A000012 -id:A000012 - cp's OEIS frontend

A152201 Triangle read by rows, A000012 * A152198.

1, 2, 3, 1, 4, 2, 5, 4, 1, 6, 6, 2, 7, 9, 5, 1, 8, 12, 8, 2, 9, 16, 14, 6, 1, 10, 20, 20, 10, 2, 11, 25, 30, 20, 7, 1, 12, 30, 40, 30, 12, 2, 13, 36, 55, 50, 27, 8, 1

Offset: 0

Gary W. Adamson, Nov 29 2008

Row sums = A027383: (1, 2, 4, 6, 10, 14, 22, 30,...).

			First few rows of the triangle =
1;
2;
3, 1;
4, 2;
5, 4, 1;
6, 6, 2;
7, 9, 5, 1;
8, 12, 8, 2;
9, 16, 14, 6, 1;
10, 10, 20, 20, 2;
11, 25, 30, 20, 7, 1;
12, 30, 40, 30, 12, 2;
13, 36, 55, 50, 27, 8, 1;
...

A152198, A027383

A000012 * A152198 = partial sums of A152198 column terms.

A168258 Triangle read by rows, A101688 * A000012 as infinite lower triangular matrices.

Original entry on oeis.org

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

Offset: 1

Gary W. Adamson, Nov 21 2009

Row sums = A001318, general pentagonal numbers: (1, 2, 5, 12, 15, 22, ...).
Eigensequence of the triangle = A168259: (1, 2, 6, 14, 38, 96, 254, 656, ...).
The operation A101688 * A000012 transforms rows of A101688 into sequence terms by taking partial sums from the right of A101688 rows. For example, row 3 of A101688 (0, 0, 1, 1) becomes (2, 2, 2, 1). - Gary W. Adamson, Nov 15 2022

			First few rows of the triangle:
  1;
  1, 1;
  2, 2, 1;
  2, 2, 2, 1;
  3, 3, 3, 2, 1;
  3, 3, 3, 3, 2, 1;
  4, 4, 4, 4, 3, 2, 1;
  4, 4, 4, 4, 4, 3, 2, 1;
  5, 5, 5, 5, 5, 4, 3, 2, 1;
  5, 5, 5, 5, 5, 5, 4, 3, 2, 1;
  6, 6, 6, 6, 6, 6, 5, 4, 3, 2, 1;
  6, 6, 6, 6, 6, 6, 6, 5, 4, 3, 2, 1;
  7, 7, 7, 7, 7, 7, 7, 6, 5, 4, 3, 2, 1;
  7, 7, 7, 7, 7, 7, 7, 7, 6, 5, 4, 3, 2, 1;
  8, 8, 8, 8, 8, 8, 8, 8, 7, 6, 5, 4, 3, 2, 1;
  ...

Cf. A001318, A101688, A000012, A168259, A004736, A204164.

T(n, k) = if(binomial(k, n-k)>0, 1, 0); \\ A101688
lista(nn) = my(ma=matrix(nn+1, nn, n, k, T(n-1, k-1)), mb=matrix(nn, nn, n, k, n>=k)); my(m=ma*mb, list=List()); for (n=1, nn, listput(list, vector(n, k, m[n,k]))); Vec(list); \\ Michel Marcus, Nov 16 2022

Triangle read by rows, A101688 * A000012 as infinite lower triangular matrices.

a(n) = min(A004736, A204164); a(n) = min(j, floor((t+2)/2)), where j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Apr 18 2013

Name corrected by Gary W. Adamson, Nov 15 2022

A191750 Dirichlet convolution of A000012 with A007947.

Original entry on oeis.org

1, 3, 4, 5, 6, 12, 8, 7, 7, 18, 12, 20, 14, 24, 24, 9, 18, 21, 20, 30, 32, 36, 24, 28, 11, 42, 10, 40, 30, 72, 32, 11, 48, 54, 48, 35, 38, 60, 56, 42, 42, 96, 44, 60, 42, 72, 48, 36, 15, 33, 72, 70, 54, 30, 72, 56, 80, 90, 60, 120, 62, 96, 56, 13, 84, 144, 68, 90, 96, 144, 72

Offset: 1

Enrique Pérez Herrero, Jun 22 2011

The squarefree kernel of n is sometimes called rad(n).
Sequence is multiplicative with a(p^e) = 1 + p*e.
Dirichlet convolution of A000005 with the function of absolute values of A097945. - R. J. Mathar, Jul 12 2011
Dirichlet convolution of phi(n)*mu(n)^2 with tau(n). - Richard L. Ollerton, May 07 2021

			The divisors of 12 are 1,2,3,4,6 and 12, the squarefree kernels of these numbers are 1,2,3,2,6 and 6, so a(12) = 1+2+3+2+6+6 = 20.

