A127093 -id:A127093 - cp's OEIS frontend

A130540 Triangle read by rows T(n,k) in which column k lists the terms of A000203 interspersed with (k-1) zeros, 1 <= k <= n.

1, 3, 1, 4, 0, 1, 7, 3, 0, 1, 6, 0, 0, 0, 1, 12, 4, 3, 0, 0, 1, 8, 0, 0, 0, 0, 0, 1, 15, 7, 0, 3, 0, 0, 0, 1, 13, 0, 4, 0, 0, 0, 0, 0, 1, 18, 6, 0, 0, 3, 0, 0, 0, 0, 1, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 28, 12, 7, 4, 0, 3, 0, 0, 0, 0, 0, 1, 14, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 24, 8, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Gary W. Adamson, Jun 03 2007

The original definition was: A127093 * A125093^(-1).
Left border = A000203, sigma(n): (1, 3, 4, 7, 6, ...). Row sums = A007429: (1, 4, 5, 11, 7, 20, 9, ...); = inverse Moebius transform applied to sigma(n); (i.e., inverse Moebius transform applied twice to natural numbers).
T(n,k) is the total number of parts congruent to 0 mod k in the partitions of n into equal parts. - Omar E. Pol, Nov 19 2019
From Omar E. Pol, Jan 01 2020: (Start)
Conjecture 1: the sum of odd-indexed terms in row n equals A327096(n).
Conjecture 2: the sum of even-indexed terms in row n equals the n-th term of the sequence formed by A000004 and A007429 interleaved.
Conjecture 3: alternating row sums give A288417. (End)

			First few rows of the triangle are:
   1;
   3,  1;
   4,  0, 1;
   7,  3, 0, 1;
   6,  0, 0, 0, 1;
  12,  4, 3, 0, 0, 1;
   8,  0, 0, 0, 0, 0, 1;
  15,  7, 0, 3, 0, 0, 0, 1;
  13,  0, 4, 0, 0, 0, 0, 0, 1;
  18,  6, 0, 0, 3, 0, 0, 0, 0, 1;
  12,  0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
  28, 12, 7, 4, 0, 3, 0, 0, 0, 0, 0, 1;
  14,  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
  24,  8, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 1;
  24,  0, 6, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
  31, 15, 0, 7, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 1;
...
Extended by _Omar E. Pol_, Nov 19 2019

Cf. A000203, A007429, A127093, A125093, A244051, A288417, A327096.

A127093 * A125093^(-1), as infinite lower triangular matrices.

New name and more terms from Omar E. Pol, Nov 19 2019

A143152 Inverse Möbius transform of the least prime factor of n: A051731 * A020639.

Original entry on oeis.org

1, 3, 4, 5, 6, 8, 8, 7, 7, 10, 12, 12, 14, 12, 12, 9, 18, 13, 20, 14, 14, 16, 24, 16, 11, 18, 10, 16, 30, 20, 32, 11, 18, 22, 18, 19, 38, 24, 20, 18, 42, 22, 44, 20, 18, 28, 48, 20, 15, 17, 24, 22, 54, 18, 22, 20, 26, 34, 60, 28, 62, 36, 20, 13, 24, 26, 68, 26, 30, 26, 72, 25, 74

Offset: 1

Gary W. Adamson, Jul 27 2008

nonn

			a(4) = 5 = (1, 1, 0, 1) dot (1, 2, 3, 2) = (1 + 2 + 0 + 2), where (1, 1, 0, 1) = row 4 of triangle A051731 and A010639 = (1, 2, 3, 2, 5, 2, 7, 2, 3, 2, 11,...).
Since a(n) = sum of least prime factors of the divisors of n, the divisors of 12 are recorded in triangle row 12 of A127093: (1, 2, 3, 4, 0, 6, 0, 0, 0, 0, 0, 12). Lpf of these terms = row 12 of triangle A143151: (1, 2, 3, 2, 0, 2, 0, 0, 0, 0, 0, 2); sum = 12.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A051731, A020639, A127093, A143151.

read transforms : A020639 := proc(n) local i ; if n = 1 then 1; else for i from 1 do if n mod ithprime(i) = 0 then RETURN(ithprime(i)) ; fi; od: fi; end: a020639 := [seq(A020639(n),n=1..100)] : a143152 := MOBIUSi(a020639) : for i from 1 to nops(a143152) do printf("%d,",op(i,a143152)) ; od: # R. J. Mathar, Aug 11 2008

A020639(n) = if(1==n,n,(factor(n)[1, 1]));
A143152(n) = sumdiv(n,d,A020639(d)); \\ Antti Karttunen, Nov 12 2021

a(p) = (p+1) for prime p.
Inverse Mobius transform of A020639, where A020639(n) = Lpf(n).
Row sums of triangle A143151.
a(n) = Sum_{d|n} A020639(d). - Antti Karttunen, Nov 12 2021

Extended beyond a(14) by R. J. Mathar, Aug 11 2008

Name amended by Antti Karttunen, Nov 12 2021

A307662 Triangle T(i,j=1..i) read by rows which contain the naturally ordered divisors-or-ones of the row number i.

