A059896 -id:A059896 - cp's OEIS frontend

A060381 a(n) = prime(n)prime(n+1)...prime(2n-1), where prime(i) is the i-th prime.

1, 2, 15, 385, 17017, 1062347, 86822723, 10131543907, 1204461778591, 198229051666003, 35224440615606707, 6295457783127226289, 1331590860773071702483, 310692537866322378582047, 78832548083496383033878901, 21381953681344611984282084241

Offset: 0

Jason Earls, Apr 03 2001

Central terms of triangle A098012. - Reinhard Zumkeller, Oct 02 2014

For n >= 0, a(n+1) is the n-th power of 15 in the monoid defined by A306697. - Peter Munn, Feb 18 2020

			a(1)=2; a(2) = 3*5 = 15; a(3) = 5*7*11 = 385.

Reinhard Zumkeller, Table of n, a(n) for n = 0..250

Cf. A002110, A098012.
Related to A006516 via A019565.
A003961, A059896, A306697 are used to express relationship between terms of this sequence.

P:=Filtered([1..200],IsPrime);;
a:=List([1..15],n->Product([0..n-1],k->P[n+k])); # Muniru A Asiru, Mar 16 2019

a060381 n = a098012 (2 * n - 1) n  -- Reinhard Zumkeller, Oct 02 2014

seq(mul(ithprime(n+k),k=0..n-1),n=0..15); # Muniru A Asiru, Mar 16 2019

Table[Times@@Prime[Range[n,2n-1]],{n,20}] (* Harvey P. Dale, Jul 19 2018 *)

a(n) = prod(k=n, 2*n-1, prime(k)); \\ Michel Marcus, Mar 16 2019

a(n) = A002110(2*n-1)/A002110(n-1). - Michel Marcus, Mar 16 2019
From Peter Munn, Feb 18 2020: (Start)
a(n) = A019565(A006516(n)).
For n >= 1, a(n) = A098012(n,n), reading A098012 as a square array.
For n > 1, a(n) = A306697(a(n-1), 15) = A059896(A003961^2(a(n-1)), A003961(a(n-1))).
(End)

a(0)=1 prepended by Alois P. Heinz, Mar 16 2019

A066616 a(1) = 1; a(n) = n*a(n-1) if n does not divide a(n-1), otherwise a(n) = a(n-1).

Original entry on oeis.org

1, 2, 6, 24, 120, 120, 840, 840, 7560, 7560, 83160, 83160, 1081080, 1081080, 1081080, 17297280, 294053760, 294053760, 5587021440, 5587021440, 5587021440, 5587021440, 128501493120, 128501493120, 3212537328000, 3212537328000

Offset: 1

Amarnath Murthy, Dec 24 2001

nonn

			a(5) = 120; as 6 divides a(5), we have a(6) = a(5) = 120. Though 9 is not coprime to a(8) but still 9 does not divide a(8) so a(9) = 9 * a(8).

Harry J. Smith, Table of n, a(n) for n = 1..200

Cf. A003418, A037992 (duplicates removed).

Replacing A059896 with A059897 in the formula gives A284567.

nxt[{n_,a_}]:={n+1,If[Mod[a,n+1]==0,a,a(n+1)]}; NestList[nxt,{1,1},30][[All,2]] (* Harvey P. Dale, Jul 01 2022 *)

{ for (n=1, 200, if (n==1, a=1, if (a%n, a=n*a)); write("b066616.txt", n, " ", a) ) } \\ Harry J. Smith, Mar 12 2010

a(1) = 1; for n > 1, a(n) = A059896(a(n-1), n). - Peter Munn, Jul 12 2022

More terms from Vladeta Jovovic, Dec 26 2001

A258211 Nonsquarefree numbers of the form 4*k + 2.

