A007188 -id:A007188 - cp's OEIS frontend

A297845 Encoded multiplication table for polynomials in one indeterminate with nonnegative integer coefficients. Symmetric square array T(n, k) read by antidiagonals, n > 0 and k > 0. See comment for details.

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

Offset: 1

Rémy Sigrist, Jan 10 2018

For any number n > 0, let f(n) be the polynomial in 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 in a single indeterminate x with nonnegative integer coefficients; let g be the inverse of f; T(n, k) = g(f(n) * f(k)).
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
From Peter Munn, Jun 16 2021: (Start)
The operation defined by this sequence can be extended to be the multiplicative operator of a ring over the positive rationals that is isomorphic to the polynomial ring Z[x]. The extended function f (described in the author's original comments) is the isomorphism we use, and it has the same relationship with the extended operation that exists between their unextended equivalents.
Denoting this extension of T(.,.) as t_Q(.,.), we get t_Q(n, 1/k) = t_Q(1/n, k) = 1/T(n, k) and t_Q(1/n, 1/k) = T(n, k) for positive integers n and k. The result for other rationals is derived from the distributive property: t_Q(q, r*s) = t_Q(q, r) * t_Q(q, s); t_Q(q*r, s) = t_Q(q, s) * t_Q(r, s). This may look unusual because standard multiplication of rational numbers takes on the role of the ring's additive group.
There are many OEIS sequences that can be shown to be a list of the integers in an ideal of this ring. See the cross-references.
There are some completely additive sequences that similarly define by extension completely additive functions on the positive rationals that can be shown to be homomorphisms from this ring onto the integer ring Z, and these functions relate to some of the ideals. For example, the extended function of A048675, denoted A048675_Q, maps i/j to A048675(i) - A048675(j) for positive integers i and j. For any positive integer k, the set {r rational > 0 : k divides A048675_Q(r)} forms an ideal of the ring; for k=2 and k=3 the integers in this ideal are listed in A003159 and A332820 respectively.
(End)

			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  -> A357852
    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
From _Peter Munn_, Jun 24 2021: (Start)
The encoding, n, of polynomials, f(n), that is used for the table is further described in A206284. Examples of encoded polynomials:
   n      f(n)        n           f(n)
   1         0       16              4
   2         1       17            x^6
   3         x       21        x^3 + x
   4         2       25           2x^2
   5       x^2       27             3x
   6     x + 1       35      x^3 + x^2
   7       x^3       36         2x + 2
   8         3       49           2x^3
   9        2x       55      x^4 + x^2
  10   x^2 + 1       64              6
  11       x^4       77      x^4 + x^3
  12     x + 2       81             4x
  13       x^5       90   x^2 + 2x + 1
  15   x^2 + x       91      x^5 + x^3
(End)

Rémy Sigrist, Table of n, a(n) for n = 1..5050
Encyclopedia of Mathematics, Additive arithmetic function
Encyclopedia of Mathematics, Isomorphism
Eric Weisstein's World of Mathematics, Distributive
Eric Weisstein's World of Mathematics, Ring.
Wikipedia, Polynomial ring

Cf. A001221, A007913, A048720, A061395, A055396, A206284, A248601, A248663.
Row n: n=1: A000012, n=2: A000027, n=3: A003961, n=4: A000290, n=5: A357852, n=6: A191002, n=8: A000578.
Main diagonal: A297473.
Functions f satisfying f(T(n,k)) = f(n) * f(k): A001222, A048675 (and similarly, other rows of A104244), 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.
Integers in the ideal of the related ring (see Jun 2021 comment) generated by S: S={3}: A005408, S={4}: A000290\{0}, S={4,3}: A003159, S={5}: A007310, S={5,4}: A339690, S={6}: A325698, S={6,4}: A028260, S={7}: A007775, S={8}: A000578\{0}, S={8,3}: A191257, S={8,6}: A332820, S={9}: A016754, S={10,4}: A340784, S={11}: A008364, S={12,8}: A145784, S={13}: A008365, S={15,4}: A345452, S={15,9}: A046337, S={16}: A000583\{0}, S={17}: A008366.
Equivalent sequence for polynomial composition: A326376.
Related binary operations: A003991, A306697/A059896, A329329/A059897.

