A050873 -id:A050873 - cp's OEIS frontend

A093057 Triangle T(j,k) read by rows, where T(j,k) = number of matrix elements remaining at fixed position in the in-situ transposition of a rectangular j X k matrix (singleton cycles).

0, 0, 1, 0, 0, 2, 0, 1, 0, 3, 0, 0, 0, 0, 4, 0, 1, 2, 1, 0, 5, 0, 0, 0, 0, 0, 0, 6, 0, 1, 0, 3, 0, 1, 0, 7, 0, 0, 2, 0, 0, 2, 0, 0, 8, 0, 1, 0, 1, 4, 1, 0, 1, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 0, 1, 2, 3, 0, 5, 0, 3, 2, 1, 0, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 0, 1, 0, 1, 0, 1, 6, 1, 0, 1, 0, 1, 0

Offset: 1

Hugo Pfoertner, Mar 22 2004

Elements (1,1) and (j,k) which always remain at their old position are not counted. See A093055 for details of storage, another example, references and links.

			a(8)=T(3,5)=1 because there is one fixed element at position 8 in the transposition from
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15) ->
(1 6 11)(2 7 12)(3 8 13)(4 9 14)(5 10 15). The fixed first and last elements 1 and 15 are not counted.

Cf. A093055 number of non-singleton cycles, A093056 length of longest cycle, A050873 GCD(u, v).

T(j, k)=gcd(j-1, k-1)-1.

A132726 Triangle read by rows: T(n,k) = length of period in decimal representation of k/n, 1<=k<=n.

Original entry on oeis.org

0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 6, 6, 6, 6, 6, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 0, 6, 6, 6, 6, 6, 6, 0, 6, 6, 6, 6, 6, 6, 0

Offset: 1

Reinhard Zumkeller, Aug 27 2007

T(n,1) = A051626(n); T(n,n) = 0;
T(n,k) = T(1,k/A050873(n,k));
T(n,k) = T(n,A132740(k)), 1<=k<=n;
T(A003592(n),k) = 0, 1<=k<=A003592(n).

Eric Weisstein's World of Mathematics, Decimal expansion
Index entries for sequences related to decimal expansion of 1/n.

A136527 Triangle read by rows: T(n,k) = greatest common divisor of n-th and k-th composite number, 1<=k<=n.

Original entry on oeis.org

4, 2, 6, 4, 2, 8, 1, 3, 1, 9, 2, 2, 2, 1, 10, 4, 6, 4, 3, 2, 12, 2, 2, 2, 1, 2, 2, 14, 1, 3, 1, 3, 5, 3, 1, 15, 4, 2, 8, 1, 2, 4, 2, 1, 16, 2, 6, 2, 9, 2, 6, 2, 3, 2, 18, 4, 2, 4, 1, 10, 4, 2, 5, 4, 2, 20, 1, 3, 1, 3, 1, 3, 7, 3, 1, 3, 1, 21, 2, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 1, 22, 4, 6, 8, 3, 2, 12, 2, 3, 8, 6, 4, 3, 2, 24

Offset: 1

Reinhard Zumkeller, Jan 03 2008

			4;
2, 6;
4, 2, 8;
1, 3, 1, 9;
2, 2, 2, 1, 10;
...

Cf. A002808, A050873, A073783.

nmax = 14;
A002808 = Select[Range[FindRoot[n == nmax + PrimePi[n] + 1, {n, nmax, 2nmax}][[1, 2]] // Ceiling], CompositeQ];
T[n_, k_] := GCD[A002808[[n]], A002808[[k]]];
Table[T[n, k], {n, 1, nmax}, {k, 1, n}] // Flatten (* Jean-François Alcover, Nov 15 2021 *)

T(n,k) = A050873(A002808(n),A002808(k));
A073783(n) = T(n-1,n) for n>1;
A002808(n) = T(n,n).

A141256 An Okazaki-like composition, see A126759.

