A327413 Minimal prime partition representation of odd integers.

1, 3, 5, 7, 36, 11, 13, 105, 17, 19, 210, 23, 300, 105300, 29, 31, 528, 314160, 37, 741, 41, 43, 990, 47, 1176, 1499400, 53, 1485, 2370060, 59, 61, 1953, 4062240, 67, 2346, 71, 73, 2775, 8119650, 79, 3240, 83, 3570, 13355370, 89, 4095, 17518410, 78219700650

Offset

1,2

Comments

A partition is prime if all parts are primes. A partition of an odd integer is minimal if it has at most one odd part and is shorter than any other such partition. The terms of the sequence are the multinomials of these partition.

Links

Table of n, a(n) for n=1..48.

Eric Weisstein's World of Mathematics, Prime Partition

Formula

If 2n + 1 is prime or n = 1 then a(n) = n. For n > 1 and not prime let a(n) be the multinomial of P where P is the partition [p, 2, ..., 2] with p the greatest prime less than 2n + 1 and 2 occurring (2*n + 1 - p)/2 times in the partition.

Example

n  2n+1  partition  a(n)
1   3 :  [3]         3
2   5 :  [5]         5
3   7 :  [7]         7
4   9 :  [7, 2]      36
5   11:  [11]        11
6   13:  [13]        13
7   15:  [13, 2]     105
8   17:  [17]        17
9   19:  [19]        19
10  21:  [19, 2]     210
11  23:  [23]        23
12  25:  [23, 2]     300
13  27:  [23, 2, 2]  105300
14  29:  [29]        29

Prog

(SageMath)
def a(n, op):
    if n <= 2: return n
    if n % 2 == 1:
        if is_prime(n): return n
        r = previous_prime(n)
        p = [r] + [2]*((n + 1 - r)//2)
    else:
        r = previous_prime(n - 1)
        p = [r, n - r]
    return op(p)
print([a(n, multinomial) for n in range(1, 74, 2)]) # is this sequence.
# print([a(n, multinomial) for n in range(0, 74, 2)]) # is A327414.
# print([a(n, mult       ) for n in range(0, 74   )]) # is A327415.
# It is remarkable how nicely the syntax here reflects the intention: 'mult' and
# 'multinomials' as exchangeable operators on partitions. Traditional mathematical
# notation obscures this simplicity.

Crossrefs

Cf. A327414.

Sequence in context: A256154 A069969 A067232 * A106115 A154544 A155034

Adjacent sequences: A327410 A327411 A327412 * A327414 A327415 A327416

Keyword

nonn

Author

Peter Luschny, Sep 07 2019

Status

approved