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])))

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 and Apr 20 2021: (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).
A104244(m, T(n, k)) = A104244(m, n) * A104244(m, k).
For example, for m = 2, the above formula is equivalent to 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)

New name from Peter Munn, Jul 17 2021

A206296 Prime factorization representation of Fibonacci polynomials: a(0) = 1, a(1) = 2, and for n > 1, a(n) = A003961(a(n-1)) * a(n-2).

Original entry on oeis.org

1, 2, 3, 10, 63, 2750, 842751, 85558343750, 2098355820117528699, 769999781728184386440152910156250, 2359414683424785920146467280333749864720543920418139851

Offset: 0

Clark Kimberling, Feb 05 2012

nonn

These are numbers matched to the Fibonacci polynomials according to the scheme explained in A206284 (see also A104244). In this case, the exponent of the k-th prime p_k in the prime factorization of a(n) indicates the coefficient of term x^(k-1) in the n-th Fibonacci polynomial. See the examples.

			n    a(n)   prime factorization    Fibonacci polynomial
------------------------------------------------------------
0       1   (empty)                F_0(x) = 0
1       2   p_1                    F_1(x) = 1
2       3   p_2                    F_2(x) = x
3      10   p_3 * p_1              F_3(x) = x^2 + 1
4      63   p_4 * p_2^2            F_4(x) = x^3 + 2x
5    2750   p_5 * p_3^3 * p_1      F_5(x) = x^4 + 3x^2 + 1
6  842751   p_6 * p_4^4 * p_2^3    F_6(x) = x^5 + 4x^3 + 3x

Eric Weisstein's World of Mathematics, Fibonacci polynomial
Wikipedia, Fibonacci polynomials

Cf. A000045, A000129, A001222, A003961, A006190, A049310, A048675, A090880, A104244, A206284.
Other such mappings:
  polynomial sequence      integer sequence
  -----------------------------------------
  x^n                      A000040
  (x+1)^n                  A007188
  n*x^(n-1)                A062457
  (1-x^n)/(1-x)            A002110
  n + (n-1)x + ... +x^n    A006939
  Stern polynomials        A260443

c[n_] := CoefficientList[Fibonacci[n, x], x]
f[n_] := Product[Prime[k]^c[n][[k]], {k, 1, Length[c[n]]}]
Table[f[n], {n, 1, 11}]  (* A206296 *)

from functools import reduce
from sympy import factorint, prime, primepi
from operator import mul
def a003961(n):
    F=factorint(n)
    return 1 if n==1 else reduce(mul, [prime(primepi(i) + 1)**F[i] for i in F])
l=[1, 2]
for n in range(2, 11):
    l.append(a003961(l[n - 1])*l[n - 2])
print(l) # Indranil Ghosh, Jun 21 2017

From Antti Karttunen, Jul 29 2015: (Start)
a(0) = 1, a(1) = 2, and for n >= 2, a(n) = A003961(a(n-1)) * a(n-2).
Other identities. For all n >= 0:
A001222(a(n)) = A000045(n). [When each polynomial is evaluated at x=1.]
A048675(a(n)) = A000129(n). [at x=2.]
A090880(a(n)) = A006190(n). [at x=3.]
(End)

a(0) = 1 prepended (to indicate 0-polynomial), Name changed, Comments and Example section rewritten by Antti Karttunen, Jul 29 2015

A066117 Triangle read by rows: T(n,k) = T(n-1,k-1)*T(n,k-1) and T(n,1) = prime(n).

Original entry on oeis.org

2, 3, 6, 5, 15, 90, 7, 35, 525, 47250, 11, 77, 2695, 1414875, 66852843750, 13, 143, 11011, 29674645, 41985913344375, 2806877704512541816406250, 17, 221, 31603, 347980633, 10326201751150285, 433555011900329243987584396875

Offset: 1

Henry Bottomley, Dec 05 2001

As a square array read by descending antidiagonals, A(n, k), n >= 1, k >= 1, gives the encoding defined in A297845 of the polynomial (x+1)^(n-1) * x^(k-1). - Peter Munn, Jul 27 2022

			T(4,3) = T(3,2)*T(4,2) = 15*35 = 525. Rows start
     2;
    3, 6;
  5, 15, 90;
7, 35, 525, 47250;
...
From _Antti Karttunen_, Sep 18 2016: (Start)
Alternatively, this table can be viewed as a square array. Then the top left 5x4 corner looks as:
    2,       3,        5,         7,         11
    6,      15,       35,        77,        143
   90,     525,     2695,     11011,      31603
47250, 1414875, 29674645, 347980633, 2255916949
(End)

Index entries for triangles and arrays related to Pascal's triangle

Cf. A000040, A006094 and A066116 (three leftmost diagonal of triangular table = three topmost rows of square array).
Cf. A007188, A267096 (two rightmost diagonals of the triangular table = two leftmost columns of square array).
Cf. A003961, A055396, A297845.
Cf. A064319, A066119.
Cf. also A099884, A255483, A276586, A276588 (other arrays derived from this one).

