3, 13,   2, -7]
[-18, -12, 15, 16, -10, -17, -13,   2, -8, -14,   6,  1,  -9, -6, -11,  7]
...

Cf. A003989, A050873, A345415, A345416, A345417, A345419, A345420, A345421, A345422.

A123275 Square array A(n,m) = largest divisor of m which is coprime to n, read by upwards antidiagonals.

Original entry on oeis.org

1, 1, 2, 1, 1, 3, 1, 2, 3, 4, 1, 1, 1, 1, 5, 1, 2, 3, 4, 5, 6, 1, 1, 3, 1, 5, 3, 7, 1, 2, 1, 4, 5, 2, 7, 8, 1, 1, 3, 1, 1, 3, 7, 1, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 1, 1, 1, 5, 1, 7, 1, 1, 5, 11, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 1, 1, 3, 1, 5, 3, 1, 1, 9, 5, 11, 3, 13, 1, 2, 1, 4, 1, 2, 7, 8, 1, 2

Offset: 1

Leroy Quet, Oct 10 2006

Read by upwards antidiagonals as A(1,1), A(2,1), A(1,2), A(3,1), A(2,2), A(1,3), etc.

Seen as a triangle, the rows appear to be the reversed rows of the regular triangle defined by t(n,k) = denominator(n*k/(n-k)) for n>=2 and 1<=kMichel Marcus, Mar 24 2022

			The top left 18 x 18 corner of the array:
  1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18
  1, 1, 3, 1, 5, 3, 7, 1, 9,  5, 11,  3, 13,  7, 15,  1, 17,  9
  1, 2, 1, 4, 5, 2, 7, 8, 1, 10, 11,  4, 13, 14,  5, 16, 17,  2
  1, 1, 3, 1, 5, 3, 7, 1, 9,  5, 11,  3, 13,  7, 15,  1, 17,  9
  1, 2, 3, 4, 1, 6, 7, 8, 9,  2, 11, 12, 13, 14,  3, 16, 17, 18
  1, 1, 1, 1, 5, 1, 7, 1, 1,  5, 11,  1, 13,  7,  5,  1, 17,  1
  1, 2, 3, 4, 5, 6, 1, 8, 9, 10, 11, 12, 13,  2, 15, 16, 17, 18
  1, 1, 3, 1, 5, 3, 7, 1, 9,  5, 11,  3, 13,  7, 15,  1, 17,  9
  1, 2, 1, 4, 5, 2, 7, 8, 1, 10, 11,  4, 13, 14,  5, 16, 17,  2
  1, 1, 3, 1, 1, 3, 7, 1, 9,  1, 11,  3, 13,  7,  3,  1, 17,  9
  1, 2, 3, 4, 5, 6, 7, 8, 9, 10,  1, 12, 13, 14, 15, 16, 17, 18
  1, 1, 1, 1, 5, 1, 7, 1, 1,  5, 11,  1, 13,  7,  5,  1, 17,  1
  1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12,  1, 14, 15, 16, 17, 18
  1, 1, 3, 1, 5, 3, 1, 1, 9,  5, 11,  3, 13,  1, 15,  1, 17,  9
  1, 2, 1, 4, 1, 2, 7, 8, 1,  2, 11,  4, 13, 14,  1, 16, 17,  2
  1, 1, 3, 1, 5, 3, 7, 1, 9,  5, 11,  3, 13,  7, 15,  1, 17,  9
  1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16,  1, 18
  1, 1, 1, 1, 5, 1, 7, 1, 1,  5, 11,  1, 13,  7,  5,  1, 17,  1
...
A(12,1) = 12 because d=12 is the largest divisor of 12 for which gcd(d,1) = 1.
A(12,2) = 3 because d=3 is the largest divisor of 12 for which gcd(d,2) = 1.
A(12,3) = 4 because d=4 is the largest divisor of 12 for which gcd(d,3) = 1.
A(12,4) = 3 because d=3 is the largest divisor of 12 for which gcd(d,4) = 1.
A(12,6) = 1 because d=1 is the largest divisor of 12 for which gcd(d,6) = 1.

Antti Karttunen, Table of n, a(n) for n = 1..10440; the first 144 antidiagonals of array

Cf. A003989, A020639, A243103.

t[n_, m_] := Last[Select[Divisors[m], GCD[ #, n] == 1 &]];Flatten[Table[t[i + 1 - j, j], {i, 15}, {j, i}]] (* Ray Chandler, Oct 17 2006 *)

# Produces the triangle when the array is read by antidiagonals (upwards)
from sympy.ntheory import divisors
from math import gcd
def T(n,m):
    return [i for i in divisors(m) if gcd(i,n)==1][-1]
for i in range(1, 16):
    print([T(i+1-j, j) for j in range(1, i+1)]) # Indranil Ghosh, Mar 22 2017

