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A123098 Multiplicative encoding of triangle formed by reading Pascal's triangle mod 2 (A047999). 8
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

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

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].

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)

EXAMPLE

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)

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 *)

PROG

(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

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

CROSSREFS

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

First column of A255483.

Sequence in context: A325237 A065799 A162582 * A136699 A033710 A243157

Adjacent sequences:  A123095 A123096 A123097 * A123099 A123100 A123101

KEYWORD

nonn,changed

AUTHOR

Jonathan Vos Post, Nov 05 2006

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

STATUS

approved

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Last modified June 17 18:55 EDT 2019. Contains 324198 sequences. (Running on oeis4.)