T[n_, 1] := Prime[n];
T[n_, k_] := T[n, k] = T[n - 1, k - 1]*T[n, k - 1];
Table[T[n, k], {n, 1, 7}, {k, 1, n}] // Flatten (* Jean-François Alcover, Nov 13 2017 *)

(define (A066117 n) (A066117bi (A002260 n) (A004736 n)))
;; Compute as a square array, with row >= 1, col >= 1:
(define (A066117bi row col) (if (= 1 row) (A000040 col) (* (A066117bi (- row 1) col) (A066117bi (- row 1) (+ col 1)))))
;; With alternative recurrence:
(define (A066117bi row col) (if (= 1 col) (A007188 (- row 1)) (A003961 (A066117bi row (- col 1)))))
;; Antti Karttunen, Sep 18 2016

From Antti Karttunen, Sep 19 2016: (Start)
When computed as a square array A(row,col), row >= 1, col >= 1:
A(1,col) = A000040(col), for row > 1, A(row,col) = A(row-1,col)*A(row-1,col+1).
A(row,1) = A007188(row-1), for col > 1, A(row,col) = A003961(A(row,col-1)).
For all row >= 1, col >= 1, A055396(A(row,col)) = col.
(End)
A(1,1) = 2; for n > 1, A(n,k) = A297845(A(n-1,k),6); for k > 1, A(n,k) = A297845(A(n,k-1),3). - Peter Munn, Jul 20 2022

A123098 Multiplicative encoding of triangle formed by reading Pascal's triangle mod 2 (A047999).

Original entry on oeis.org

2, 6, 10, 210, 22, 858, 1870, 9699690, 46, 4002, 7130, 160660290, 20746, 1008940218, 2569288370, 32589158477190044730, 118, 21594, 39530, 3595293030, 94754, 17808161514, 44788794490, 7074421030108255253430, 263258, 141108130806, 281595235990, 296987147493893719182390, 944729501606

Offset: 0

Jonathan Vos Post, Nov 05 2006

nonn

This is to A047999 "Triangle formed by reading Pascal's triangle mod 2" as A007188 "Multiplicative encoding of Pascal triangle: Product p(i+1)^C(n,i)" is to A007318 "Pascal's triangle read by rows." a(2^n - 1) = primorial(2^n) = A002110(A000079(n)). In row(n) the primes with exponent 1 form row(n) of a Sierpinski sieve, so this sequence is a kind of Gödelization of a Sierpinski sieve.
All terms are divisible by 2 and the n-th term, a(n-1), is also divisible by prime(n). This sequence appears as first column of the square array A255483; its second column A276804 is very similar, with all prime factors shifted to the net larger prime (cf. A003961). - M. F. Hasler, Sep 17 2016
a(n) is the n-th power of 6 in the ring defined in A329329. - Peter Munn, Jan 04 2020