Original entry on oeis.org

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

Offset: 1

I. V. Serov, Apr 20 2019

Replace 0 with 1 in A127093. - Omar E. Pol, Apr 21 2019

			Triangle begins:
  1,
  1, 2,
  1, 1, 3,
  1, 2, 1, 4,
  1, 1, 1, 1, 5,
  1, 2, 3, 1, 1, 6,
  1, 1, 1, 1, 1, 1, 7,
  1, 2, 1, 4, 1, 1, 1, 8,
  1, 1, 3, 1, 1, 1, 1, 1, 9,
  1, 2, 1, 1, 5, 1, 1, 1, 1,10,
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1,11,
  1, 2, 3, 4, 1, 6, 1, 1, 1, 1, 1,12,
  ...

I. V. Serov, Rows n = 1..131 of triangle, flattened

Cf. A100995, A127093, A307641, A027750, A002024, A002260.

Table[Map[If[Mod[n, #] == 0, #, 1] &, Range@ n], {n, 13}] // Flatten (* Michael De Vlieger, Apr 23 2019 *)

row(n) = vector(n, k, if ((n % k) == 0, k, 1)); \\ Michel Marcus, Apr 21 2019

For j=1..floor(i/2), T(i,j)=T(i-j,j).
For j=floor(i/2)+1..i-1, T(i,j)=1.
T(i,i) = i.
Let i = A002024(n) and j = A002260(n):
a(n) = a((i-j-1)*(i-j)/2+j) if j=1..floor(i/2).
a(n) = 1 if j=floor(i/2)+1..i-1.
a(n) = i if j=i.
T(i,j) = a(n).

A127108 Triangle read by rows, A127099 * A000012.

Original entry on oeis.org

1, 5, 2, 7, 3, 3, 17, 10, 4, 4, 11, 5, 5, 5, 5, 35, 23, 15, 6, 6, 6, 15, 7, 7, 7, 7, 7, 7, 49, 34, 20, 20, 8, 8, 8, 8, 34, 21, 21, 9, 9, 9, 9, 9, 9, 55, 37, 25, 25, 25, 10, 10, 10, 10, 10, 23, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 119, 91, 67, 46, 30, 30, 12, 12, 12, 12, 12, 12

Offset: 0

Gary W. Adamson, Jan 05 2007, Jul 27 2008

The operation A000012 * A127099 generates n-th row of the triangle by taking partial sums of n-th row of triangle A127099. Row 4 of A127099 (7, 6, 0, 4) becomes row 4 of A127108: (17, 10, 4, 4).
Row sums = A001001: (1, 7, 13, 35, 31, 91, ...).
Left column of the triangle = A060640: (1, 5, 7, 17, 11, 35, ...).

			First few rows of the triangle:
   1;
   5,  2;
   7,  3,  3;
  17, 10,  4,  4;
  11,  5,  5,  5,  5;
  35, 23, 15,  6,  6,  6;
  15,  7,  7,  7,  7,  7,  7;
  49, 34, 20, 20,  8,  8,  8,  8;
  34, 21, 21,  9,  9,  9,  9,  9,  9;
  55, 37, 25, 25, 25, 10, 10, 10, 10, 10;
  ...

Cf. A060640, A001001, A126988, A127093, A000203, A127094, A123229, A127096, A127098, A127099.

Triangle read by rows, A127099 * A000012.

Edited by N. J. A. Sloane, Aug 13 2008 at the suggestion of R. J. Mathar

A127573 Triangle T(n, k) = k*sigma(k) if k divides n, else 0.