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1837 from Vincenzo Librandi)
Wikipedia, Dirichlet convolution.
Wikipedia, Radical of an integer.

Cf. A007947, A000012 (all 1's sequence), A005117, A073355.

A007947:=func< n | &*PrimeDivisors(n) >; A191750:=func< n | &+[ A007947(d): d in Divisors(n) ] >; [ A191750(n): n in [1..80] ]; // Klaus Brockhaus, Jun 27 2011

with(numtheory): A191750 := n -> add(ilcm(op(factorset(k))),k=divisors(n)):
seq(A191750(i), i=1..80); # Peter Luschny, Jun 23 2011

rad[n_]:=Times@@(FactorInteger[n][[All,1]]); A191750[n_]:=Plus@@rad/@Divisors[n]; Array[A191750,50]
a[1] = 1; a[n_] := Times @@ ((1 + First[#] * Last[#])& /@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Sep 21 2020 *)

rad(n)=local(p); p=factor(n)[, 1]; prod(i=1, length(p), p[i]);
A191750(n)=sumdiv(n, d, rad(d))

for(n=1, 100, print1(direuler(p=2, n, (1 + p*X - X)/(1 - X)^2)[n], ", ")) \\ Vaclav Kotesovec, Jun 19 2020

a(n) = Sum_{d|n} rad(d) = Sum_{d|n} A007947(d).
a(n) <= sigma_1(n) = A000203(n); equality holds if n is a squarefree number (A005117).
Dirichlet g.f.: zeta^2(s)*Product_{primes p} (1+p^(1-s)-p^(-s)). - R. J. Mathar, Jul 12 2011
G.f.: Sum_{k>=1} rad(k)*x^k/(1 - x^k). - Ilya Gutkovskiy, Nov 06 2018
a(n) = Sum_{d|n} mu(d)^2*phi(d)*tau(n/d). - Ridouane Oudra, Nov 19 2019
From Vaclav Kotesovec, Jun 19 2020: (Start)
Dirichlet g.f.: zeta(s)^2 * zeta(s-1) / zeta(2*s-2) * Product_{primes p} (1 - 1/(p^s + p)).
Dirichlet g.f.: zeta(s)^2 * zeta(s-1) * Product_{primes p} (1 + p^(1-2*s) - p^(2-2*s) - p^(-s)).
Sum_{k=1..n} a(k) ~ c * Pi^2 * n^2 / 12, where c = A065463 = Product_{p prime} (1 - 1/(p*(p+1))) = 0.70444220099916559... (End)
From Richard L. Ollerton, May 07 2021: (Start)
a(n) = Sum_{k=1..n} mu(n/gcd(n,k))^2*tau(gcd(n,k)).
a(n) = Sum_{k=1..n} mu(gcd(n,k))^2*tau(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). (End)

A230849 A075526 and A000012 interleaved.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 1, 4, 1, 2, 1, 4, 1, 2, 1, 4, 1, 6, 1, 2, 1, 6, 1, 4, 1, 2, 1, 4, 1, 6, 1, 6, 1, 2, 1, 6, 1, 4, 1, 2, 1, 6, 1, 4, 1, 6, 1, 8, 1, 4, 1, 2, 1, 4, 1, 2, 1, 4, 1, 14, 1, 4, 1, 6, 1, 2, 1, 10, 1, 2, 1, 6, 1, 6, 1, 4, 1, 6, 1, 6, 1, 2, 1, 10, 1, 2, 1

Offset: 1

Omar E. Pol, Nov 01 2013

nonn

a(n) is also the length of the n-th edge of a staircase which represents the function pi(x) on the first quadrant of the square grid, see A000720.
a(2n-1) is the length of the n-th horizontal edge in the staircase.
a(2n) is the length of the n-th vertical edge in the staircase.
For another version see A230850.

