A297845 -id:A297845 - cp's OEIS frontend

A331287 a(n) = gcd(n, A225546(n)).

1, 2, 1, 1, 1, 2, 1, 2, 9, 2, 1, 12, 1, 2, 1, 1, 1, 18, 1, 4, 1, 2, 1, 24, 1, 2, 9, 4, 1, 2, 1, 2, 1, 2, 1, 9, 1, 2, 1, 8, 1, 2, 1, 4, 9, 2, 1, 4, 1, 2, 1, 4, 1, 18, 1, 8, 1, 2, 1, 12, 1, 2, 9, 1, 1, 2, 1, 4, 1, 2, 1, 18, 1, 2, 3, 4, 1, 2, 1, 80, 1, 2, 1, 12, 1, 2, 1, 8, 1, 18, 1, 4, 1, 2, 1, 8, 1, 2, 9, 1, 1, 2, 1, 8, 1

Offset: 1

Antti Karttunen, Jan 20 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..11250
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A225546, A225547 (fixed points), A297845, A331288, A331310, A331311.

PARI
```
A331287(n) = gcd(n, A225546(n));
```

A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
A331287(n) = if(1==n,1,my(f=factor(n),u=#binary(vecmax(f[, 2])),prods=vector(u,x,1),m=1,e); for(i=1,u,if(!(n%prime(i)),for(k=1,#f~, if(bitand(f[k,2],m),prods[i] *= f[k,1]))); m<<=1); prod(i=1,u,prime(i)^min(valuation(n,prime(i)),A048675(prods[i]))));

a(n) = gcd(n, A225546(n)).
a(n) = A331310(n) * A331311(n).
a(A297845(n,9)) = A297845(a(n),9). - Peter Munn, Jan 24 2020

A297473 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; a(n) = g(f(n)^2).

Original entry on oeis.org

1, 2, 5, 16, 11, 90, 17, 512, 625, 550, 23, 6480, 31, 1666, 2695, 65536, 41, 101250, 47, 110000, 10285, 5566, 59, 1866240, 14641, 10478, 1953125, 653072, 67, 1212750, 73, 33554432, 19435, 23698, 31603, 65610000, 83, 33934, 44795, 88000000, 97, 9071370, 103

Offset: 1

Rémy Sigrist, Dec 30 2017

nonn

This sequence is the main diagonal of A297845.

This sequence has similarities with A296857.

			For n = 12:
- 12 = 2^2 * 3 = prime(1+0)^2 * prime(1+1),
- f(12) = 2 + x,
- f(12)^2 = 4 + 4*x + x^2,
- a(12) = prime(1+0)^4 * prime(1+1)^4 * prime(1+2) = 2^4 * 3^4 * 5 = 6480.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Colored logarithmic scatterplot of the first 100000 terms (where the color is function of A001222(n))

Cf. A000040, A001221, A001222, A031368, A055396, A061395, A296857, A297845.

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

For any n > 0 and k > 0:
- A001221(a(n)) <= A001221(n)^2,
- A001222(a(n)) = A001222(n)^2,
- A055396(a(n)) = 2*A055396(n)-1 + [n=1],
- A061395(a(n)) = 2*A061395(n)-1 + [n=1],
- a(A000040(n)) = A031368(n),
- a(A000040(n)^k) = A031368(n)^(k^2).

A326376 Square array T(n, k) read by antidiagonals upwards, 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) o f(k)) (where o denotes function composition).

Original entry on oeis.org

1, 2, 1, 1, 2, 1, 4, 2, 2, 1, 1, 4, 3, 2, 1, 2, 2, 4, 4, 2, 1, 1, 4, 5, 4, 5, 2, 1, 8, 2, 6, 16, 4, 6, 2, 1, 1, 8, 7, 8, 11, 4, 7, 2, 1, 2, 4, 8, 256, 10, 90, 4, 8, 2, 1, 1, 4, 9, 8, 17, 12, 17, 4, 9, 2, 1, 4, 2, 10, 16, 8, 47250, 14, 512, 4, 10, 2, 1, 1, 8

Offset: 1

Rémy Sigrist, Jul 02 2019

This sequence has connections with A297845.

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.

