A138994 -id:A138994 - cp's OEIS frontend

A151898 First differences of Frobenius numbers for 7 successive numbers A138987.

1, 1, 1, 1, 1, 9, 2, 2, 2, 2, 2, 16, 3, 3, 3, 3, 3, 23, 4, 4, 4, 4, 4, 30, 5, 5, 5, 5, 5, 37, 6, 6, 6, 6, 6, 44, 7, 7, 7, 7, 7, 51, 8, 8, 8, 8, 8, 58, 9, 9, 9, 9, 9, 65, 10, 10, 10, 10, 10, 72, 11, 11, 11, 11, 11, 79, 12, 12, 12, 12, 12, 86, 13, 13, 13, 13, 13, 93, 14, 14, 14, 14, 14, 100, 15

Offset: 1

Artur Jasinski, Apr 05 2008

First differences of Frobenius numbers for 2 successive numbers see A005843
First differences of Frobenius numbers for 3 successive numbers see A014682
First differences of Frobenius numbers for 4 successive numbers see A138995
First differences of Frobenius numbers for 5 successive numbers see A138996
First differences of Frobenius numbers for 6 successive numbers see A138997
First differences of Frobenius numbers for 7 successive numbers see A151898
First differences of Frobenius numbers for 8 successive numbers see A138999

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,2,0,0,0,0,0,-1).

Cf. A028387, A037165, A079326, A138985, A138986, A138987, A138988.
Cf. A138989, A138990, A138991, A138992, A138993, A138994.
Cf. A005843, A014682, A138995, A138996, A138997, A138999.

a = {}; Do[AppendTo[a, FrobeniusNumber[{n + 1, n + 2, n + 3, n + 4, n + 5, n + 6, n + 7}]], {n, 1, 100}]; Differences[a]
Differences[Table[FrobeniusNumber[Range[n,n+6]],{n,2,90}]] (* or *) LinearRecurrence[ {0,0,0,0,0,2,0,0,0,0,0,-1},{1,1,1,1,1,9,2,2,2,2,2,16},90] (* Harvey P. Dale, Jul 26 2024 *)

a(n) = A138987(n+1)-A138987(n).

G.f.: -x*(2*x^11-9*x^5-x^4-x^3-x^2-x-1) / ((x-1)^2*(x+1)^2*(x^2-x+1)^2*(x^2+x+1)^2). [Colin Barker, Dec 13 2012]

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.

Original entry on oeis.org

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

Gordon Hamilton, Jul 09 2025

nonn

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

			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.

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

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).

More terms from David A. Corneth, Jul 09 2025

cp's OEIS Frontend

A151898 First differences of Frobenius numbers for 7 successive numbers A138987.

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