A327414 Minimal prime partition representation of even integers.

0, 2, 6, 20, 56, 120, 792, 364, 560, 8568, 1140, 1540, 42504, 2600, 98280, 2035800, 4960, 5984, 376992, 12620256, 9880, 850668, 13244, 15180, 1712304, 19600, 2598960, 177100560, 27720, 4582116, 386206920, 37820, 41664, 8936928, 969443904, 54740, 13991544

Offset

0,2

Comments

A partition is prime if all parts are primes. A partition of an even integer n > 2 is minimal if it has at most two parts, one of which is the greatest prime less than n - 1. The terms of the sequence are the multinomials of these partition. By convention a(0) = 0 and a(1) = 2.

Links

Table of n, a(n) for n=0..36.

Eric Weisstein's World of Mathematics, Prime Partition

Formula

For n >= 2 let a(n) be the multinomial of P where P is the partition [p, 2n - p] with p the greatest prime less than 2n - 1.

Example

n   2n  partition a(n)
2   4 :  [2,  2]   6
3   6 :  [3,  3]   20
4   8 :  [5,  3]   56
5   10:  [7,  3]   120
6   12:  [7,  5]   792
7   14:  [11, 3]   364
8   16:  [13, 3]   560
9   18:  [13, 5]   8568
10  20:  [17, 3]   1140
11  22:  [19, 3]   1540
12  24:  [19, 5]   42504
13  26:  [23, 3]   2600
14  28:  [23, 5]   98280
15  30:  [23, 7]   2035800

Prog

(SageMath)
def a(n):
    if n < 2: return 2*n
    p = previous_prime(2*n - 1)
    return multinomial([p, 2*n - p])
print([a(n) for n in range(40)])

Crossrefs

Cf. A327413.

Sequence in context: A045655 A303307 A321192 * A110295 A027294 A231538

Adjacent sequences: A327411 A327412 A327413 * A327415 A327416 A327417

Keyword

nonn

Author

Peter Luschny, Sep 07 2019

Status

approved