Original entry on oeis.org

1, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 4, 2, 5, 3, 2, 2, 6, 2, 7, 2, 3, 4, 8, 2, 2, 5, 2, 3, 9, 2, 10, 2, 4, 6, 3, 2, 11, 7, 5, 2, 12, 3, 13, 4, 2, 8, 14, 2, 15, 2, 6, 5, 16, 2, 4, 3, 7, 9, 17, 2, 18, 10, 3, 2, 5, 4, 19, 6, 8, 3, 20, 2, 21, 11, 2, 7, 22, 5, 23, 2, 2, 12, 24, 3, 6, 13, 9, 4, 25, 2, 26, 8, 10, 14

Offset: 0

Reinhard Zumkeller, Jun 17 2008

nonn

a(5*n) = a(3*n) = a(2*n) = a(n);
for n with GCD(n,30)=1: a(n+30*k)=a(n)+8*k, note: 30=2*3*5,
A000010(30)=8;
for k>1: a(A007775(k-1))=k and a(m)A007775(k-1).

R. Zumkeller, Table of n, a(n) for n = 0..10000

a(n) = if n=0 then 1 else if GCD(n,30)>1 then a(LPD(n)) else 2*floor(n/30) + (if n mod 30 = 1 then 2 else IP(n)-1), with IP=A049084, LPD=A032742 and GCD=A050873.

A320043 Row sums of the triangle A322550.

Original entry on oeis.org

1, 6, 13, 50, 37, 196, 189, 384, 351, 1210, 601, 2366, 1471, 2156, 2941, 6936, 3277, 10830, 5563, 9022, 9681, 23276, 9897, 26300, 19267, 30030, 23043, 58870, 21087, 76880, 46717, 59296, 57801, 83546, 50281, 156066, 90973, 117968, 90539, 235340, 86179, 284746

Offset: 1

Stefano Spezia, Dec 16 2018

nonn

Conjecture: a(n) is not a perfect square except for n = 1, 6 and 96.

Stefano Spezia, Table of n, a(n) for n = 1..10000

Cf. A000027, A000290, A000578, A050873, A322550.

List([1..50], n->Sum([1..n], k->(n+1-k)^2*k/GcdInt(n+1-k,k)^3));

[(&+[(n+1-k)^2*k/Gcd(n+1-k,k)^3: k in [1..n]]): n in [1..50]];

a := n -> sum((n+1-k)^2*k/gcd(n+1-k, k)^3, k = 1 .. n): seq(a(n), n = 1 .. 50);

a[n_]:=Sum[(n+1-k)^2*k/GCD[n+1-k,k]^3, {k, 1, n}]; Array[a, 50]

a(n):=sum((n+1-k)^2*k/gcd(n+1-k,k)^3, k, 1, n)$ makelist(a(n), n, 1, 50);

a(n) = sum(k=1, n, (n+1-k)^2*k/gcd(n+1-k,k)^3);
vector(50, n, a(n))

a(n) = Sum_{k=1..n} (n + 1 - k)^2*k/gcd(n + 1 - k, k)^3.

a(n) = Sum_{k=1..n} A000290(n + 1 - k)*A000027(k)/A000578(A050873(n + 1 - k, k)).

A322482 Table read by downward antidiagonals: T(n,k) is the greatest divisor of n which is a unitary divisor of k.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Dec 11 2018

This relation was defined by Cohen in 1960.
The common notation for T(n,k) is (n,k)*.
If T(n,k) = 1 then n is said to be semi-prime to k.
In general T(n,k) != T(k,n).
The relation is used to define semi-unitary divisors (A322483).

			The table starts
  1  1  1  1  1  1  1  1  1  1 ...
  1  2  1  1  1  2  1  1  1  2 ...
  1  1  3  1  1  3  1  1  1  1 ...
  1  2  1  4  1  2  1  1  1  2 ...
  1  1  1  1  5  1  1  1  1  5 ...
  1  2  3  1  1  6  1  1  1  2 ...
  1  1  1  1  1  1  7  1  1  1 ...
  1  2  1  4  1  2  1  8  1  2 ...
  1  1  3  1  1  3  1  1  9  1 ...
  1  2  1  1  5  2  1  1  1 10 ...
  ...
The triangle formed by the antidiagonals starts
  1
  1 1
  1 2 1
  1 1 1 1
  1 1 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 1 1 2 5 1 1 2 1
  ...

J. Sandor and B. Crstici, Handbook of Number Theory, II, Springer Verlag, 2004, chapter 3.6, pp. 281.

Eckford Cohen, Arithmetical functions associated with the unitary divisors of an integer, Mathematische Zeitschrift, Vol. 74, No. 1 (1960), pp. 66-80.
M. V. Subbarao On some arithmetic convolutions, The theory of arithmetic functions, Springer, Berlin, Heidelberg, 1972, pp. 247-271.
D. Suryanarayana and V. Siva Rama Prasad, Sum functions of k-ary and semi-k-ary divisors, Journal of the Australian Mathematical Society, Vol. 15, No. 2 (1973), pp. 148-162.

