A104244 Suppose m = Product_{i=1..k} p_i^e_i, where p_i is the i-th prime number and each e_i is a nonnegative integer. Then we can define P_m(x) = Sum_{i=1..k} e_i*x^(i-1). The sequence is the square array A(n,m) = P_m(n) read by descending antidiagonals.

0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 1, 2, 3, 1, 0, 2, 4, 2, 4, 1, 0, 1, 3, 9, 2, 5, 1, 0, 3, 8, 4, 16, 2, 6, 1, 0, 2, 3, 27, 5, 25, 2, 7, 1, 0, 2, 4, 3, 64, 6, 36, 2, 8, 1, 0, 1, 5, 6, 3, 125, 7, 49, 2, 9, 1, 0, 3, 16, 10, 8, 3, 216, 8, 64, 2, 10, 1, 0, 1, 4, 81, 17, 10, 3, 343, 9, 81, 2, 11, 1, 0, 2, 32, 5

Offset: 1

Author

Olaf Voß, Feb 26 2005

Keywords

• easy
• nonn
• tabl

Comments

From Antti Karttunen, Jul 29 2015: (Start)

The square array A(row,col) is read by downwards antidiagonals as: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

A(n,m) (entry at row=n, column=m) gives the evaluation at x=n of the polynomial (with nonnegative integer coefficients) bijectively encoded in the prime factorization of m. See A206284, A206296 for the details of that encoding. (The roles of variables n and m were accidentally swapped in this description, corrected by Antti Karttunen, Oct 30 2016)

(End)

Each row is a completely additive sequence, row n mapping prime(m) to n^(m-1). - Peter Munn, Apr 22 2022

Examples

			a(13) = 3 because 3 = p_1^0 * p_2^1 * p_3^0 * ..., so P_3(x) = 0*x^(1-1) + 1*x^(2-1) + 0*x^(3-1) + ... = x. Hence a(13) = A(3,3) = P_3(3) = 3. [Elaborated by _Peter Munn_, Aug 13 2022]
The top left corner of the array:
0, 1,  1, 2,   1,  2,   1,  3,  2,   2,     1,  3,      1,    2,   2, 4
0, 1,  2, 2,   4,  3,   8,  3,  4,   5,    16,  4,     32,    9,   6, 4
0, 1,  3, 2,   9,  4,  27,  3,  6,  10,    81,  5,    243,   28,  12, 4
0, 1,  4, 2,  16,  5,  64,  3,  8,  17,   256,  6,   1024,   65,  20, 4
0, 1,  5, 2,  25,  6,  125, 3, 10,  26,   625,  7,   3125,  126,  30, 4
0, 1,  6, 2,  36,  7,  216, 3, 12,  37,  1296,  8,   7776,  217,  42, 4
0, 1,  7, 2,  49,  8,  343, 3, 14,  50,  2401,  9,  16807,  344,  56, 4
0, 1,  8, 2,  64,  9,  512, 3, 16,  65,  4096, 10,  32768,  513,  72, 4
0, 1,  9, 2,  81, 10,  729, 3, 18,  82,  6561, 11,  59049,  730,  90, 4
0, 1, 10, 2, 100, 11, 1000, 3, 20, 101, 10000, 12, 100000, 1001, 110, 4
...
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..5050; the first 100 antidiagonals of the array
• Index entries for sequences computed from exponents in factorization of n

Crossrefs

Cf. A000720.

Transpose: A104245.

Main diagonal: A090883.

Row 1: A001222, row 2: A048675, row 3: A090880, row 4: A090881, row 5: A090882, row 10: A054841; and, in the extrapolated table, row 0: A007814, row -1: A195017.

Other completely additive sequences with prime(k) mapped to a function of k include k: A056239, k-1: A318995, k+1: A318994, k^2: A289506, 2^k-1: A293447, k!: A276075, F(k-1): A265753, F(k-2): A265752.

For completely additive sequences with primes p mapped to a function of p, see A001414.

For completely additive sequences where some primes are mapped to 1, the rest to 0 (notably, some ruler functions) see the cross-references in A249344.

For completely additive sequences, s, with primes p mapped to a function of s(p-1) and maybe s(p+1), see A352957.

See the formula section for the relationship to A073133, A206296.

See the comments for the relevance of A206284.

A297845 represents multiplication of the relevant polynomials.

Cf. A090884, A248663, A265398, A265399 for other related sequences.

A167219 lists columns that contain their own column number.

Formula

A(n,A206296(k)) = A073133(n,k). [This formula demonstrates how this array can be used with appropriately encoded polynomials. Note that A073133 reads its antidiagonals by ascending order, while here the order is opposite.] - Antti Karttunen, Oct 30 2016

From Peter Munn, Apr 05 2021: (Start)

The sequence is defined by the following identities:

A(n, 3) = n;

A(n, m*k) = A(n, m) + A(n, k);

A(n, A297845(m, k)) = A(n, m) * A(n, k).

(End)

Extensions

Starting offset changed from 0 to 1 by Antti Karttunen, Jul 29 2015

Name edited (and aligned with rest of sequence) by Peter Munn, Apr 23 2022