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

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

Author

Jonathan Vos Post, Nov 05 2006

Keywords

• nonn

Comments

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

Examples

			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)
		

Links

• 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.

Crossrefs

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

First column of A255483.

Programs

Maple

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

Mathematica

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 *)

PARI

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

Python

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

Scheme

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

Formula

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

Extensions

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