Cf. A050873 (gcd), A165430 (unitary gcd).

udiv[n_] := Select[Divisors[n], GCD[#,n/#] == 1 &]; semiuGCD[a_, b_] := Max[ Intersection[Divisors[a], udiv[b]]]; Table[semiuGCD[n, k], {n,1,20}, {k, n-1, 1, -1 }] // Flatten

udivisors(n) = {my(d=divisors(n)); select(x->(gcd(x, n/x)==1), d);}
T(n,k) = {my(dn = divisors(n), udk = udivisors(k)); vecmax(setintersect(dn, udk));} \\ Michel Marcus, Dec 14 2018

T(1,n) = T(n,1) = 1.

T(n,n) = n.

A322550 Table read by ascending antidiagonals: T(n, k) is the minimum number of cubes necessary to fill a right square prism with base area n^2 and height k.

Original entry on oeis.org

1, 4, 2, 9, 1, 3, 16, 18, 12, 4, 25, 4, 1, 2, 5, 36, 50, 48, 36, 20, 6, 49, 9, 75, 1, 45, 3, 7, 64, 98, 4, 100, 80, 2, 28, 8, 81, 16, 147, 18, 1, 12, 63, 4, 9, 100, 162, 192, 196, 180, 150, 112, 72, 36, 10, 121, 25, 9, 4, 245, 1, 175, 2, 3, 5, 11, 144, 242, 300, 324, 320, 294, 252, 200, 144, 90, 44, 12

Offset: 1

Stefano Spezia, Dec 15 2018

			The table T starts in row n = 1 with columns k >= 1 as:
   1     2     3     4     5     6     7     8     9 ...
   4     1    12     2    20     3    28     4    36 ...
   9    18     1    36    45     2    63    72     3 ...
  16     4    48     1    80    12   112     2   144 ...
  25    50    75   100     1   150   175   200   225 ...
  36     9     4    18   180     1   252    36    12 ...
  49    98   147   196   245   294     1   392   441 ...
  64    16   192     4   320    48   448     1   576 ...
  81   162     9   324   405    18   567   648     1 ...
...
The triangle X(n, k) begins
  n\k|   1     2     3     4     5     6     7     8     9
  ---+----------------------------------------------------
   1 |   1
   2 |   4     2
   3 |   9     1     3
   4 |  16    18    12     4
   5 |  25     4     1     2     5
   6 |  36    50    48    36    20     6
   7 |  49     9    75     1    45     3     7
   8 |  64    98     4   100    80     2    28     8
   9 |  81    16   147    18     1    12    63     4     9
...

Stefano Spezia, First 150 antidiagonals of the table, flattened

Cf. A000012 (main diagonal of the table), A000027 (1st row of the table or diagonal of the triangle), A000290 (k=1), A000578, A011379 (superdiagonal of the table), A045991 (subdiagonal of the table), A050873, A119619, A320043 (row sums of the triangle).

Flat(List([1..12], n->List([1..n], k->(n+1-k)^2*k/GcdInt(n+1-k,k)^3)));

[[(n+1-k)^2*k/Gcd(n+1-k,k)^3: k in [1..n]]: n in [1..12]]; // triangle output

a := (n, k) -> (n+1-k)^2*k/gcd(n+1-k, k)^3: seq(seq(a(n, k), k = 1 .. n), n = 1 .. 12)

T[n_,k_]:=n^2*k/GCD[n,k]^3; Flatten[Table[T[n-k+1,k], {n, 12}, {k, n}]]

sjoin(v, j) := apply(sconcat, rest(join(makelist(j, length(v)), v)))$ display_triangle(n) := for i from 1 thru n do disp(sjoin(makelist((i+1-j)^2*j/gcd(i+1-j,j)^3, j, 1, i), " ")); display_triangle(12);

T(n, k) = (n+1-k)^2*k/gcd(n+1-k,k)^3;
tabl(nn) = for(i=1, nn, for(j=1, i, print1(T(i, j), ", ")); print);
tabl(12) \\ triangle output

T(n, k) = n^2*k/gcd(n, k)^3.
T(n, k) = A000290(n)*k/A000578(A050873(n,k)).
X(n, k) = T(n + 1 - k, k).
X(2*n - 1, n) = A000012(n).
Product_{k=1..n} X(n, k)^(1/3) = A119619(n+1). - Stefano Spezia, Jun 24 2024

A384245 Triangle read by rows: T(n, k) for 1 <= k <= n is the largest divisor of k that is an infinitary divisor of n.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, May 23 2025

First differs from A384047 at n = 30.