Original entry on oeis.org

18, 50, 54, 90, 98, 126, 150, 162, 198, 234, 242, 250, 270, 294, 306, 338, 342, 350, 378, 414, 450, 486, 490, 522, 550, 558, 578, 594, 630, 650, 666, 686, 702, 722, 726, 738, 750, 774, 810, 846, 850, 882, 918, 950, 954, 990, 1014, 1026, 1050, 1058, 1062, 1078, 1098, 1134, 1150

Offset: 1

Juri-Stepan Gerasimov, May 23 2015

The asymptotic density of this sequence is 1/4 - 2/Pi^2 = 0.047357... (A190357) - Amiram Eldar, Feb 10 2021
From Peter Munn, Jan 20 2022: (Start)
Even numbers whose square part is odd (and nontrivial).
If m is in the sequence then every odd multiple of m is in the sequence.
Closed under the operation A059896(.,.).
(End)

			18 is in this sequence because 4 * 4 + 2 = 18 = 2 * 3^2.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Square Part.

Intersection of A016825 and either A013929 or A335437.

Cf. A039956, A053850, A059896, A190357.

[n*4+2: n in [1..300] | not IsSquarefree(4*n+2)];

remove(numtheory:-issqrfree, [4*i+2 $ i=0..1000]); # Robert Israel, May 27 2015

Select [Range[300], ! SquareFreeQ[(4 # - 2)] &] 4 - 2 (* Vincenzo Librandi, May 24 2015 *)

select(n->!issquarefree(n), vector(50,n,2*n+9))*2 \\ Charles R Greathouse IV, May 26 2015

a(n) = 2*A053850(n). - Charles R Greathouse IV, May 26 2015

A336882 a(0) = 1; for k >= 0, 0 <= i < 2^k, a(2^k + i) = m_k * a(i), where m_k is the least odd number not in terms 0..2^k - 1.

Original entry on oeis.org

1, 3, 5, 15, 7, 21, 35, 105, 9, 27, 45, 135, 63, 189, 315, 945, 11, 33, 55, 165, 77, 231, 385, 1155, 99, 297, 495, 1485, 693, 2079, 3465, 10395, 13, 39, 65, 195, 91, 273, 455, 1365, 117, 351, 585, 1755, 819, 2457, 4095, 12285, 143, 429, 715, 2145, 1001

Offset: 0

Peter Munn, Aug 16 2020

nonn

A permutation of the odd numbers.
Every positive integer, m, is the product of a unique subset of the terms of A050376. The members of the subset are often known as the Fermi-Dirac factors of m. In this sequence, the odd numbers appear lexicographically according to their Fermi-Dirac factors (with those factors listed in decreasing order). The equivalent sequence for all positive integers is A052330.
The sequence has a conditional exponential identity shown in the formula section. This relies on the offset being 0, as in related sequences, notably A019565 and A052330.

			a(0) = 1, as specified explicitly.
m_0 = 3, the least odd number not in terms 0..0.
So a(1) = a(2^0 + 0) = m_0 * a(0) = 3 * 1 = 3.
m_1 = 5, the least odd number not in terms 0..1.
So a(2) = a(2^1 + 0) = m_1 * a(0) = 5 * 1 = 5;
and a(3) = a(2^1 + 1) = m_1 * a(1) = 5 * 3 = 15.
The initial terms are tabulated below, equated with the product of their Fermi-Dirac factors to exhibit the lexicographic order. We start with 1, since 1 is factored as the empty product and the empty list is first in lexicographic order.
   n     a(n)
   0    1,
   1    3 = 3,
   2    5 = 5,
   3   15 = 5 * 3,
   4    7 = 7,
   5   21 = 7 * 3,
   6   35 = 7 * 5,
   7  105 = 7 * 5 * 3,
   8    9 = 9,
   9   27 = 9 * 3,
  10   45 = 9 * 5,
  11  135 = 9 * 5 * 3,
  12   63 = 9 * 7.

Sean A. Irvine, Java program (github)

Permutation of A005408.
Subsequence of A052330.
Subsequences: A062090, A332382 (squarefree terms).
A003986, A003987, A004198, A059896, A059897 are used to express relationship between terms of this sequence.
Cf. A019565, A050376.

a(2^k) = min({ 2*m+1 : m >= 0, 2*m+1 <> a(j), 0 <= j < 2^k }) = A062090(k+2).
If x AND y = 0, a(x+y) = a(x) * a(y), where AND denotes the bitwise operation, A004198(.,.).
a(x XOR y) = A059897(a(x), a(y)), where XOR denotes bitwise exclusive-or, A003987(.,.).
a(x OR y) = A059896(a(x), a(y)), where OR denotes the bitwise operation, A003986(.,.).