			Illustration of initial terms, n = 1..22:
.
1                                                            _ _|
1                                                _ _ _ _ _ _|
1                                        _ _ _ _|
1                                    _ _|
1                            _ _ _ _|
1                        _ _|
1                _ _ _ _|
1            _ _|
1        _ _|
1      _|
1    _|
.
.    1 1   2   2       4   2       4   2       4           6   2
.
Drawing vertical line segments below the staircase (as shown below) we have that the number of cells in the vertical bars gives A000720.
Drawing horizontal line segments above the staircase we have that the number of cells in the k-th horizontal bar is A006093(k).
.    _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
30  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
28  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | |
22  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | | | | | | | |
18  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | | | | | | | | | | | |
16  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| | | | | | | | | | | | | | |
12  |_ _ _ _ _ _ _ _ _ _ _ _| | | | | | | | | | | | | | | | | | |
10  |_ _ _ _ _ _ _ _ _ _| | | | | | | | | | | | | | | | | | | | |
6   |_ _ _ _ _ _| | | | | | | | | | | | | | | | | | | | | | | | |
4   |_ _ _ _| | | | | | | | | | | | | | | | | | | | | | | | | | |
2   |_ _| | | | | | | | | | | | | | | | | | | | | | | | | | | | |
1   |_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|
.
.    0 1 2 2 3 3 4 4 4 4 5 5 6 6 6 6 7 7 8 8 8 8 9 9 9 9 9 9 10 10
.

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

Cf. A000012, A000040, A000720, A001223, A006093, A007504, A046992, A075526, A141042, A152535, A230850.

Riffle[Join[{1},Differences[Prime[Range[100]]]],1] (* Paolo Xausa, Oct 31 2023 *)

A230849(n) = if((n%2)&&(n>1),prime((n+1)/2)-prime(((n+1)/2)-1),1); \\ Antti Karttunen, Dec 23 2018

A104726 Triangle generated as the matrix product of A026729 and A000012 (triangular views), read by rows.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 3, 3, 3, 1, 5, 5, 5, 4, 1, 8, 8, 8, 8, 5, 1, 13, 13, 13, 13, 12, 6, 1, 21, 21, 21, 21, 21, 17, 7, 1, 34, 34, 34, 34, 34, 33, 23, 8, 1, 55, 55, 55, 55, 55, 55, 50, 30, 9, 1, 89, 89, 89, 89, 89, 89, 88, 73, 38

Offset: 0

Gary W. Adamson, Mar 20 2005

If the triangular factors A026729 and A000012 are commuted in the product, A004070 results.

Riordan array (1/(1-x-x^2), x*(1+x)). - Philippe Deléham, Mar 06 2013

			First few rows of the triangle are
1;
1, 1;
2, 2, 1;
3, 3, 3, 1;
5, 5, 5, 4, 1;
8, 8, 8, 8, 5, 1;
13, 13, 13, 13, 12, 6, 1;
21, 21, 21, 21, 21, 17, 7, 1;
...
Production array begins
1, 1
1, 1, 1
-1, -1, 1, 1
2, 2, -1, 1, 1
-5, -5, 2, -1, 1, 1
14, 14, -5, 2, -1, 1, 1
-42, -42, 14, -5, 2, -1, 1, 1
132, 132, -42, 14, -5, 2, -1, 1, 1
-429, -429, 132, -42, 14, -5, 2, -1, 1, 1
... which is based on A000108 or A168491. - _Philippe Deléham_, Mar 06 2013

Cf. A001629 (row sums), A026729, A004070, A000071.

A104726 := proc(n,k)
        add( binomial(j,n-j),j=k..n) ;
end proc:
seq(seq(A104726(n,k),k=0..n),n=0..10) ; # R. J. Mathar, Oct 30 2011

T(n,k) = sum_{j=k..n} binomial(j,n-j). - R. J. Mathar, Oct 30 2011
T(n,0) = T(n-1,0) + T(n-2,0), T(n,k) = T(n-1,k-1) + T(n-2,k-1) for k>0. - Philippe Deléham, Mar 06 2013
T(2*n,n) = A000045(2n+1) = A001519(n+1) = A122367(n). - Philippe Deléham, Mar 06 2013

A128316 Triangle read by rows: A000012 * A128315 as infinite lower triangular matrices.