			Triangle begins:
  1
  1, 2
  1, 1, 3
  1, 1, 1, 4
  1, 1, 1, 1, 5
  1, 2, 3, 2, 1, 6
  1, 1, 1, 1, 1, 1, 7
  1, 2, 1, 4, 1, 2, 1, 8
  1, 1, 1, 1, 1, 1, 1, 1, 9
  1, 2, 1, 2, 5, 2, 1, 2, 1, 10

Amiram Eldar, Table of n, a(n) for n = 1..10153 (first 142 rows flattened)

Cf. A050873, A064379, A077609, A384047, A384246 (positions of 1's).

infdivs[n_] := If[n == 1, {1}, Sort@ Flatten@ Outer[Times, Sequence @@ (FactorInteger[n] /. {p_, m_Integer} :> p^Select[Range[0, m], BitOr[m, #] == m &])]];  (* Michael De Vlieger at A077609 *)
T[n_, k_] := Max[Intersection[infdivs[n], Divisors[k]]];
Table[T[n, k], {n, 1, 13}, {k, 1, n}] // Flatten

isidiv(d, f) = {if (d==1, return (1)); for (k=1, #f~, bne = binary(f[k, 2]); bde = binary(valuation(d, f[k, 1])); if (#bde < #bne, bde = concat(vector(#bne-#bde), bde)); for (j=1, #bne, if (! bne[j] && bde[j], return (0)); ); ); return (1); }
infdivs(n) = {my(f = factor(n), d = divisors(f), idiv = []); for (k=1, #d, if (isidiv(d[k], f), idiv = concat(idiv, d[k])); ); idiv; } \\ Michel Marcus at A077609
T(n, k) = vecmax(setintersect(infdivs(n), divisors(k)));

A137324 a(n) = Sum_{prime p < n} gcd(n,p).

Original entry on oeis.org

1, 3, 2, 6, 3, 5, 6, 9, 4, 8, 5, 13, 12, 7, 6, 10, 7, 13, 16, 19, 8, 12, 13, 22, 11, 16, 9, 17, 10, 12, 23, 28, 21, 14, 11, 31, 26, 17, 12, 22, 13, 25, 20, 37, 14, 18, 21, 20, 33, 28, 15, 19, 30, 23, 36, 45, 16, 24, 17, 49, 26, 19, 34, 31, 18, 36, 43, 30, 19, 23, 20, 58, 27, 40, 37

Offset: 3

Max Sills, Apr 06 2008

			a(10) = 9 because gcd(10,2) = 2, gcd(10,3) = 1, gcd(10,5) = 5, gcd(10,7) = 1; 2 + 1 + 5 + 1 = 9.
The underlying irregular table of gcd(n,2), gcd(n,3), gcd(n,5), gcd(n,7), etc., for which a(n) provides row sums, is obtained by deleting columns from A050873(n,k) and looks as follows for n=3,4,5,...:
  1
  2 1
  1 1
  2 3 1
  1 1 1
  2 1 1 1
  1 3 1 1
  2 1 5 1
  1 1 1 1
  2 3 1 1 1
  1 1 1 1 1
  2 1 1 7 1 1
  1 3 5 1 1 1
  2 1 1 1 1 1
  1 1 1 1 1 1
  2 3 1 1 1 1 1
  1 1 1 1 1 1 1
  2 1 5 1 1 1 1 1

Cf. A000720, A006579, A048865, A105221.

