A037165 -id:A037165 - cp's OEIS frontend

A069763 Frobenius number of the numerical semigroup generated by consecutive cubes.

181, 1637, 7811, 26659, 73529, 174761, 372007, 727271, 1328669, 2296909, 3792491, 6023627, 9254881, 13816529, 20114639, 28641871, 39988997, 54857141, 74070739, 98591219, 129531401, 168170617, 215970551, 274591799, 345911149

Offset: 2

Victoria A Sapko (vsapko(AT)canes.gsw.edu), Apr 18 2002

The Frobenius number of a numerical semigroup generated by relatively prime integers a_1,...,a_n is the largest positive integer that is not a nonnegative linear combination of a_1,...,a_n. Since consecutive cubes are relatively prime, they generate a numerical semigroup with a Frobenius number. The Frobenius number of a 2-generated semigroup has the formula ab-a-b.

			a(2)=181 because 181 is not a nonnegative linear combination of 8 and 27, but all integers greater than 181 are.

R. Fröberg, C. Gottlieb and R. Häggkvist, On numerical semigroups, Semigroup Forum, 35 (1987), 63-83 (for definition of Frobenius number).

Cf. A000578, A037165, A059769, A069755-A069764.

a(n) = n^3*(n+1)^3-n^3-(n+1)^3 = n^6+3*n^5+3*n^4-n^3-3*n^2-3*n-1.

G.f.: x^2*(181+370*x+153*x^2+24*x^3-13*x^4+6*x^5-x^6)/(1-x)^7. [Colin Barker, Feb 14 2012]

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

Original entry on oeis.org

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

A069758 Frobenius number of the numerical semigroup generated by three consecutive hexagonal numbers.

Original entry on oeis.org

65, 377, 395, 797, 1589, 6029, 3347, 4571, 6035, 10997, 10979, 12212, 19409, 47246, 24023, 29003, 35357, 52112, 50603, 50411, 73049, 158207, 78155, 90203, 102005, 144443, 138467, 131474, 183077

Offset: 2

Victoria A Sapko (vsapko(AT)canes.gsw.edu), Apr 08 2002

The Frobenius number of a numerical semigroup generated by relatively prime integers a_1,...,a_n is the largest positive integer that is not a nonnegative linear combination of a_1,...,a_n. Since three consecutive hexagonal numbers are relatively prime, they generate a numerical semigroup with a Frobenius number.

			a(2)=65 because 65 is not a nonnegative linear combination of 6, 15 and 28, but all integers greater than 65 are.

R. Fröberg, C. Gottlieb and R. Häggkvist, On numerical semigroups, Semigroup Forum, 35 (1987), 63-83 (for definition of Frobenius number).

Cf. A000384, A037165, A059769, A069755-A069764.

FrobeniusNumber/@Partition[Table[n(2n-1),{n,2,35}],3,1] (* Harvey P. Dale, Jul 25 2011 *)

A069760 Frobenius number of the numerical semigroup generated by consecutive centered square numbers.

Original entry on oeis.org

47, 287, 959, 2399, 5039, 9407, 16127, 25919, 39599, 58079, 82367, 113567, 152879, 201599, 261119, 332927, 418607, 519839, 638399, 776159, 935087, 1117247, 1324799, 1559999, 1825199, 2122847

Offset: 1

Victoria A Sapko (vsapko(AT)canes.gsw.edu), Apr 09 2002

The Frobenius number of a numerical semigroup generated by relatively prime integers a_1,...,a_n is the largest positive integer that is not a nonnegative linear combination of a_1,...,a_n. Since consecutive centered squares are relatively prime, they generate a numerical semigroup with a Frobenius number. The Frobenius number of a 2-generator semigroup is ab-a-b.

			a(1)=47 because 47 is not a nonnegative linear combination of 5 and 13, but all integers greater than 47 are.

R. Fröberg, C. Gottlieb and R. Häggkvist, On numerical semigroups, Semigroup Forum, 35 (1987), 63-83 (for definition of Frobenius number).
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A001844, A037165, A059769, A069755-A069764.

