A001212 -id:A001212 - cp's OEIS frontend

A005342 a(n) = solution to the postage stamp problem with n denominations and 7 stamps.

7, 23, 69, 165, 345, 664, 1137, 1911

Offset: 1

Fred Lunnon [W. F. Lunnon] defines "solution" to be the smallest value not obtainable by the best set of stamps. The solutions given are one lower than this, that is, the sequence gives the largest number obtainable without a break using the best set of stamps.

R. K. Guy, Unsolved Problems in Number Theory, C12.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. Alter and J. A. Barnett, A postage stamp problem, Amer. Math. Monthly, 87 (1980), 206-210.
Erich Friedman, Postage stamp problem
R. L. Graham and N. J. A. Sloane, On Additive Bases and Harmonious Graphs
R. L. Graham and N. J. A. Sloane, On Additive Bases and Harmonious Graphs, SIAM J. Algebraic and Discrete Methods, 1 (1980), 382-404.
W. F. Lunnon, A postage stamp problem, Comput. J. 12 (1969) 377-380.

Postage stamp sequences: A001208, A001209, A001210, A001211, A001212, A001213, A001214, A001215, A001216, A005342, A005343, A005344, A014616, A053346, A053348, A075060, A084192, A084193.

Entry improved by comments from John Seldon (johnseldon(AT)onetel.com), Sep 15 2004

a(8) from Challis and Robinson by Robert Price, Jul 19 2013

A005343 a(n) = solution to the postage stamp problem with n denominations and 8 stamps.

Original entry on oeis.org

8, 28, 89, 234, 512, 1045, 2001, 3485

Offset: 1

N. J. A. Sloane

Fred Lunnon [W. F. Lunnon] defines "solution" to be the smallest value not obtainable by the best set of stamps. The solutions given are one lower than this, that is, the sequence gives the largest number obtainable without a break using the best set of stamps.

R. K. Guy, Unsolved Problems in Number Theory, C12.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. Alter and J. A. Barnett, A postage stamp problem, Amer. Math. Monthly, 87 (1980), 206-210.
M. F. Challis and J. P. Robinson, Some Extremal Postage Stamp Bases, J. Integer Seq., 13 (2010), Article 10.2.3. [From John P Robinson (john-robinson(AT)uiowa.edu), Feb 18 2010]
Erich Friedman, Postage stamp problem
R. L. Graham and N. J. A. Sloane, On Additive Bases and Harmonious Graphs
R. L. Graham and N. J. A. Sloane, On Additive Bases and Harmonious Graphs, SIAM J. Algebraic and Discrete Methods, 1 (1980), 382-404.
W. F. Lunnon, A postage stamp problem, Comput. J. 12 (1969) 377-380.

Postage stamp sequences: A001208, A001209, A001210, A001211, A001212, A001213, A001214, A001215, A001216, A005342, A005343, A005344, A014616, A053346, A053348, A075060, A084192, A084193.

Entry improved by comments from John Seldon (johnseldon(AT)onetel.com), Sep 15 2004

a(8) from Challis and Robinson. John P Robinson (john-robinson(AT)uiowa.edu), Feb 18 2010

A005344 a(n) = solution to the postage stamp problem with n denominations and 9 stamps.

Original entry on oeis.org

9, 34, 112, 326, 797, 1617, 3191

Offset: 1

N. J. A. Sloane

Fred Lunnon [W. F. Lunnon] defines "solution" to be the smallest value not obtainable by the best set of stamps. The solutions given are one lower than this, that is, the sequence gives the largest number obtainable without a break using the best set of stamps.

R. K. Guy, Unsolved Problems in Number Theory, C12.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. Alter and J. A. Barnett, A postage stamp problem, Amer. Math. Monthly, 87 (1980), 206-210.
Erich Friedman, Postage stamp problem
R. L. Graham and N. J. A. Sloane, On Additive Bases and Harmonious Graphs
R. L. Graham and N. J. A. Sloane, On Additive Bases and Harmonious Graphs, SIAM J. Algebraic and Discrete Methods, 1 (1980), 382-404.
W. F. Lunnon, A postage stamp problem, Comput. J. 12 (1969) 377-380.