Original entry on oeis.org

1, 1, 1, 3, -1, 1, 2, 3, -2, 1, 4, -1, 4, -3, 1, 4, 3, -5, 7, -4, 1, 6, -3, 10, -13, 11, -5, 1, 4, 8, -14, 23, -24, 16, -6, 1, 7, -2, 15, -33, 46, -40, 22, -7, 1, 7, 4, -15, 47, -79, 86, -62, 29, -8, 1, 9, -6, 30, -73, 131, -166, 148, -91, 37, -9, 1, 7, 12, -37, 103, -204, 297, -314, 239, -128, 46, -10, 1

Offset: 1

Gary W. Adamson, Feb 25 2007

A128316 * [1,2,3...] = A000034: [1,2,1,2,...].

			First few rows of the triangle:
  1;
  1,  1;
  3, -1,   1;
  2,  3   -2,   1;
  4, -1,   4,  -3,   1;
  4,  3,  -5,   7,  -4,  1;
  6, -3,  10, -13,  11, -5,  1;
  4,  8, -14,  23, -24, 16, -6, 1;
  ...

G. C. Greubel, Rows n = 1..100 of the triangle, flattened

Cf. A000012, A000034, A014300, A026641, A059851, A128315.

Sums include: A000027 (row), A032766, A047215, A344817 (alternating sign).

A128316:= func< n,k | (&+[(-1)^(j+k)*Floor(n/j)*Binomial(j-1,k-1): j in [k..n]]) >;
[A128316(n,k): k in [1..n], n in [1..15]]; // G. C. Greubel, Jun 23 2024

T[n_, k_]:= Sum[(-1)^(j+k)*Floor[n/j]*Binomial[j-1,k-1], {j,k,n}];
Table[T[n,k], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Jun 23 2024 *)

def A128316(n,k): return sum((-1)^(j+k)*int(n//j)*binomial(j-1,k-1) for j in range(k,n+1))
flatten([[A128316(n,k) for k in range(1,n+1)] for n in range(1,16)]) # G. C. Greubel, Jun 23 2024

Sum_{k=1..n} T(n, k) = A000027(n) (row sums).
T(n, 1) = A059851(n).
From G. C. Greubel, Jun 23 2024: (Start)
T(n, k) = A010766(n,k) * AA130595(n-1, k-1) as infinite lower triangular matrices.
T(n, k) = Sum_{j=k..n} (-1)^(j+k) * floor(n/j) * binomial(j-1, k-1).
T(2*n-1, n) = (-1)^(n-1)*A026641(n).
T(2*n-2, n-1) = (-1)^n*A014300(n-1), for n >= 2.
Sum_{k=1..n} (-1)^(k-1)*T(n, k) = A344817(n).
Sum_{k=1..n} k*T(n, k) = A032766(n-1).
Sum_{k=1..n} (k+1)*T(n, k) = A047215(n). (End)

a(28) = 1 inserted and more terms from Georg Fischer, Jun 06 2023

A128489 Triangle read by rows: A000012 * A126988 as infinite lower triangular matrices.

Original entry on oeis.org

1, 3, 1, 6, 1, 1, 10, 3, 1, 1, 15, 3, 1, 1, 1, 21, 6, 3, 1, 1, 1, 28, 6, 3, 1, 1, 1, 1, 36, 10, 3, 3, 1, 1, 1, 1, 45, 10, 6, 3, 1, 1, 1, 1, 1, 55, 15, 6, 3, 3, 1, 1, 1, 1, 1, 66, 15, 6, 3, 3, 1, 1, 1, 1, 1, 1, 78, 21, 10, 6, 3, 3, 1, 1, 1, 1, 1, 1, 91, 21, 10, 6, 3, 3, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Gary W. Adamson, Mar 04 2007

Row sums = A024916: (1, 4, 8, 15, 21, 33, ...).

			First few rows of the triangle:
   1;
   3,  1;
   6,  1, 1;
  10,  3, 1, 1;
  15,  3, 1, 1, 1;
  21,  6, 3, 1, 1, 1;
  28,  6, 3, 1, 1, 1, 1;
  36, 10, 3, 3, 1, 1, 1, 1;
  45, 10, 6, 3, 1, 1, 1, 1, 1;
  ...

Cf. A000012, A000217, A024916, A126988.

By columns, k=1,2,3,...; k repeated terms of the triangular series, (1, 3, 6, 10, ...) in the k-th column.

a(11) = 1 inserted and more terms from Georg Fischer, May 29 2023

A130127 Triangle defined by A000012 * A130125, read by rows.