[&+[Gcd(n,p):p in PrimesInInterval(1,n-1)]:n in [3..77]]; // Marius A. Burtea, Aug 07 2019

A137324 := proc(n) local a,i; a :=0 ; for i from 1 to numtheory[pi](n-1) do a := a+gcd(n,ithprime(i)) ; od: a; end: seq(A137324(n),n=3..80) ; # R. J. Mathar, Apr 09 2008

Table[Plus @@ GCD[n, Select[Range[n - 1], PrimeQ[ # ] &]], {n, 3, 70}] (* Stefan Steinerberger, Apr 09 2008 *)

a(n) = sum(k=1, n-1, gcd(n,k)*isprime(k)); \\ Michel Marcus, Nov 07 2014

from math import gcd
from sympy import primerange
def a(n): return sum(gcd(n, p) for p in primerange(1, n))
print([a(n) for n in range(3, 78)]) # Michael S. Branicky, Nov 21 2021

a(p) = A000720(p) - 1 for prime p. - R. J. Mathar, Apr 09 2008

a(n) = A048865(n) + A105221(n). - Wesley Ivan Hurt, Nov 21 2021

Corrected and extended by R. J. Mathar and Stefan Steinerberger, Apr 09 2008

A350900 Triangle read by rows: T(n, k) = Sum_{i=1..n} gcd(i,n) / gcd(gcd(i,k),n) for 1 <= k <= n.

Original entry on oeis.org

1, 3, 2, 5, 5, 3, 8, 5, 8, 4, 9, 9, 9, 9, 5, 15, 10, 9, 10, 15, 6, 13, 13, 13, 13, 13, 13, 7, 20, 12, 20, 9, 20, 12, 20, 8, 21, 21, 11, 21, 21, 11, 21, 21, 9, 27, 18, 27, 18, 15, 18, 27, 18, 27, 10, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 11, 40, 25, 24, 20, 40, 15, 40, 20, 24, 25, 40, 12

Offset: 1

Werner Schulte, Jan 21 2022

Subtriangle (triangle without main diagonal) is symmetrical.

Conjecture: Let f be an arbitrary arithmetic function. Define for n > 0 the sequence a(f; n) = Sum_{i=1..n, k=1..n} f(gcd(i,n)/gcd(gcd(i,k),n)); a(f; n) equals Dirichlet convolution of f(n)*A000010(n) and A057660(n); if f is multiplicative, then a(f; n) is multiplicative; row sums of this triangle use f(n) = n (see formula section).

			The triangle T(n, k) for 1 <= k <= n starts:
n \k :   1   2   3   4   5   6   7   8   9  10  11  12
======================================================
   1 :   1
   2 :   3   2
   3 :   5   5   3
   4 :   8   5   8   4
   5 :   9   9   9   9   5
   6 :  15  10   9  10  15   6
   7 :  13  13  13  13  13  13   7
   8 :  20  12  20   9  20  12  20   8
   9 :  21  21  11  21  21  11  21  21   9
  10 :  27  18  27  18  15  18  27  18  27  10
  11 :  21  21  21  21  21  21  21  21  21  21  11
  12 :  40  25  24  20  40  15  40  20  24  25  40  12
  etc.

Row sums gives A373059.

Cf. A000010, A002618, A018804, A057660, A050873.

T(n, k) = sum(i=1, n, gcd(i,n) / gcd(gcd(i,k),n));
row(n) = vector(n, k, T(n,k)); \\ Michel Marcus, Jan 22 2022

T(n, 1) = A018804(n); T(n, n) = n.
T(n, k) = T(n, n-k) for 1 <= k < n.
Conjecture: Row sums equal Dirichlet convolution of A002618 and A057660.

cp's OEIS Frontend

A093057 Triangle T(j,k) read by rows, where T(j,k) = number of matrix elements remaining at fixed position in the in-situ transposition of a rectangular j X k matrix (singleton cycles).

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A132726 Triangle read by rows: T(n,k) = length of period in decimal representation of k/n, 1<=k<=n.

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A136527 Triangle read by rows: T(n,k) = greatest common divisor of n-th and k-th composite number, 1<=k<=n.

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A141256 An Okazaki-like composition, see A126759.

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A320043 Row sums of the triangle A322550.

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A322482 Table read by downward antidiagonals: T(n,k) is the greatest divisor of n which is a unitary divisor of k.

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A322550 Table read by ascending antidiagonals: T(n, k) is the minimum number of cubes necessary to fill a right square prism with base area n^2 and height k.

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A384245 Triangle read by rows: T(n, k) for 1 <= k <= n is the largest divisor of k that is an infinitary divisor of n.

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