Postage stamp sequences: A001208, A001209, A001210, A001211, A001212, A001213, A001214, A001215, A001216, A005342, A005343, A005344, A014616, A053346, A053348, A075060, A084192, A084193.

Entry improved by comments from John Seldon (johnseldon(AT)onetel.com), Sep 15 2004

a(7) from Challis and Robinson by Robert Price, Jul 19 2013

A053346 a(n) = solution to the postage stamp problem with 7 denominations and n stamps.

Original entry on oeis.org

7, 26, 70, 162, 336, 638, 1137, 2001, 3191, 5047, 7820, 11568, 17178

Offset: 1

N. J. A. Sloane, Jun 20 2003

Fred Lunnon [W. F. Lunnon] defines "solution" to be the smallest value not obtainable by the best set of stamps. The solutions given are one lower than this, that is, the sequence gives the largest number obtainable without a break using the best set of stamps.

R. K. Guy, Unsolved Problems in Number Theory, C12.

R. Alter and J. A. Barnett, A postage stamp problem, Amer. Math. Monthly, 87 (1980), 206-210.
M. F. Challis, Two new techniques for computing extremal h-bases A_kComp. J. 36(2) (1993) 117-126
M. F. Challis, J. P. Robinson, Some extremal postage stamp bases, JIS 13 (2010) #10.2.3.
Erich Friedman, Postage stamp problem
W. F. Lunnon, A postage stamp problem, Comput. J. 12 (1969) 377-380.
Eric Weisstein's World of Mathematics, Postage stamp problem

Postage stamp sequences: A001208, A001209, A001210, A001211, A001212, A001213, A001214, A001215, A001216, A005342, A005343, A005344, A014616, A053346, A053348, A075060, A084192, A084193.

a(9) from Challis by R. J. Mathar, Apr 01 2006
Entry improved by comments from John Seldon (johnseldon(AT)onetel.com), Sep 15 2004
a(10)-a(13) from Challis and Robinson by Robert Price, Jul 19 2013

A053348 a(n) = solution to the postage stamp problem with 8 denominations and n stamps.

Original entry on oeis.org

8, 32, 93, 228, 524, 1007, 1911, 3485

Offset: 1

N. J. A. Sloane, Jun 20 2003

Fred Lunnon [W. F. Lunnon] defines "solution" to be the smallest value not obtainable by the best set of stamps. The solutions given are one lower than this, that is, the sequence gives the largest number obtainable without a break using the best set of stamps.

R. K. Guy, Unsolved Problems in Number Theory, C12.

R. Alter and J. A. Barnett, A postage stamp problem, Amer. Math. Monthly, 87 (1980), 206-210.
M. F. Challis, Two new techniques for computing extremal h-bases A_kComp. J. 36(2) (1993) 117-126
M. F. Challis, J. P. Robinson, Some extremal postage stamp bases, JIS 13 (2010) #10.2.3.
Erich Friedman, Postage stamp problem
W. F. Lunnon, A postage stamp problem, Comput. J. 12 (1969) 377-380.
Eric Weisstein's World of Mathematics, Postage stamp problem

Postage stamp sequences: A001208, A001209, A001210, A001211, A001212, A001213, A001214, A001215, A001216, A005342, A005343, A005344, A014616, A053346, A053348, A075060, A084192, A084193.

a(6) from Challis by R. J. Mathar, Apr 01 2006
Entry improved by comments from John Seldon (johnseldon(AT)onetel.com), Sep 15 2004
a(7)-a(8) from Challis and Robinson by Robert Price, Jul 19 2013

A075060 a(n) = solution to the postage stamp problem with n denominations and 10 stamps.

Original entry on oeis.org

10, 40, 146, 427, 1055, 2510, 5047

Offset: 1

N. J. A. Sloane, Jun 20 2003

Fred Lunnon [W. F. Lunnon] defines "solution" to be the smallest value not obtainable by the best set of stamps. The solutions given are one lower than this, that is, the sequence gives the largest number obtainable without a break using the best set of stamps.