Original entry on oeis.org

1, 1, 6, 1, 0, 12, 1, 6, 0, 28, 1, 0, 0, 0, 30, 1, 6, 12, 0, 0, 72, 1, 0, 0, 0, 0, 0, 56, 1, 6, 0, 28, 0, 0, 0, 120, 1, 0, 12, 0, 0, 0, 0, 0, 117, 1, 6, 0, 0, 30, 0, 0, 0, 0, 180, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 132, 1, 6, 12, 28, 0, 72, 0, 0, 0, 0, 0, 336

Offset: 1

Gary W. Adamson, Jan 19 2007

Right border = n*sigma(n), A064987; row sums = A001001.

			First few rows of the triangle:
  1;
  1, 6;
  1, 0, 12;
  1, 6,  0, 28;
  1, 0,  0,  0, 30;
  ...

Cf. A064987, A127093, A001001, A000203, A127574.

Corrected, extended and edited by Andrey Zabolotskiy, Sep 18 2022

A127574 Triangle T(n,k) = k*sigma(n) if k divides n, else 0.

Original entry on oeis.org

1, 3, 6, 4, 0, 12, 7, 14, 0, 28, 6, 0, 0, 0, 30, 12, 24, 36, 0, 0, 72, 8, 0, 0, 0, 0, 0, 56, 15, 30, 0, 60, 0, 0, 0, 120, 13, 0, 39, 0, 0, 0, 0, 0, 117, 18, 36, 0, 0, 90, 0, 0, 0, 0, 180, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 132, 28, 56, 84, 112, 0, 168, 0, 0, 0, 0, 0, 336

Offset: 1

Gary W. Adamson, Jan 19 2007

			First few rows of the triangle are:
   1;
   3,  6;
   4,  0, 12;
   7, 14,  0, 28;
   6,  0,  0,  0, 30;
  12, 24, 36,  0,  0, 72;
  ...

Cf. A127093, A127573, A064987, A000203, A072861 (row sums).

T(n,k) = Sum_{j=k..n} A130208(n,j)*A127093(j,k), product of the two infinite lower triangular matrices.
T(n,1) = A000203(n).
T(n,n) = A064987(n).

A127626 Triangle T(n,k) = A018804(k) if k|n, else T(n,k)=0.

Original entry on oeis.org

1, 1, 3, 1, 0, 5, 1, 3, 0, 8, 1, 0, 0, 0, 9, 1, 3, 5, 0, 0, 15, 1, 0, 0, 0, 0, 0, 13, 1, 3, 0, 8, 0, 0, 0, 20, 1, 0, 5, 0, 0, 0, 0, 0, 21, 1, 3, 0, 0, 9, 0, 0, 0, 0, 27, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 21, 1, 3, 5, 8, 0, 15, 0, 0, 0, 0, 0, 40, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 25

Offset: 1

Gary W. Adamson, Jan 20 2007

Inverse Mobius transform of a matrix with A018804 in the main diagonal and the rest zeros.

			First few rows of the triangle are:
1;
1, 3;
1, 0, 5;
1, 3, 0, 8;
1, 0, 0, 0, 9;
1, 3, 5, 0, 0, 15;
...

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened

Cf. A051731, A038040 (row sums), A018804 (diagonal).

Cf. A127093.

a127626 n k = a127626_tabl !! (n-1) !! (k-1)
a127626_row n = a127626_tabl !! (n-1)
a127626_tabl = map
   (map (\x -> if x == 0 then 0 else a018804 x)) a127093_tabl
-- Reinhard Zumkeller, Jan 21 2014

A127651 Triangle T(n,k) = n*k if k|n, 0 otherwise; 1<=k<=n.

Original entry on oeis.org

1, 2, 4, 3, 0, 9, 4, 8, 0, 16, 5, 0, 0, 0, 25, 6, 12, 18, 0, 0, 36, 7, 0, 0, 0, 0, 0, 49, 8, 16, 0, 32, 0, 0, 0, 64, 9, 0, 27, 0, 0, 0, 0, 0, 81, 10, 20, 0, 0, 50, 0, 0, 0, 0, 100, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 121, 12, 24, 36, 48, 0, 72, 0, 0, 0, 0, 0, 144

Offset: 1

Gary W. Adamson, Jan 22 2007

Equals the matrix product A127648 * A127093 as infinite lower triangular matrices.

			First few rows of the triangle are:
1;
2, 4;
3, 0, 9;
4, 8, 0, 16;
5, 0, 0, 0, 25;
6, 12, 18, 0, 0, 36;
7, 0, 0, 0, 0, 0, 49;
8, 16, 0, 32, 0, 0, 0, 64;
...

Cf. A127648, A127093, A064987 (row sums).