A305720 Square array T(n, k) read by antidiagonals, n > 0 and k > 0; for any prime number p, the p-adic valuation of T(n, k) is the product of the p-adic valuations of n and of k.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 4, 3, 4, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 16, 1, 2, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 8, 1, 4, 5, 4, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 9, 64, 1, 6, 1, 64, 9, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 4, 1, 8, 7, 8

Offset: 1

Rémy Sigrist, Jun 09 2018

The array T is completely multiplicative in both parameters.
For any n > 0 and prime number p, T(n, p) is the highest power of p dividing n.
For any function f associating a nonnegative value to any pair of nonnegative values and such that f(0, 0) = 0, we can build an analog of this sequence, say P_f, such that for any prime number p and any n and k > 0 with p-adic valuations i and j, the p-adic valuation of P_f(n, k) equals f(i, j):
   f(i, j)    P_f
   -------    ---
   i * j      T (this sequence)
   i + j      A003991 (product)
   abs(i-j)   A089913
   min(i, j)  A003989 (GCD)
   max(i, j)  A003990 (LCM)
   i AND j    A059895
   i OR j     A059896
   i XOR j    A059897
If log(N) denotes the set {log(n) : n is in N, the set of the positive integers}, one can define a binary operation on log(N): with prime factorizations n = Product p_i^e_i and k = Product p_i^f_i, set log(n) o log(k) = Sum_{i} (e_i*f_i) * log(p_i). o has the premises of a scalar product even if log(N) isn't a vector space. T(n, k) can be viewed as exp(log(n) o log(k)). - Luc Rousseau, Oct 11 2020

			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    1    4    1    2    1    8    1    2  -> A006519
    3|    1    1    3    1    1    3    1    1    9    1  -> A038500
    4|    1    4    1   16    1    4    1   64    1    4
    5|    1    1    1    1    5    1    1    1    1    5  -> A060904
    6|    1    2    3    4    1    6    1    8    9    2  -> A065331
    7|    1    1    1    1    1    1    7    1    1    1  -> A268354
    8|    1    8    1   64    1    8    1  512    1    8
    9|    1    1    9    1    1    9    1    1   81    1
   10|    1    2    1    4    5    2    1    8    1   10  -> A132741

Cf. A003989, A003990, A003991, A006519, A007947, A038500, A054496, A059895, A059896, A059897, A060904, A065331, A089913, A132741, A268354, A268357.

T[n_, k_] := With[{p = FactorInteger[GCD[n, k]][[All, 1]]}, If[p == {1}, 1, Times @@ (p^(IntegerExponent[n, p] * IntegerExponent[k, p]))]];
Table[T[n-k+1, k], {n, 1, 15}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 11 2018 *)

T(n, k) = my (p=factor(gcd(n, k))[,1]); prod(i=1, #p, p[i]^(valuation(n, p[i]) * valuation(k, p[i])))

T(n, k) = T(k, n) (T is commutative).
T(m, T(n, k)) = T(T(m, n), k) (T is associative).
T(n, k) = 1 iff gcd(n, k) = 1.
T(n, n) = A054496(n).
T(n, A007947(n)) = n.
T(n, 1) = 1.
T(n, 2) = A006519(n).
T(n, 3) = A038500(n).
T(n, 4) = A006519(n)^2.
T(n, 5) = A060904(n).
T(n, 6) = A065331(n).
T(n, 7) = A268354(n).
T(n, 8) = A006519(n)^3.
T(n, 9) = A038500(n)^2.
T(n, 10) = A132741(n).
T(n, 11) = A268357(n).

A306446 a(n) is the number of connected components in the Fermi-Dirac factorization of n (see Comments for precise definition).