			a(0) = 2^T(0,0) = 2^1 = 2.
a(1) = 2^T(1,0) * 3^T(1,1) = 2^1 * 3^1 = 6.
a(2) = 2^T(2,0) * 3^T(2,1) * 5^T(2,2) = 2^1 * 3^0 * 5^1 = 10.
a(3) = 2^T(3,0) * 3^T(3,1) * 5^T(3,2) * 7^T(3,3) = 2^1 * 3^1 * 5^1 * 7^1 = 210.
a(4) = 2^1 * 3^0 * 5^0 * 7^0 * 11^1 = 22.
a(5) = 2^1 * 3^1 * 5^0 * 7^0 * 11^1 * 13^1 = 858.
a(6) = 2^1 * 3^0 * 5^1 * 7^0 * 11^1 * 13^0 * 17^1 = 1870.
a(7) = 2^1 * 3^1 * 5^1 * 7^1 * 11^1 * 13^1 * 17^1 * 19^1 = 9699690.
a(8) = 2^1 * 3^0 * 5^0 * 7^0 * 11^0 * 13^0 * 17^0 * 19^0 * 23^1 = 46.
a(9) = 2^1 * 3^1 * 5^0 * 7^0 * 11^0 * 13^0 * 17^0 * 19^0 * 23^1 * 29^1 = 4002.
a(10) = 2^1 * 3^0 * 5^1 * 7^0 * 11^0 * 13^0 * 17^0 * 19^0 * 23^1 * 29^0 * 31^1 = 7130.
a(11) = 2^1 * 3^1 * 5^1 * 7^1 * 11^0 * 13^0 * 17^0 * 19^0 * 23^1 * 29^1 * 31^1 * 37^1 = 160660290.
a(12) = 2^1 * 3^0 * 5^0 * 7^0 * 11^1 * 13^0 * 17^0 * 19^0 * 23^1 * 29^0 * 31^0 * 37^0 * 41^1 = 20746.
From _N. J. A. Sloane_, Feb 28 2015: (Start)
Factorizations of initial terms, from Cobeli-Zaharescu paper:
                     2 = 2
                     6 = 2*3
                    10 = 2*5
                   210 = 2*3*5*7
                    22 = 2*11
                   858 = 2*3*11*13
                  1870 = 2*5*11*17
               9699690 = 2*3*5*7*11*13*17*19
                    46 = 2*23
                  4002 = 2*3*23*29
                  7130 = 2*5*23*31
             160660290 = 2*3*5*7*23*29*31*37
                 20746 = 2*11*23*41
            1008940218 = 2*3*11*13*23*29*41*43
            2569288370 = 2*5*11*17*23*31*41*47
  32589158477190044730 = 2*3*5*7*11*13*17*19*23*29*31*37*41*43*47*53
  ... (End)
From _Jon E. Schoenfield_, Jun 09 2019: (Start)
   n | Factorization of a(n)
  ---+-----------------------------------------------
   0 | 2
   1 | 2* 3
   2 | 2   * 5
   3 | 2* 3* 5* 7
   4 | 2         *11
   5 | 2* 3      *11*13
   6 | 2   * 5   *11   *17
   7 | 2* 3* 5* 7*11*13*17*19
   8 | 2                     *23
   9 | 2* 3                  *23*29
  10 | 2   * 5               *23   *31
  11 | 2* 3* 5* 7            *23*29*31*37
  12 | 2         *11         *23         *41
  13 | 2* 3      *11*13      *23*29      *41*43
  14 | 2   * 5   *11   *17   *23   *31   *41   *47
  15 | 2* 3* 5* 7*11*13*17*19*23*29*31*37*41*43*47*53
  ... (End)

Chai Wah Wu, Table of n, a(n) for n = 0..510
C. Cobeli, A. Zaharescu, A game with divisors and absolute differences of exponents, Journal of Difference Equations and Applications, Vol. 20, #11 (2014) pp. 1489-1501, DOI: 10.1080/10236198.2014.940337. Also available as arXiv:1411.1334 [math.NT], 2014.

Cf. A000040, A000120, A001316, A001317, A007188, A007318, A007913, A047999, A019565, A255484, A329329.

First column of A255483.

f:=n->mul(ithprime(k+1)^(binomial(n,k) mod 2),k=0..n);
[seq(f(n),n=0..40)];

a[n_] := Product[Prime[k+1]^Mod[Binomial[n, k], 2], {k, 0, n}];
Table[a[n], {n, 0, 28}] (* Jean-François Alcover, Oct 01 2018, from Maple *)

a(n) = prod (k=0, n, if (binomial(n,k)%2, prime(k+1), 1)) \\ Rémy Sigrist, Jun 09 2019

from operator import mul
from functools import reduce
from sympy import prime
def A123098(n):
    return reduce(mul,(1 if ~(n-1) & k else prime(k+1) for k in range(n))) # Chai Wah Wu, Feb 08 2016