A127651 := proc(n,k)
        if n mod k =0 then
                n*k;
        else
                0 ;
        end if;
end proc: # R. J. Mathar, Oct 01 2011

T(n,k) = n*A127093(n,k). - R. J. Mathar, Oct 01 2011

A138045 Triangle read by rows: largest proper divisor of n as a table, ones excluded.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Mats Granvik, Mar 02 2008

The numbers in the triangle form lines that begin at T(A001248,A000040). The first line of numbers from the right, is T(A005843,A000027). The second line is T(A016945,A005408). The third line is T(A084967,A007310).

			The first few terms of the table are:
  0
  0,0
  0,0,0
  0,2,0,0
  0,0,0,0,0
  0,0,3,0,0,0
  0,0,0,0,0,0,0
  0,0,0,4,0,0,0,0
  0,0,3,0,0,0,0,0,0

Antti Karttunen, Table of n, a(n) for n = 1..23220 (the first 215 rows of the triangle).
Eric Weisstein's World of Mathematics, Proper Divisor.

Cf. A001248, A016945, A084967, A007310, A127093, A032742, A007775.

up_to = 23220; \\ binomial(215+1,2)
A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
A138045tr(n, k) = if((k>1) && (A032742(n)==k), k, 0);
A138045list(up_to) = { my(v = vector(up_to), i=0); for(n=1,oo, for(k=1,n, i++; if(i > up_to, return(v)); v[i] = A138045tr(n,k))); (v); };
v138045 = A138045list(up_to);
A138045(n) = v138045[n]; \\ Antti Karttunen, Dec 24 2018

T(n,k) = if k==A032742(n) and n(T(n,k))==n(A032742(n)) and k>1 then k else 0 (1<=k<=n), T(1,1)=0.

A143151 Triangle read by rows, A051731 * (A020639 * 0^(n-k)), 1<=k<=n.

Original entry on oeis.org

1, 1, 2, 1, 0, 3, 1, 2, 0, 2, 1, 0, 0, 0, 5, 1, 2, 3, 0, 0, 2, 1, 0, 0, 0, 0, 0, 7, 1, 2, 0, 2, 0, 0, 0, 2, 1, 0, 3, 0, 0, 0, 0, 0, 3, 1, 2, 0, 0, 5, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 1, 2, 3, 2, 0, 2, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13, 1, 2, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 2

Offset: 1

Gary W. Adamson and Mats Granvik, Jul 27 2008

Row sums = A143152: (1, 3, 4, 5, 6, 8, 8, 7, 7, 10, 12, 12, 14, 12, ...).

			First few rows of the triangle are:
  1;
  1, 2;
  1, 0, 3;
  1, 2, 0, 2;
  1, 0, 0, 0, 5;
  1, 2, 3, 0, 0, 2;
  1, 0, 0, 0, 0, 0, 7;
  1, 2, 0, 2, 0, 0, 0, 2;
  1, 0, 3, 0, 0, 0, 0, 0, 3;
  1, 2, 0, 0, 5, 0, 0, 0, 0, 2;
  1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11;
  ...
Row 12 = (1, 2, 3, 2, 0, 2, 0, 0, 0, 0, 0, 2) since the divisors of 12 are shown in row 12 of triangle A127093: (1, 2, 3, 4, 0, 6, 0, 0, 0, 0, 0, 12).
Lpf of these terms = row 12 of A143152.

Cf. A020639, A127093, A143152.

Triangle read by rows, A051731 * (A020639 * 0^(n-k)), 1<=k<=n; where A020639 = Lpf(n). By rows, least prime factors of the divisors of n, where the divisors of n are recorded in triangle A127093.

cp's OEIS Frontend

A130540 Triangle read by rows T(n,k) in which column k lists the terms of A000203 interspersed with (k-1) zeros, 1 <= k <= n.

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A143152 Inverse Möbius transform of the least prime factor of n: A051731 * A020639.

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A307662 Triangle T(i,j=1..i) read by rows which contain the naturally ordered divisors-or-ones of the row number i.

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Mathematica

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A127108 Triangle read by rows, A127099 * A000012.

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A127573 Triangle T(n, k) = k*sigma(k) if k divides n, else 0.

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A127574 Triangle T(n,k) = k*sigma(n) if k divides n, else 0.

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A127626 Triangle T(n,k) = A018804(k) if k|n, else T(n,k)=0.

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A127651 Triangle T(n,k) = n*k if k|n, 0 otherwise; 1<=k<=n.

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A138045 Triangle read by rows: largest proper divisor of n as a table, ones excluded.

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