

A304794


a(n) is the least possible difference between the greatest and the smallest prime in any partition of prime(n) into three primes p+q+r (p <= q <= r); n >= 4.


0



1, 2, 2, 2, 2, 6, 4, 6, 2, 6, 4, 4, 2, 6, 4, 10, 10, 12, 12, 8, 2, 8, 12, 8, 12, 10, 10, 2, 6, 4, 10, 10, 18, 18, 18, 8, 8, 2, 2, 12, 8, 12, 10, 6, 8, 12, 8, 12, 10, 4, 10, 10, 14, 14, 24, 14, 14, 8, 14, 2, 6, 4, 4, 4, 24, 20, 18, 14, 28, 14, 28, 26, 26, 4, 12
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OFFSET

4,2


COMMENTS

From Goldbach's (weak) conjecture, any prime >= 7 can be expressed as the sum of three primes. For n >= 5 at least 2 different partitions of this kind are possible for the same prime. Many primes (>90% in the above data) exhibit the property of the maximum value of the smallest prime p (=A302607(n)) occurring in the same partition as the minimum value of the greatest prime, r (=A302756(n)). The exceptions in these data are 103, 193, 229, 271, 281, ... where the greatest value of the smallest prime and the smallest value of the greatest prime appear in different partitions. In such cases a(n) is chosen from the partition with the smallest difference rp, see examples.


LINKS

Table of n, a(n) for n=4..78.


EXAMPLE

Prime(6) = 13 = 3+5+5 = 3+3+7, so a(6) = 53 = 2.
Prime(27) = 103 = 31+31+41 = 29+37+37, A302607(27) = 31, and A302756(27) = 37; a(27) = 3729 = 8.
Prime(44) = 193 = 61+61+71 = 59+67+67, A302607(44) = 61, A302756(44) = 67; a(44) = 6759 = 8.


MATHEMATICA

Array[Min@ Map[First@ #  Last@ # &, Select[IntegerPartitions[Prime@ #, {3}], AllTrue[#, PrimeQ] &]] &, 75, 4] (* Michael De Vlieger, Jul 19 2018 *)


PROG

(PARI) a(n) = {my(pn = prime(n), res = oo); forprime(p=2, pn, forprime(q=p, pn, forprime(r=q, pn, if (p+q+r == pn, res = min(res, rp)); ); ); ); res; } \\ Michel Marcus, Jul 05 2018


CROSSREFS

Cf. A302756, A302607.
Sequence in context: A260983 A103222 A319100 * A175809 A061033 A075094
Adjacent sequences: A304791 A304792 A304793 * A304795 A304796 A304797


KEYWORD

nonn


AUTHOR

David James Sycamore, May 18 2018


STATUS

approved



