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A327415 Representation of integers by the product of prime partitions. 1
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified July 30 12:08 EDT 2021. Contains 346359 sequences. (Running on oeis4.)