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

1,2

Comments

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

Links

Table of n, a(n) for n=1..60.

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

Example

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.

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).

Sequence in context: A239993 A332335 A310961 * A181901 A310962 A310963

Adjacent sequences: A385826 A385827 A385828 * A385830 A385831 A385832

Keyword

nonn

Author

Gordon Hamilton, Jul 09 2025

Extensions

More terms from David A. Corneth, Jul 09 2025

Status

approved