R. K. Guy, Unsolved Problems in Number Theory, C12.

R. Alter and J. A. Barnett, A postage stamp problem, Amer. Math. Monthly, 87 (1980), 206-210.
M. F. Challis and J. P. Robinson, Some Extremal Postage Stamp Bases, J. Integer Seq., 13 (2010), Article 10.2.3. [From John P Robinson (john-robinson(AT)uiowa.edu), Feb 18 2010]
Erich Friedman, Postage stamp problem
W. F. Lunnon, A postage stamp problem, Comput. J. 12 (1969) 377-380.
Eric Weisstein's World of Mathematics, Postage stamp problem

Postage stamp sequences: A001208, A001209, A001210, A001211, A001212, A001213, A001214, A001215, A001216, A005342, A005343, A005344, A014616, A053346, A053348, A075060, A084192, A084193.

Entry improved by comments from John Seldon (johnseldon(AT)onetel.com), Sep 15 2004

a(7) from Challis and Robinson by John P Robinson (john-robinson(AT)uiowa.edu), Feb 18 2010

A084192 Array read by antidiagonals: T(n,k) = solution to postage stamp problem with n stamps and k denominations (n >= 1, k >= 1).

Original entry on oeis.org

1, 2, 2, 3, 4, 3, 4, 8, 7, 4, 5, 12, 15, 10, 5, 6, 16, 24, 26, 14, 6, 7, 20, 36, 44, 35, 18, 7, 8, 26, 52, 70, 71, 52, 23, 8, 9, 32, 70, 108, 126, 114, 69, 28, 9, 10, 40, 93, 162, 211, 216, 165, 89, 34, 10, 11, 46, 121, 228, 336, 388, 345, 234, 112, 40, 11, 12, 54, 154, 310, 524

Offset: 0

N. J. A. Sloane, Jun 20 2003

Fred Lunnon [W. F. Lunnon] defines "solution" to be the smallest value not obtainable by the best set of stamps. The solutions given in this sequence and in A001208, A001209, A001210, A001211, A001212, ... are one lower than this, that is, the sequence gives the largest number obtainable without a break using the best set of stamps.

			Array begins:
   1,   2,   3,   4,   5,   6,   7,   8,   9,  10,  11, ...
   2,   4,   7,  10,  14,  18,  23,  28,  34,  40, ...
   3,   8,  15,  26,  35,  52,  69,  89, 112, ...
   4,  12,  24,  44,  71, 114, 165, 234, ...
   5,  16,  36,  70, 126, 216, 345, ...
   6,  20,  52, 108, 211, 388, ...
   7,  26,  70, 162, 336, ...
   8,  32,  93, 228, ...
   9,  40, 121, ...
  10,  46, ...
  11, ...
  ...

Postage stamp sequences: A001208 A001209 A001210 A001211 A001212 A001213 A001214 A001215 A001216 A005342 A005343 A005344 A014616 A053346 A053348 A075060 A084192 A084193

A084193 gives transposed array. Rows and columns give rise to A014616, A001208, A001209, A001210, A001211, A053346, A053348, A001212, A001213, A001214, A001215, A001216, A005342, A005343, A005344, A075060.

Entry improved by comments from John Seldon (johnseldon(AT)onetel.com), Sep 15 2004
More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jun 26 2003
Comments corrected by Shawn Pedersen, Apr 17 2012

A084193 Array read by antidiagonals: T(k,n) = solution to postage stamp problem with n stamps and k denominations (n >= 1, k >= 1).