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 2, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 2, 2, 1, 1, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 1, 2, 2, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 2, 2, 1, 1, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 3, 2, 2, 2

Offset: 1

Rémy Sigrist, Feb 16 2019

nonn

For any n > 0:
- let F(n) be the set of distinct Fermi-Dirac primes (A050376) with product n,
- let G(n) be the undirected graph with vertices F(n) and the following connection rules:  for any k >= 0 and any pair of consecutive prime numbers (p, q):
   - p^(2^k) and p^(2^(k+1)) are connected,
   - p^(2^k) and q^(2^k) are connected,
- a(n) is the number of connected components in G(n).
The sequence may be specified algebraically by formulas (1) to (2c) in my contemporary entry in the formula section. - Peter Munn, Jan 05 2021

			For n = 67!:
- the Fermi-Dirac primes p^(2^k) in F(67!) can be depicted as:
    6|@
    5|
    4| @
    3| @@@
    2| @@ @@
    1| @@@@ @@@@@
    0| @@  @@@   @@@@@@@@
  ---+-------------------
  k/p|    111122334445566
     |2357137939171373917
- G(67!) has 4 connected components:
    6|A
    5|
    4| B
    3| BBB
    2| BB BB
    1| BBBB CCCCC
    0| BB  CCC   DDDDDDDD
  ---+-------------------
  k/p|    111122334445566
     |2357137939171373917
- hence a(67!) = 4.

Antti Karttunen, Table of n, a(n) for n = 1..100000
OEIS Wiki, "Fermi-Dirac representation" of n
Rémy Sigrist, PARI program for A306446
Wikipedia, Connected component (graph theory)

Cf. A064547, A069010.
A050376, A059895, A059896, A306697 are used in a formula defining this sequence.
A329050 corresponds to the array depicted in the first example, with prime(n+1) = p.
The formula section details how the sequence maps the terms of A002110, A066205.
A003961, A225546, A340346 are used to express relationship between terms of this sequence.

PARI
```
See Links section.
```

If m and n are coprime, then a(m * n) <= a(m) + a(n).
a(p^k) = A069010(k) for any k >= 0 and any prime number p.
a(n) <= A064547(n).
a(A002110(k)) = 1 for any k > 0.
a(A066205(k)) = k for any k > 0.
From Peter Munn, Jan 05 2021: (Start)
(1) a(1) = 0, otherwise a(n) > 0.
For any k, n > 0:
(2a) a(A050376(k)) = 1;
(2b) a(A059896(n,k)) <= a(n) + a(k);
(2c) a(A059896(n,k)) = a(n) + a(k) if and only if A059895(A306697(n,24), k) = 1 and A059895(n, A306697(k,24)) = 1.
For any n > 0, write n = j * k^2 * m^4, j, k squarefree, m > 0:
(3a) a(n) <= a(j) + a(k) + a(m);
(3b) if gcd(j, k) = 1, a(n) = a(j) + a(n/j);
(3c) if gcd(j, k) = j, a(n) = a(n/j);
(3d) if gcd(k, m) = 1, a(n) = a(n/m^4) + a(m^4);
(3e) if gcd(j, k) = k and gcd(k, m) = 1, a(n) = a(j) + a(m).
For any n > 0:
(4a) a(n^2) = a(A003961(n)) = a(A225546(n)) = a(n);
(4b) a(n) = a(A340346(n)) + a(n/A340346(n)).
For any odd n > 0 (with k >= 0, m >= 0):
(5) If n = 9^k * (6m + 1) or n = 9^k * (6m + 5) then a(2n) = a(n) + 1; otherwise a(2n) = a(n).
(End)

cp's OEIS Frontend

A060381 a(n) = prime(n)*prime(n+1)*...*prime(2*n-1), where prime(i) is the i-th prime.

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A066616 a(1) = 1; a(n) = n*a(n-1) if n does not divide a(n-1), otherwise a(n) = a(n-1).

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A258211 Nonsquarefree numbers of the form 4*k + 2.

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Magma

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A336882 a(0) = 1; for k >= 0, 0 <= i < 2^k, a(2^k + i) = m_k * a(i), where m_k is the least odd number not in terms 0..2^k - 1.

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A305720 Square array T(n, k) read by antidiagonals, n > 0 and k > 0; for any prime number p, the p-adic valuation of T(n, k) is the product of the p-adic valuations of n and of k.

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A306446 a(n) is the number of connected components in the Fermi-Dirac factorization of n (see Comments for precise definition).

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Formula

A060381 a(n) = prime(n)prime(n+1)...prime(2n-1), where prime(i) is the i-th prime.