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A297845 Square array T(n, k) read by antidiagonals, n > 0 and k > 0: for any number n > 0, let f(n) be the polynomial of a single indeterminate x where the coefficient of x^e is the prime(1+e)-adic valuation of n (where prime(k) denotes the k-th prime); f establishes a bijection between the positive numbers and the polynomials of a single indeterminate x with nonnegative integer coefficients; let g be the inverse of f; T(n, k) = g(f(n) * f(k)). 15
1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 5, 4, 1, 1, 5, 9, 9, 5, 1, 1, 6, 7, 16, 7, 6, 1, 1, 7, 15, 25, 25, 15, 7, 1, 1, 8, 11, 36, 11, 36, 11, 8, 1, 1, 9, 27, 49, 35, 35, 49, 27, 9, 1, 1, 10, 25, 64, 13, 90, 13, 64, 25, 10, 1, 1, 11, 21, 81, 125, 77, 77, 125, 81 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

This table has many similarities with A248601.

For any n > 0 and m > 0, f(n * m) = f(n) + f(m).

Also, f(1) = 0 and f(2) = 1.

The function f can be naturally extended to the set of positive rational numbers: if r = u/v (not necessarily in reduced form), then f(r) = f(u) - f(v); as such, f is a homomorphism from the multiplicative group of positive rational numbers to the additive group of polynomials of a single indeterminate x with integer coefficients.

See A297473 for the main diagonal of T.

As a binary operation, T(.,.) is related to A306697(.,.) and A329329(.,.). When their operands are terms of A050376 (sometimes called Fermi-Dirac primes) the three operations give the same result. However the rest of the multiplication table for T(.,.) can be derived from these results because T(.,.) distributes over integer multiplication (A003991), whereas for A306697 and A329329, the equivalent derivation uses distribution over A059896(.,.) and A059897(.,.) respectively. - Peter Munn, Mar 25 2020

LINKS

Rémy Sigrist, Table of n, a(n) for n = 1..5050

Eric Weisstein's World of Mathematics, Distributive

FORMULA

T is completely multiplicative in both parameters:

- for any n > 0

- and k > 0 with prime factorization Prod_{i > 0} prime(i)^e_i:

- T(prime(n), k) = T(k, prime(n)) = Prod_{i > 0} prime(n + i - 1)^e_i.

For any m > 0, n > 0 and k > 0:

- T(n, k) = T(k, n) (T is commutative),

- T(m, T(n, k)) = T(T(m, n), k) (T is associative),

- T(n, 1) = 1 (1 is an absorbing element for T),

- T(n, 2) = n (2 is an identity element for T),

- T(n, 2^i) = n^i for any i >= 0,

- T(n, 4) = n^2 (A000290),

- T(n, 8) = n^3 (A000578),

- T(n, 3) = A003961(n),

- T(n, 3^i) = A003961(n)^i for any i >= 0,

- T(n, 6) = A191002(n),

- A001221(T(n, k)) <= A001221(n) * A001221(k),

- A001222(T(n, k)) = A001222(n) * A001222(k),

- A055396(T(n, k)) = A055396(n) + A055396(k) - 1 when n > 1 and k > 1,

- A061395(T(n, k)) = A061395(n) + A061395(k) - 1 when n > 1 and k > 1,

- T(A000040(n), A000040(k)) = A000040(n + k - 1),

- T(A000040(n)^i, A000040(k)^j) = A000040(n + k - 1)^(i * j) for any i >= 0 and j >= 0.

From Peter Munn, Mar 13 2020:(Start)

T(A329050(i_1, j_1), A329050(i_2, j_2)) = A329050(i_1+i_2, j_1+j_2).

T(n, m*k) = T(n, m) * T(n, k); T(n*m, k) = T(n, k) * T(m, k) (T distributes over multiplication).

A048675(T(n, k)) = A048675(n) * A048675(k).

A195017(T(n, k)) = A195017(n) * A195017(k).

A248663(T(n, k)) = A048720(A248663(n), A248663(k)).

T(n, k) = A306697(n, k) if and only if T(n, k) = A329329(n, k).

A007913(T(n, k)) = A007913(T(A007913(n), A007913(k))) = A007913(A329329(n, k)).

(End)

EXAMPLE

Array T(n, k) begins:

  n\k|  1   2   3    4    5    6    7     8    9    10

  ---+------------------------------------------------

    1|  1   1   1    1    1    1    1     1    1     1  -> A000012

    2|  1   2   3    4    5    6    7     8    9    10  -> A000027

    3|  1   3   5    9    7   15   11    27   25    21  -> A003961

    4|  1   4   9   16   25   36   49    64   81   100  -> A000290

    5|  1   5   7   25   11   35   13   125   49    55

    6|  1   6  15   36   35   90   77   216  225   210  -> A191002

    7|  1   7  11   49   13   77   17   343  121    91

    8|  1   8  27   64  125  216  343   512  729  1000  -> A000578

    9|  1   9  25   81   49  225  121   729  625   441

   10|  1  10  21  100   55  210   91  1000  441   550

PROG

(PARI) T(n, k) = my (f=factor(n), p=apply(primepi, f[, 1]~), g=factor(k), q=apply(primepi, g[, 1]~)); prod (i=1, #p, prod(j=1, #q, prime(p[i]+q[j]-1)^(f[i, 2]*g[j, 2])))

CROSSREFS

Cf. A000012, A000027, A000290, A000578, A001221, A003961, A007913, A048720, A061395, A055396, A191002, A248601, A248663, A297473.

Functions f satisfying f(T(n,k)) = f(n) * f(k): A001222, A048675, A195017.

Powers of k: k=3: A000040, k=4: A001146, k=5: A031368, k=6: A007188 (see also A066117), k=7: A031377, k=8: A023365, k=9: main diagonal of A329050.

Related binary operations: A003991, A306697/A059896, A329329/A059897.

Sequence in context: A196416 A329329 A306697 * A183456 A296313 A183342

Adjacent sequences:  A297842 A297843 A297844 * A297846 A297847 A297848

KEYWORD

nonn,tabl,mult

AUTHOR

Rémy Sigrist, Jan 10 2018

STATUS

approved

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Last modified June 21 04:10 EDT 2021. Contains 345354 sequences. (Running on oeis4.)