A104887 -id:A104887 - cp's OEIS frontend

A098012 Triangle read by rows in which the k-th term in row n (n >= 1, k = 1..n) is Product_{i=0..k-1} prime(n-i).

2, 3, 6, 5, 15, 30, 7, 35, 105, 210, 11, 77, 385, 1155, 2310, 13, 143, 1001, 5005, 15015, 30030, 17, 221, 2431, 17017, 85085, 255255, 510510, 19, 323, 4199, 46189, 323323, 1616615, 4849845, 9699690, 23, 437, 7429, 96577, 1062347, 7436429, 37182145, 111546435, 223092870

Offset: 1

Alford Arnold, Sep 09 2004

Also, square array A(m,n) in which row m lists all products of m consecutive primes (read by falling antidiagonals). See also A248164. - M. F. Hasler, May 03 2017

			2
3 3*2
5 5*3 5*3*2
7 7*5 7*5*3 7*5*3*2
Or, as an infinite square array:
     2     3     5     7  ... : row 1 = A000040,
     6    15    35    77  ... : row 2 = A006094,
    30   105   385  1001  ... : row 3 = A046301,
   210  1155  5005 17017  ... : row 4 = A046302,
   ..., with col.1 = A002110, col.2 = A070826, col.3 = A059865\{1}. - _M. F. Hasler_, May 03 2017

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

Cf. A000040, A002110, A006094, A046301, A046302, A046303.

Cf. A060381 (central terms), A104887, A248147.

P:=Filtered([1..200],IsPrime);;
T:=Flat(List([1..9],n->List([1..n],k->Product([0..k-1],i->P[n-i])))); # Muniru A Asiru, Mar 16 2019

a098012 n k = a098012_tabl !! (n-1) !! (k-1)
a098012_row n = a098012_tabl !! (n-1)
a098012_tabl = map (scanl1 (*)) a104887_tabl
-- Reinhard Zumkeller, Oct 02 2014

T:=(n,k)->mul(ithprime(n-i),i=0..k-1): seq(seq(T(n,k),k=1..n),n=1..9); # Muniru A Asiru, Mar 16 2019

Flatten[ Table[ Product[ Prime[i], {i, n, j, -1}], {n, 9}, {j, n, 1, -1}]] (* Robert G. Wilson v, Sep 21 2004 *)

T098012(n,k)=prod(i=0,k-1,prime(n-i)) \\ "Triangle" variant
A098012(m,n)=prod(i=0,m-1,prime(n+i)) \\ "Square array" variant. - M. F. Hasler, May 03 2017

n-th row = partial products of row n in A104887. - Reinhard Zumkeller, Oct 02 2014

More terms from Robert G. Wilson v, Sep 21 2004

A272172 Triangle read by rows: T(n,k) in which row n lists the first n terms of A000203 in reverse order.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Apr 21 2016

			Triangle begins:
1;
3,   1;
4,   3,  1;
7,   4,  3,  1;
6,   7,  4,  3,  1;
12,  6,  7,  4,  3,  1;
8,  12,  6,  7,  4,  3,  1;
15,  8, 12,  6,  7,  4,  3,  1;
13, 15,  8, 12,  6,  7,  4,  3,  1;
18, 13, 15,  8, 12,  6,  7,  4,  3,  1;
12, 18, 13, 15,  8, 12,  6,  7,  4,  3,  1;
28, 12, 18, 13, 15,  8, 12,  6,  7,  4,  3,  1;
...

Mirror of A245093.
Column k gives A000203 starting in row k.
Row sums give A024916.
Cf. A004736, A104887, A272171.

a(n) = A000203(A004736(n)).

T(n,k) = A000203(n-k+1).

A272171 Triangle read by rows: T(n,k) in which row n lists the first n terms of A000005 in reverse order.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Apr 21 2016

			Triangle begins:
1;
2, 1;
2, 2, 1;
3, 2, 2, 1;
2, 3, 2, 2, 1;
4, 2, 3, 2, 2, 1;
2, 4, 2, 3, 2, 2, 1;
4, 2, 4, 2, 3, 2, 2, 1;
3, 4, 2, 4, 2, 3, 2, 2, 1;
4, 3, 4, 2, 4, 2, 3, 2, 2, 1;
2, 4, 3, 4, 2, 4, 2, 3, 2, 2, 1;
6, 2, 4, 3, 4, 2, 4, 2, 3, 2, 2, 1;
...

Column k gives A000005 starting in row k.
Row sums give A006218, n >= 1.
Cf. A004736, A104887, A272172.

a(n) = A000005(A004736(n)).

T(n,k) = A000005(n-k+1).

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A098012 Triangle read by rows in which the k-th term in row n (n >= 1, k = 1..n) is Product_{i=0..k-1} prime(n-i).

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A272172 Triangle read by rows: T(n,k) in which row n lists the first n terms of A000203 in reverse order.

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A272171 Triangle read by rows: T(n,k) in which row n lists the first n terms of A000005 in reverse order.

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