Original entry on oeis.org

1, 1, 2, 2, 2, 4, 2, 4, 4, 8, 3, 4, 8, 8, 16, 3, 6, 8, 16, 16, 32, 4, 6, 12, 16, 32, 32, 64, 4, 8, 12, 24, 32, 64, 64, 128, 5, 8, 16, 24, 48, 64, 128, 128, 256, 5, 10, 16, 32, 48, 96, 128, 256, 256, 512, 6, 10, 20, 32, 64, 96, 192, 256, 512, 512, 1024, 6, 12, 20, 40, 64, 128, 192, 384, 512, 1024, 1024, 2048

Offset: 1

Gary W. Adamson, May 11 2007

Row sums = A011377: (1, 3, 8, 18, 39, ...). A130126 = A130125 * A000012.

			First few rows of the triangle:
  1;
  1, 2;
  2, 2,  4;
  2, 4,  4,  8;
  3, 4,  8,  8, 16;
  3, 6,  8, 16, 16, 32;
  4, 6, 12, 16, 32, 32, 64;
  ...

G. C. Greubel, Rows n = 1..100 of triangle, flattened

Cf. A000012, A130125, A130126, A011377.

[[2^(k-1)*Floor((n-k+2)/2): k in [1..n]]: n in [1..12]]; // G. C. Greubel, Jun 06 2019

Table[2^(k-1)*Floor[(n-k+2)/2], {n,1,12}, {k,1,n}]//Flatten (* G. C. Greubel, Jun 06 2019 *)

{T(n,k) = 2^(k-1)*floor((n-k+2)/2)}; \\ G. C. Greubel, Jun 06 2019

[[2^(k-1)*floor((n-k+2)/2) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Jun 06 2019

T(n,k) = 2^(k-1) * floor((n-k+2)/2). - G. C. Greubel, Jun 06 2019

More terms added by G. C. Greubel, Jun 06 2019

A130211 Triangle read by rows: matrix product A054522 * A000012.

Original entry on oeis.org

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

Offset: 1

Gary W. Adamson, May 17 2007

			First few rows of the triangle are:
1;
2, 1;
3, 2, 2;
4, 3, 2, 2;
5, 4, 4, 4, 4;
6, 5, 4, 2, 2, 2;
7, 6, 6, 6, 6, 6, 6;
8, 7, 6, 6, 4, 4, 4, 4;
...

Cf. A000010, A054522, A130212 (product with swapped order), A057660 (row sums).

A130211 := proc(n,k)
    add( A054522(n,j),j=k..n) ;
end proc:
seq(seq(A130211(n,k),k=1..n),n=1..15) ; # R. J. Mathar, Aug 06 2016

A054522 * A000012 as infinite lower triangular matrices.

T(n,n) = A000010(n).

A130302 A000012 * A130296.

Original entry on oeis.org

1, 3, 1, 6, 2, 1, 10, 3, 2, 1, 15, 4, 3, 2, 1, 21, 5, 4, 3, 2, 1, 28, 6, 5, 4, 3, 2, 1, 36, 7, 6, 5, 4, 3, 2, 1

Offset: 1

Gary W. Adamson, May 20 2007

Row sums = n^2. A130303 = A130296 * A000012.

			First few rows of the triangle are:
1;
3, 1;
6, 2, 1;
10, 3, 2, 1;
15, 4, 3, 2, 1;
21, 5, 4, 3, 2, 1;
...

Cf. A130296, A130303.

A000012 * A130296 as infinite lower triangular matrices. Triangular series as the left border; (1,2,3...) in all other columns.

cp's OEIS Frontend

A152201 Triangle read by rows, A000012 * A152198.

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A168258 Triangle read by rows, A101688 * A000012 as infinite lower triangular matrices.

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A191750 Dirichlet convolution of A000012 with A007947.

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A230849 A075526 and A000012 interleaved.

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A104726 Triangle generated as the matrix product of A026729 and A000012 (triangular views), read by rows.

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A128316 Triangle read by rows: A000012 * A128315 as infinite lower triangular matrices.

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A128489 Triangle read by rows: A000012 * A126988 as infinite lower triangular matrices.

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A130127 Triangle defined by A000012 * A130125, read by rows.