Original entry on oeis.org

1, 2, 2, 3, 4, 3, 4, 7, 8, 4, 5, 10, 15, 12, 5, 6, 14, 26, 24, 16, 6, 7, 18, 35, 44, 36, 20, 7, 8, 23, 52, 71, 70, 52, 26, 8, 9, 28, 69, 114, 126, 108, 70, 32, 9, 10, 34, 89, 165, 216, 211, 162, 93, 40, 10, 11, 40, 112, 234, 345, 388, 336, 228, 121, 46, 11, 12, 47, 146, 326, 512

Offset: 0

N. J. A. Sloane, Jun 20 2003

Fred Lunnon [W. F. Lunnon] defines "solution" to be the smallest value not obtainable by the best set of stamps. The solutions given in this sequence and in A001208, A001209, A001210, A001211, A001212, ... are one lower than this, that is, the sequence gives the largest number obtainable without a break using the best set of stamps.

			Array begins:
   1,   2,   3,   4,   5,   6,   7,   8,   9,  10,  11, ...
   2,   4,   8,  12,  16,  20,  26,  32,  40,  46, ...
   3,   7,  15,  24,  36,  52,  70,  93, 121, ...
   4,  10,  26,  44,  70, 108, 162, 228, ...
   5,  14,  35,  71, 126, 211, 336, ...
   6,  18,  52, 114, 216, 388, ...
   7,  23,  69, 165, 345, ...
   8,  28,  89, 234, ...
   9,  34, 112, ...
  10,  40, ...
  11, ...
  ...

Index to sequences related to the Postage Stamp Problem

A084192 gives transposed array. Rows and columns give rise to A014616, A001208, A001209, A001210, A001211, A053346, A053348, A001212, A001213, A001214, A001215, A001216, A005342, A005343, A005344, A075060.

Entry improved by comments from John Seldon (johnseldon(AT)onetel.com), Sep 15 2004
More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jun 26 2003
Comments corrected by Shawn Pedersen, Apr 17 2012

A196416 Table read by antidiagonals: V(n,m) = solution to postage stamp problem with n stamps in set, m stamps on letter.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 5, 4, 1, 1, 5, 8, 9, 5, 1, 1, 6, 11, 16, 13, 6, 1, 1, 7, 15, 27, 25, 17, 7, 1, 1, 8, 19, 36, 45, 37, 21, 8, 1, 1, 9, 24, 53, 72, 71, 53, 27, 9, 1, 1, 10, 29, 70, 115, 127, 109, 71, 33, 10, 1

Offset: 0

N. J. A. Sloane, Oct 01 2011

Given n, m, the postage stamp problem is to choose a set of n nonnegative integers such that the sums of m or fewer of these integers can realize the numbers 1, 2, ..., N-1, where N is as large as possible. V(n,m) denotes the value of N.

			Array begins:
m\n 0 1 2 3 4 5 6 ...
---------------------
0...1 1 1 1 1 1 1 ...
1...1 2 3 4 5 6 7  ...
2...1 3 5 9 13 17 21  ...
3...1 4 8 16 25 37 53 ...
4...1 5 11 27 45 71 109  ...
5...1 6 15 36 72 127 212  ...
6...1 7 19 53 115 217 389  ...
...

W. F. Lunnon, A postage stamp problem, Comput. J. 12 (1969) 377-380.

Rows and columns include: A014616/A024206, A195618/A001208, A196069/A001209, A001210, A001211, A196094/A001212, A001213, A001214, A001215, A001216.

A123509 Rohrbach's problem: a(n) is the largest integer such that there exists a set of n integers that is a basis of order 2 for (0, 1, ..., a(n)-1).

Original entry on oeis.org

1, 3, 5, 9, 13, 17, 21, 27, 33, 41, 47, 55, 65, 73, 81, 93, 105, 117, 129, 141, 153, 165, 181, 197, 213

Offset: 1

Warren D. Smith, Oct 02 2006

Notation: N[q] = the set of q+1 elements inside {0,1,...,N-1}
Length of the longest sequence of consecutive integers that can be obtained from a set of n distinct integers by summing any two integers in the set or doubling any one. - Jon E. Schoenfield, Jul 16 2017
According to Zhining Yang, Jul 08 2017, a(13) to a(20) are 65, 70, 79, 90, 101, 112, 123, 134, but there is some doubt about these terms, and they should be confirmed before they are accepted. They do not agree with the conjecture, so perhaps the VBA program is not correct.
The definition of Rohrbach's Problem in the paper of S. Gunturk and M. B. Nathanson in the links is different from the one here. In the paper, the set should contain n nonnegative integers instead of integers. The result should be equal to A001212(n-1)+1 according to the definition in the paper since adding one 0 before any set for A001212(n-1) provides a set of the problem. The data provided by Zhining Yang is obviously wrong since a(n) >= A001212(n-1)+1. And A302648 provides another lower bound of this array since a(n) >= 2*A302648(n)+1. - Zhao Hui Du, Apr 13 2018