Table[4n^4+16n^3+20n^2+8n-1,{n,30}] (* or *) LinearRecurrence[ {5,-10,10,-5,1},{47,287,959,2399,5039},30] (* Harvey P. Dale, Apr 25 2011 *)

a(n) = 4*n^4+16*n^3+20*n^2+8*n-1.
a(n) = 5*a(n-1)-10*a(n-2) +10*a(n-3)-5*a(n-4)+a(n-5). - Harvey P. Dale, Apr 25 2011
G.f.: x*(47+52*x-6*x^2+4*x^3-x^4)/(1-x)^5. - Colin Barker, Feb 14 2012

A165401 Antidiagonally reading the array, formed via: first, writing the primes in the first row (row_1), and forming all successive rows' elements using the previous rows' elements as: row_2(j) = row_1(j)*row_1(j+1) - row_1(j) - row_1(j+1), and so on. The first 'column' of the array, 2 1 -1 -1 -1 -1 -1 -1 ... is converted to its absolute value.

Original entry on oeis.org

2, 1, 3, 1, 7, 5, 1, 131, 23, 7, 1, 165619, 1275, 59, 11, 1, 1443643414307, 8716707, 6843, 119, 13, 1, 1930155333520916730618052608, 1337002832135523, 153383955, 22419, 191, 17, 1

Offset: 1

Umut Uludag, Sep 17 2009

The actual array that leads to the list given above is:
2 3 5 7 11 13 17 19 ...
1 7 23 59 119 191 287 395 ...
1 131 1275 6843 22419 54339 112683 241915 ...
1 165619 8716707 ...
1 1443643414307 ...
It can be observed that:
1) Obviously, all primes will appear in the list at least once, as the first row of the generating array is composed of all the primes.
2) There exist primes that appear more than once (e.g., 23, 59, 131, 191...).
3) When we have two successive primes in a row, prime(i) & prime(i+1) -as in all of the first row, and occasionally in other rows- the element just below these two is, obviously, Euler_tot( prime(i) * prime (i+1) ) - 1.

Cf. A000040: The list of prime numbers is the first row of the array, that is converted to the list. Cf. A037165: This list, "Prime(n)*prime(n+1)-prime(n)-prime(n+1)" is the second row of the array, that is converted to the list.

A329857 Positive integers which can be represented as p*q - p - q where p and q are distinct odd primes.

Original entry on oeis.org

7, 11, 19, 23, 31, 35, 39, 43, 47, 55, 59, 63, 71, 79, 83, 87, 91, 95, 103, 107, 111, 115, 119, 131, 139, 143, 155, 159, 163, 167, 175, 179, 183, 191, 199, 203, 207, 211, 215, 219, 223, 231, 239, 251, 259, 263, 271, 275, 279, 287, 295, 299, 311, 323, 327, 331, 335, 343, 347, 351, 355, 359

Offset: 1

Craig J. Beisel, Nov 22 2019

Cf. A037165 (a subsequence), A046388, A091305, A096345, A137367 (subsequence with twin primes), A218862.

Select[Range[360], {} != Solve[p*q-p-q  == # && p >q> 2, {p,q}, Primes] &] (* Giovanni Resta, Jan 16 2020 *)

lim=1000; x=[]; forprime(p=3, lim/3, forprime(q=p+2, lim/3, if(setsearch(x,p*q-q-p),, x=setunion(x,[p*q-q-p])))); for(i=1, length(x), if(x[i]<(lim), print1(x[i], ", ")))

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A069763 Frobenius number of the numerical semigroup generated by consecutive cubes.

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

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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.

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A069758 Frobenius number of the numerical semigroup generated by three consecutive hexagonal numbers.

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A069760 Frobenius number of the numerical semigroup generated by consecutive centered square numbers.

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A329857 Positive integers which can be represented as p*q - p - q where p and q are distinct odd primes.

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