			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
    2|  2  2   2    2   2      2   2          2        2        2
    3|  1  2   3    4   5      6   7          8        9       10
    4|  4  4   4    4   4      4   4          4        4        4
    5|  1  2   5   16  11     90  17        512      625      550
    6|  2  4   6    8  10     12  14         16       18       20
    7|  1  2   7  256  17  47250  29  134217728  5764801  5656750
    8|  8  8   8    8   8      8   8          8        8        8
    9|  1  4   9   16  25     36  49         64       81      100
   10|  2  4  10   32  22    180  34       1024     1250     1100
The corresponding polynomials are:
  f(n)\f(k)| 0 1 x     2 x^2   x+1             x^3   3  2*x     x^2+1
  ---------+---------------------------------------------------------------------
          0| 0 0 0     0 0     0               0     0  0       0
          1| 1 1 1     1 1     1               1     1  1       1
          x| 0 1 x     2 x^2   x+1             x^3   3  2*x     x^2+1
          2| 2 2 2     2 2     2               2     2  2       2
        x^2| 0 1 x^2   4 x^4   x^2+2*x+1       x^6   9  4*x^2   x^4+2*x^2+1
        x+1| 1 2 x+1   3 x^2+1 x+2             x^3+1 4  2*x+1   x^2+2
        x^3| 0 1 x^3   8 x^6   x^3+3*x^2+3*x+1 x^9   27 8*x^3   x^6+3*x^4+3*x^2+1
          3| 3 3 3     3 3     3               3     3  3       3
        2*x| 0 2 2*x   4 2*x^2 2*x+2           2*x^3 6  4*x     2*x^2+2
      x^2+1| 1 2 x^2+1 5 x^4+1 x^2+2*x+2       x^6+1 10 4*x^2+1 x^4+2*x^2+2

See A326377 for the main diagonal of T.

Cf. A006519, A061142, A297473, A297845, A319525.

g(p) = my (c=Vecrev(Vec(p))); prod (i=1, #c, if (c[i], prime(i)^c[i], 1))
f(n, v='x) = my (f=factor(n)); sum (i=1, #f~, f[i,2] * v^(primepi(f[i,1]) - 1))
T(n,k) = g(f(n, f(k)))

For any m, n, k > 0 and any i >= 0:
- T(1, k) = 1,
- T(2^i, k) = 2^i,
- T(3, k) = k,
- T(3^i, k) = k^i,
- T(5, k) = A297473(k),
- T(6, k) = 2*k,
- T(n, 1) = A006519(n),
- T(n, 2) = A061142(n),
- T(n, 3) = n,
- T(n, 5) = A319525(n),
- T(m*n, k) = T(m, k) * T(n, k).

A324592 Square array T(n, k) read by diagonals, n > 0, k > 0; for any number m > 0 with prime factorization Product_{i > 0} prime(i)^e(i), let f(m) = Sum_{i > 0} e(i) * sqrt(A005117(i)); f establishes a bijection between the positive numbers and the finite sums of square roots of squarefree numbers; let g be the inverse of f; T(n, k) = g(f(n) * f(k)).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 4, 4, 1, 1, 5, 9, 9, 5, 1, 1, 6, 11, 16, 11, 6, 1, 1, 7, 12, 25, 25, 12, 7, 1, 1, 8, 17, 36, 8, 36, 17, 8, 1, 1, 9, 27, 49, 55, 55, 49, 27, 9, 1, 1, 10, 16, 64, 31, 72, 31, 64, 16, 10, 1, 1, 11, 33, 81, 125, 119, 119, 125

Offset: 1

Rémy Sigrist, Sep 03 2019

The set of square roots of squarefree numbers, { sqrt(A005117(i)), i > 0 }, is Q-linearly independent. The set of finite sums of square roots of squarefree numbers is closed under multiplication, hence the sequence is well defined.
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).
This sequence has similarities with A297845.

			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
    2|  1   2   3    4    5    6    7     8     9    10
    3|  1   3   4    9   11   12   17    27    16    33
    4|  1   4   9   16   25   36   49    64    81   100
    5|  1   5  11   25    8   55   31   125   121    40
    6|  1   6  12   36   55   72  119   216   144   330
    7|  1   7  17   49   31  119   32   343   289   217
    8|  1   8  27   64  125  216  343   512   729  1000
    9|  1   9  16   81  121  144  289   729   256  1089
   10|  1  10  33  100   40  330  217  1000  1089   400
For n = 3 and k = 5:
- f(3) = f(prime(2)) = sqrt(A005117(2)) = sqrt(2),
- f(5) = f(prime(3)) = sqrt(A005117(3)) = sqrt(3),
- f(3) * f(5) = sqrt(6) = sqrt(A005117(5)),
- hence T(3, 5) = prime(5) = 11.

Eric Jaffe, Linearly Independent Integer Roots over the Scalar Field Q
Rémy Sigrist, PARI program for A324592

Cf. A001221, A005117, A297845.

PARI
```
See Links section.
```

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(m, n*k) = T(m, n) * T(m, k) and T(n*k, m) = T(n, m) * T(k, m) (T is completely multiplicative in both parameters),
- 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,
- A001221(T(n, k)) <= A001221(n) * A001221(k),
- T(prime(n), prime(n)) = 2^A005117(n) (where prime(n) denotes the n-th prime number).

cp's OEIS Frontend

A331287 a(n) = gcd(n, A225546(n)).

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