A327415 Representation of integers by the product of prime partitions.

0, 1, 2, 3, 4, 5, 9, 7, 15, 14, 21, 11, 35, 13, 33, 26, 39, 17, 65, 19, 51, 38, 57, 23, 95, 46, 69, 92, 115, 29, 161, 31, 87, 62, 93, 124, 155, 37, 217, 74, 111, 41, 185, 43, 123, 86, 129, 47, 215, 94, 141, 188, 235, 53, 329, 106, 159, 212, 265, 59, 371, 61

Offset

0,3

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. 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 products of these partitions. For n in {0, 1, 2} a(n) = n by convention.

Links

Table of n, a(n) for n=0..61.

Eric Weisstein's World of Mathematics, Prime Partition

Example

n   a(n)  partition
2     2   [2]
3     3   [3]
4     4   [2, 2]
5     5   [5]
6     9   [3, 3]
7     7   [7]
8    15   [5, 3]
9    14   [7, 2]
10   21   [7, 3]
11   11   [11]
12   35   [7, 5]
13   13   [13]
14   33   [11, 3]
15   26   [13, 2]
16   39   [13, 3]
17   17   [17]
18   65   [13, 5]
19   19   [19]
20   51   [17, 3]

Maple

a := proc(n) local r, p;
    if n <= 2 then return n fi;
    if n::odd then
        if isprime(n) then return n fi;
        r := prevprime(n);
        p := [seq(2, i=1..(n + 1 - r)/2), r]
    else
        r := prevprime(n - 1);
        p := [n - r, r]
    fi;
    return mul(k, k in p)
end: seq(a(n), n = 0..61);

Prog

(SageMath)
def a(n):
    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 mul(p)
print([a(n) for n in range(40)])

Crossrefs

Sequence in context: A222257 A327456 A238535 * A072501 A092975 A164340

Adjacent sequences: A327412 A327413 A327414 * A327416 A327417 A327418

Keyword

nonn

Author

Peter Luschny, Sep 08 2019

Status

approved