(define (A123098 n) (A019565 (A001317 n))) ;; Antti Karttunen, Sep 18 2016

a(n) = Product_{i=0..n} p(i+1)^(C(n,i) mod 2).
a(n) = Product_{i=0..n} p(i+1)^T(n,i), where T(n,i) are as in A047999 and where Sum_{k>=0} T(n, k) = A001316(n) = 2^A000120(n).
From Antti Karttunen, Sep 18 2016: (Start)
a(n) = A007913(A007188(n)). [From the first comment.]
a(n) = A019565(A001317(n)).
(End)
a(0) = 2, and for n > 0, a(n) = A329329(a(n-1), 6). - Peter Munn, Jan 04 2020

Further terms from N. J. A. Sloane, Feb 28 2015

Changed offset from 1 to 0, corresponding changes to formulas and examples from Antti Karttunen, Sep 18 2016

A136104 A007318 * A002110; a(n) = Sum_{k=0..n} binomial(n,k)*A002110(k).

Original entry on oeis.org

1, 3, 11, 55, 375, 3731, 47743, 777771, 14770535, 331611235, 9205305591, 285781156343, 10308779559631, 418386835375575, 18097509979840775, 846748292083023991, 44182142790019823943, 2570069981187508600331, 157428743473326543397855, 10449715795107936675445215, 739751959772798881608189731

Offset: 0

Gary W. Adamson, Dec 14 2007

nonn

			a(3) = 55 = (1, 3, 3, 1) dot (1, 2, 6, 30) = (1 + 6 + 18 + 30), where A002110 = (1, 2, 6, 30, 210, 2310, ...).

Cf. A002110, A007188, A007318, A276085.
Leftmost column of A276586.
Cf. also A001339, A121572.

b:= proc(n) option remember; `if`(n=0, 1, ithprime(n)*b(n-1)) end:
a:= n-> add(binomial(n, k)*b(k), k=0..n):
seq(a(n), n=0..20);  # Alois P. Heinz, Sep 20 2016

b[n_] := b[n] = If[n==0, 1, Prime[n]*b[n-1]]; a[n_] := Sum[Binomial[n, k]*b[k], {k, 0, n}]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Feb 22 2017, translated from Maple *)

Binomial transform of primorial numbers, A002110.

a(n) = A276085(A007188(n)). - Antti Karttunen, Sep 18 2016

A few more terms from L. Edson Jeffery, Apr 11 2011

Explicit binomial sum formula added to the name by Antti Karttunen, Sep 19 2016

A277322 a(n) = number of irreducible polynomial factors (counted with multiplicity) in the polynomial (with nonnegative integral coefficients) constructed from the prime factorization of n.

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 3, 0, 1, 1, 4, 1, 5, 2, 2, 0, 6, 1, 7, 1, 2, 1, 8, 1, 2, 2, 1, 1, 9, 1, 10, 0, 3, 2, 3, 1, 11, 2, 2, 1, 12, 1, 13, 1, 2, 1, 14, 1, 3, 1, 3, 1, 15, 1, 3, 1, 3, 3, 16, 1, 17, 2, 2, 0, 4, 1, 18, 1, 3, 1, 19, 1, 20, 2, 2, 1, 4, 2, 21, 1, 1, 2, 22, 2, 3, 2, 2, 1, 23, 2, 4, 1, 4, 2, 4, 1, 24, 1, 2, 1, 25, 1, 26, 1, 2

Offset: 1

Antti Karttunen, Oct 09 2016

nonn

Let p(n,x) be the completely additive polynomial-valued function such that p(prime(n),x) = x^(n-1) as defined by Clark Kimberling in A206284. Then this sequence is the number of irreducible factors in p(n,x), counted with multiplicity.

			For n = 7 = prime(4), the corresponding polynomial is x^3, which factorizes as (x)(x)(x), thus a(7) = 3.
For n = 14 = prime(4) * prime(1), the corresponding polynomial is x^3 + 1, which factorizes as (x + 1)(x^2 - x + 1), thus a(14) = 2.
For n = 90 = prime(3) * prime(2)^2 * prime(1), the corresponding polynomial is x^2 + 2x + 1, which factorizes as (x + 1)^2, thus a(90) = 2.
pfps(660) = pfps(2^2*3*5*11) = pfps(2^2) + pfps(3) + pfps(5) + pfps(11) = 2 + x + x^2 + x^4 which is irreducible, so a(660) = 1.
For n = 30030 = Product_{i=1..6} prime(i), the corresponding polynomial is x^5 + x^4 + x^3 + x^2 + x + 1, which factorizes as (x+1)(x^2 - x + 1)(x^2 + x + 1), thus a(30030) = 3.