;; A naive implementation of A020639 given under that entry. The result of (A123275bi b a) is a product of all those prime factors of a (possibly occurring multiple times) that are not prime factors of b:
(define (A123275 n) (A123275bi (A004736 n) (A002260 n)))
(define (A123275bi b a) (let loop ((a a) (m 1)) (let ((s (A020639 a))) (cond ((= 1 a) m) ((zero? (modulo b s)) (loop (/ a s) m)) (else (loop (/ a s) (* s m)))))))
;; Antti Karttunen, Mar 22 2017

Extended by Ray Chandler, Oct 17 2006

Name and Example section edited by Antti Karttunen, Mar 22 2017

A178340 Triangle T(n,m) read by rows: denominator of the coefficient [x^m] of the umbral inverse Bernoulli polynomial B^{-1}(n,x).

Original entry on oeis.org

1, 2, 1, 3, 1, 1, 4, 1, 2, 1, 5, 1, 1, 1, 1, 6, 1, 2, 3, 2, 1, 7, 1, 1, 1, 1, 1, 1, 8, 1, 2, 1, 4, 1, 2, 1, 9, 1, 1, 3, 1, 1, 3, 1, 1, 10, 1, 2, 1, 1, 5, 1, 1, 2, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 12, 1, 2, 3, 4, 1, 1, 1, 4, 3, 2, 1, 13

Offset: 0

Paul Curtz, May 25 2010

This is the triangle of denominators associated with the numerators of A178252.
(Unlike the coefficients of the Bernoulli Polynomials, the coefficients of the umbral inverse Bernoulli polynomials are all positive.)
Usually T(n,m) = A003989(n-m+1,m) for m>=1, but since we are tabulating denominators of reduced fractions here, this formula may be wrong by a cancelling integer factor.

			The triangle T(n,m) begins:
n\m  0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 ...
0:   1
1:   2 1
2:   3 1 1
3:   4 1 2 1
4:   5 1 1 1 1
5:   6 1 2 3 2 1
6:   7 1 1 1 1 1 1
7:   8 1 2 1 4 1 2 1
8:   9 1 1 3 1 1 3 1 1
9:  10 1 2 1 1 5 1 1 2 1
10: 11 1 1 1 1 1 1 1 1 1  1
11: 12 1 2 3 4 1 1 1 4 3  2  1
12: 13 1 1 1 1 1 1 1 1 1  1  1  1
13: 14 1 2 1 2 1 2 7 2 1  2  1  2  1
14: 15 1 1 3 1 5 3 1 1 3  5  1  3  1  1
... reformatted. - _Wolfdieter Lang_, Aug 25 2015
-------------------------------------------------
The rational triangle TinvB(n,m):= A178252(n,m) / T(n,m) begins:
n\m    0 1   2    3    4     5    6  7   8  9 10
0:     1
1:   1/2 1
2:   1/3 1   1
3    1/4 1 3/2    1
4:   1/5 1   2    2    1
5:   1/6 1 5/2 10/3  5/2     1
6:   1/7 1   3    5    5     3    1
7:   1/8 1 7/2    7 35/4     7  7/2  1
8:   1/9 1   4 28/3   14    14 28/3  4   1
9:  1/10 1 9/2   12   21 126/5   21 12 9/2  1
10: 1/11 1   5   15   30    42   42 30  15  5  1
... - _Wolfdieter Lang_, Aug 25 2015
Recurrence from the Sheffer a-sequence:
Tinv(3,2) = (3/2)*TinvB(2,1) = (3/2)*1 = 3/2.
From the z-sequence: Tinv(3,0) = 3*Sum_{j=0..2} z_j*TinvB(2,j) = 3*((1/2)*(1/3) -(1/12)*1 + 0*1) = 3*(1/6 - 1/12) = 1/4. - _Wolfdieter Lang_, Aug 25 2015

Cf. A178252.

max = 13; coes = Table[ PadRight[ CoefficientList[ BernoulliB[n, x], x], max], {n, 0, max-1}]; inv = Inverse[coes]; Table[ Take[inv[[n]], n], {n, 1, max}] // Flatten // Denominator (* Jean-François Alcover_, Aug 09 2012 *)

T(n,0) = n+1.
Recurrence for the rational triangle
TinvB(n,m):= A178252(n,m) / T(n,m) from the Sheffer a-sequence, which is 1, (repeat 0), see the comment under A178252: TinvB(n,m) = (n/m)*TinvB(n-1,m-1), for n >= m >= 1. From the z-sequence: TinvB(n,0) = n*Sum_{j=0..n-1} z_j * TinvB(n-1,j), n >= 1, TinvB(0,0) = 1. - Wolfdieter Lang, Aug 25 2015

A180172 a(n) = gcd(prime(n)+2, n).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 11, 3, 1, 1, 1, 1, 1, 9, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 5, 1, 1, 1, 1, 1, 9, 1, 1, 13, 5, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 5, 11, 1, 1, 1, 1, 71, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 15, 7, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1

Offset: 1

Zak Seidov, Aug 15 2010

nonn

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A003989, A052147, A084309, A084310, A180173.

