A385829 Numbers k that are the largest k such that k cannot be partitioned into parts that are a set of at least two consecutive primes.

1, 4, 7, 9, 13, 16, 23, 27, 30, 31, 35, 41, 42, 49, 53, 54, 59, 63, 64, 65, 66, 67, 79, 80, 83, 85, 95, 101, 102, 105, 107, 110, 113, 114, 116, 117, 119, 121, 125, 131, 135, 136, 138, 143, 145, 150, 160, 162, 163, 169, 174, 175, 178, 187, 191, 194, 197, 199, 200, 203

Offset: 1

Author

Gordon Hamilton, Jul 09 2025

Keywords

• nonn

Comments

If we consider partitions into one distinct prime then no such largest number k exists.

Examples

			1 is a term as it is the largest positive integer that cannot be partitioned into parts 2 and 3. We have 2 = 2, 3 = 3 and so any positive integer at least two can be partitioned into parts 2 and 3.
30 is a term as 30 is the largest number that cannot be partitions into parts 7, 11 and 13. Proof:
30 cannot be written as a partition of 7, 11, 13 and we have 31 = 7 + 11 + 13, 32 = 3*7 + 11, 33 = 3*11, 34 = 3*7 + 13, 35 = 5*7, 36 = 2*7 + 2*11, 37 = 11 + 2*13 which proves that the next 7 positive integers after 30 can be partitioned into parts 7, 11, 13. Any larger number than that can have more sevens added.
		

Links

• David A. Corneth, Terms with their corresponding list of consecutive primes

Crossrefs

Frobenius numbers for k successive primes: A037165 (k=2), A138989 (k=3), A138990 (k=4), A138991 (k=5), A138992 (k=6), A138993 (k=7), A138994 (k=8).

Extensions

More terms from David A. Corneth, Jul 09 2025