Antti Karttunen, Table of n, a(n) for n = 1..30033

Cf. A206442 (gives the number of irreducible polynomial factors without multiplicity), A206284 (positions of 1's, i.e., irreducible polynomials).

Cf. A000040, A007188, A064989, A260443, A277013.

allocatemem(2^29);
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
pfps(n) = if(1==n, 0, if(!(n%2), 1 + pfps(n/2), 'x*pfps(A064989(n))));
A277322 = n -> if(!bitand(n,(n-1)), 0, vecsum(factor(pfps(n))[,2]));
for(n=1, 121121, write("b277322.txt", n, " ", A277322(n)));

pfps(n)=my(f=factor(n)); sum(i=1,#f~, f[i,2] * 'x^(primepi(f[i,1])-1))
A277322(n) = if(1==n, 0, vecsum(factor(pfps(n))[, 2])); \\ Charles R Greathouse IV, test for one added by Antti Karttunen, Oct 09 2016

a(2^n) = 0. [By an explicit convention.]
a(A000040(n)) = n-1.
a(A007188(n)) = n.
a(A260443(n)) = A277013(n).

A285101 a(0) = 2, for n > 0, a(n) = a(n-1)*A242378(n,a(n-1)), where A242378(n,a(n-1)) shifts the prime factorization of a(n-1) n primes towards larger primes with A003961.

Original entry on oeis.org

2, 6, 210, 3573570, 64845819350301990, 28695662573739152697846686144187168109530, 1038300112150956151877699324649731518883355380534272386781875587619359740733888844803014212990

Offset: 0

Antti Karttunen, Apr 15 2017

nonn

Multiplicative encoding of irregular table A053632 (in style of A007188 and A260443).

Indranil Ghosh, Table of n, a(n) for n = 0..9

Cf. A001222, A003961, A007913, A028362, A048675, A053632, A068052, A242378, A248663, A285102.

Cf. also A007188, A260443.

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A242378(k,n) = { while(k>0,n = A003961(n); k = k-1); n; };
A285101(n) = { if(0==n,2,A285101(n-1)*A242378(n,A285101(n-1))); };

from sympy import factorint, prime, primepi
from operator import mul
from functools import reduce
def a003961(n):
    f=factorint(n)
    return 1 if n==1 else reduce(mul, [prime(primepi(i) + 1)**f[i] for i in f])
def a242378(k, n):
    while k>0:
        n=a003961(n)
        k-=1
    return n
l=[2]
for n in range(1, 7):
    x=l[n - 1]
    l.append(x*a242378(n, x))
print(l) # Indranil Ghosh, Jun 27 2017

(definec (A285101 n) (if (zero? n) 2 (* (A285101 (- n 1)) (A242378bi n (A285101 (- n 1)))))) ;; For A242378bi see A242378.

a(0) = 2, for n > 0, a(n) = a(n-1)*A242378(n,a(n-1)).
Other identities. For all n >= 0:
A001222(a(n)) = A000079(n).
A048675(a(n)) = A028362(1+n).
A248663(a(n)) = A068052(n).
A007913(a(n)) = A285102(n).

A267096 a(n) = Product_{i=0..n} prime(i+2)^binomial(n,i).

Original entry on oeis.org

3, 15, 525, 1414875, 41985913344375, 433555011900329243987584396875, 3514495551481947615680580256869117013417604971088496013610671875

Offset: 0

Antti Karttunen, Feb 06 2016

nonn

			Terms are obtained by exponentiating the odd primes in range [3 .. prime(2+n)] with the binomial coefficients obtained from row n of Pascal's triangle (A007318) and then multiplying the factors together:
            3^1
         3^1 * 5^1
      3^1 * 5^2 * 7^1
   3^1 * 5^3 * 7^3 * 11^1
3^1 * 5^4 * 7^6 * 11^4 * 13^1
etc.

Index entries for triangles and arrays related to Pascal's triangle

Second column (or diagonal from right) in A066117.

Cf. A000040, A003961, A007188, A007318, A252738, A276804.