[GCD(n,NthPrime(n) +2): n in [1..110]]; // G. C. Greubel, Mar 12 2023

Mathematica
```
Table[GCD[n,Prime[n]+2],{n,200}]
```

[gcd(nth_prime(n) + 2, n) for n in range(1,111)] # G. C. Greubel, Mar 12 2023

A261176 Minimum value of (1/2)Sum_{i=1..n} Sum_{j=1..n} Sum_{k=1..n} Sum_{l=1..n} gcd(b(i,j),b(k,l)) ((i-k)^2+(j-l)^2) for an n X n matrix b filled with the integers 1 to n^2.

Original entry on oeis.org

0, 9, 126, 802, 3158, 10040, 25464, 58837, 123422, 238203, 429467, 733923, 1200319, 1912928, 2945116, 4369570, 6338678, 9053512, 12622814, 17359779, 23503546, 31347788, 41161317

Offset: 1

Hugo Pfoertner, Aug 15 2015

In one of his programming contests, Al Zimmermann coined the term "Delacorte Numbers" (after G. T. Delacorte, Jr., a New York City philantropist and benefactor) for the sum of D(a,b) = gcd(a,b) * distance^2(a,b), taken over all distinct pairs of integers (a,b) in a rectangular matrix.
The challenge in the contest was to find two kinds of arrangements of 1 to n^2, one minimizing the combined sum (this sequence) and the other maximizing the combined sum (A261177).
All terms beyond a(5) are conjectured based on numerical results. Terms up to a(17) have at least 5 independent verifications.
Upper bounds for the next terms are a(24)<=53670478, a(25)<=68938808, a(26)<=87777189, a(27)<=110759499.

			a(2)=9, because the matrix ((1 2)(3 4)) has Delacorte Number
D(1,2) + D(1,3) + D(1,4) + D(2,3) + D(2,4) + D(3,4) =
gcd(1,2)*(1^2 + 0^2) +
gcd(1,3)*(0^2 + 1^2) +
gcd(1,4)*(1^2 + 1^2) +
gcd(2,3)*(1^2 + 1^2) +
gcd(2,4)*(0^2 + 1^2) +
gcd(3,4)*(1^2 + 0^2) = 1*1 + 1*1 + 1*2 + 1*2 + 2*1 + 1*1 = 9.
Putting (2,4) in a row or column gives the minimum value of the matrix, whereas putting this pair in one of the diagonals gives the maximum.
a(3)=126, because no arrangement of the matrix elements exists that produces a smaller Delacorte Number than e.g. ((1 2 4)(3 6 8)(5 9 7)).

Al Zimmermann's Programming Contests, Delacorte Numbers, Description, October 2014.
Al Zimmermann's Programming Contests, Delacorte Numbers, Final Report, January 2015.
The New York Community Trust: George T. Delacorte.
Arch D. Robison, Computing Delacorte Numbers with Julia, January 21, 2015.

Cf. A261177, A003989, A018782.

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A186230 Triangle T(n,k), n>=1, 1<=k<=n, read by rows: T(n,k) is the number of positive integers j

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A258820 Reversed rows of A178252 presented as diagonals of an irregular triangle.

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A285724 Square array read by descending antidiagonals: If n > k, A(n,k) = T(lcm(n,k), gcd(n,k)), otherwise A(n,k) = T(gcd(n,k), lcm(n,k)), where T(n,k) is sequence A000027 considered as a two-dimensional table.

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A287957 Table read by antidiagonals: T(n, k) = greatest common recursive divisor of n and k; n > 0 and k > 0.

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A331306 Lexicographically earliest infinite sequence such that a(i) = a(j) => A285732(i) = A285732(j) for all i, j.

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A345418 Table read by upward antidiagonals: Given m, n >= 1, write gcd(prime(m),prime(n)) as d = u*prime(m)+v*prime(n) where u, v are minimal; T(m,n) = v.

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A123275 Square array A(n,m) = largest divisor of m which is coprime to n, read by upwards antidiagonals.

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A178340 Triangle T(n,m) read by rows: denominator of the coefficient [x^m] of the umbral inverse Bernoulli polynomial B^{-1}(n,x).

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A345418 Table read by upward antidiagonals: Given m, n >= 1, write gcd(prime(m),prime(n)) as d = uprime(m)+vprime(n) where u, v are minimal; T(m,n) = v.

A261176 Minimum value of (1/2)Sum_{i=1..n} Sum_{j=1..n} Sum_{k=1..n} Sum_{l=1..n} gcd(b(i,j),b(k,l)) ((i-k)^2+(j-l)^2) for an n X n matrix b filled with the integers 1 to n^2.