(define (A267096 n) (mul (lambda (k) (expt (A000040 (+ 2 k)) (A007318tr n k))) 0 n)) ;; Where A007318tr gives binomial coefficients, as in A007318.
(define (mul intfun lowlim uplim) (let multloop ((i lowlim) (res 1)) (cond ((> i uplim) res) (else (multloop (1+ i) (* res (intfun i)))))))

a(n) = Product_{i=0..n} prime(i+2)^C(n,i).

a(n) = A003961(A007188(n)).

A276080 a(n) = A276075(A206296(n)).

Original entry on oeis.org

0, 1, 2, 7, 28, 139, 822, 5677, 44888, 400021, 3966970, 43328131, 516782292, 6682867087, 93130824878, 1391321096089, 22181459914672, 375880800693097, 6746469047955378, 127851581333528191, 2551039715319388940, 53457519928692619411, 1173770856436282074982, 26948387795024752862917, 645694707721735535710728, 16117771962578155161812989

Offset: 0

Antti Karttunen, Aug 18 2016

nonn

Cf. A000045, A000142, A206296, A276075.

Cf. also A276081, A001338, A007188.

A276080 := proc (n) add((n-2*k)*factorial(n-k-1)/factorial(k), k = 0..floor((1/2)*n-1/2)) end proc:
seq(A276080(n), n = 0..25); # Peter Bala, Dec 24 2017

Map[If[# == 1, 0, Total[FactorInteger[#] /. {p_, e_} /; p > 1 :> e PrimePi[p]!]] &, Nest[Append[#, (Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}] - Boole[# == 1] &@ #[[-1]]) #[[-2]]] &, {1, 2}, 24]] (* Michael De Vlieger, Dec 24 2017 *)

from sympy import factorint, factorial as f, prime, primepi
from operator import mul
from functools import reduce
def a003961(n):
    F=factorint(n)
    return 1 if n==1 else reduce(mul, [prime(primepi(i) + 1)**F[i] for i in F])
def a276075(n):
    F=factorint(n)
    return 0 if n==1 else sum([F[i]*f(primepi(i)) for i in F])
l=[1, 2]
L=[0, 1]
for n in range(2, 11):
    l.append(a003961(l[n - 1])*l[n - 2])
    L.append(a276075(l[n]))
print(L) # Indranil Ghosh, Jun 21 2017

(define (A276080 n) (A276075 (A206296 n)))
;; A more practical standalone program, that uses memoization-macro definec:
(define (A276080 n) (sum_factorials_times_elements_in (A206296as_index_lists n)))
(definec (A206296as_index_lists n) (cond ((zero? n) (list)) ((= 1 n) (list 1)) (else (map + (cons 0 (A206296as_index_lists (- n 1))) (append (A206296as_index_lists (- n 2)) (list 0 0))))))
(define (sum_factorials_times_elements_in nums) (let loop ((s 0) (nums nums) (i 2) (f 1)) (cond ((null? nums) s) (else (loop (+ s (* (car nums) f)) (cdr nums) (+ 1 i) (* i f))))))

a(n) = A276075(A206296(n)).
From Peter Bala, Dec 24 2017: (Start)
a(n) = Sum_{k = 0..floor((n-1)/2)} (n-2*k)!*binomial(n-k-1,k).
O.g.f.: Sum_{n >= 1} n!*x^n/(1 - x^2)^n = x + 2*x^2 + 7*x^3 + 28*x^4 + ....
Cf. A001339(n) = A276075(A007188(n)) for n >= 1, with o.g.f. Sum_{n >= 0} n!*x^n/(1 - x)^n. (End)

cp's OEIS Frontend

A206295 (1/6)*A007188(n).

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A297845 Encoded multiplication table for polynomials in one indeterminate with nonnegative integer coefficients. Symmetric square array T(n, k) read by antidiagonals, n > 0 and k > 0. See comment for details.

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A206296 Prime factorization representation of Fibonacci polynomials: a(0) = 1, a(1) = 2, and for n > 1, a(n) = A003961(a(n-1)) * a(n-2).

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A066117 Triangle read by rows: T(n,k) = T(n-1,k-1)*T(n,k-1) and T(n,1) = prime(n).

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A123098 Multiplicative encoding of triangle formed by reading Pascal's triangle mod 2 (A047999).

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A136104 A007318 * A002110; a(n) = Sum_{k=0..n} binomial(n,k)*A002110(k).

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A277322 a(n) = number of irreducible polynomial factors (counted with multiplicity) in the polynomial (with nonnegative integral coefficients) constructed from the prime factorization of n.

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