			Example: 8[3]: 0,1,3,4 means {0,1,2,...,8} is covered thus: 0=0+0, 1=0+1, 2=1+1, 3=0+3, 4=0+4=1+3, 5=1+4, 6=3+3, 7=3+4, 8=4+4.
N[q]: set
------------------------------
3[2]: 0,1,
4[3]: 0,1,2,
5[3]: 0,1,2,
6[3]: 0,2,3,
7[4]: 0,1,2,3,
8[4]: 0,1,3,4,
9[4]: 0,1,3,4,
10[5]: 0,1,2,4,5,
11[5]: 0,1,2,4,5,
12[5]: 0,1,3,5,6,
13[5]: 0,1,3,5,6,
14[6]: 0,1,2,4,6,7,
15[6]: 0,1,2,4,6,7,
16[6]: 0,1,3,5,7,8,
17[6]: 0,1,3,5,7,8,
18[6]: 0,2,3,7,8,10,
19[7]: 0,1,2,4,6,8,9,
20[7]: 0,1,3,5,7,9,10,
21[7]: 0,1,3,5,7,9,10,
22[7]: 0,2,3,7,8,10,11,
23[8]: 0,1,2,4,6,8,10,11,
24[8]: 0,1,3,5,7,9,11,12,
25[8]: 0,1,3,5,7,9,11,12,
26[8]: 0,2,3,7,8,10,12,13,
27[8]: 0,1,3,4,9,10,12,13,
28[8]: 0,2,3,7,8,12,13,15,
29[9]: 0,1,3,5,7,9,11,13,14,
30[9]: 0,2,3,7,8,10,12,14,15,
31[9]: 0,1,3,4,9,10,12,14,15,
32[9]: 0,2,3,7,8,12,13,15,16,
a(5)=13 because we can obtain at most a total of 13 consecutive integers from a set of 5 integers by summing any two integers in the set or doubling any one; from the 5-integer set {1,2,4,6,7}, we can obtain all 13 integers in the interval [2..14] as follows: 2=1+1, 3=1+2, 4=2+2, 5=1+4, 6=2+4, 7=1+6, 8=2+6, 9=2+7, 10=4+6, 11=4+7, 12=6+6, 13=6+7, 14=7+7.
a(16)=90 because we can obtain at most a total of 90 consecutive integers from a set of 16 integers by summing any two integers in the set or doubling any one: from the 16-integer set {1,2,4,5,8,9,10,17,18,22,25,36,47,58,69,80}, we can obtain all 90 integers in the interval [2..91]. - _Jon E. Schoenfield_, Jul 16 2017

S. Gunturk and M. B. Nathanson, A new upper bound for finite additive bases, Acta Arithmetica, Vol. 124, No. 3 (2006) 235-255.
Jürgen Herzog, Shinya Kumashiro, and Dumitru I. Stamate, The tiny trace ideals of the canonical modules in Cohen-Macaulay rings of dimension one, arXiv:2106.09404 [math.AC], 2021. See p. 9.
Jürgen Herzog, Shinya Kumashiro, and Dumitru I. Stamate, The far-flung Gorenstein property for numerical semigroups, Extended abstract for IMNS 2024. See p. 3.
Kagawa, VBA program
H. Rohrbach, Ein Beitrag zur additiven Zahlentheorie, Math. Z. 42 (1937) 1-30. DOI
W. D. Smith, More information

Cf. A001212, A008932.

a(n) = A001212(n-1)+1 (conjecture). - R. J. Mathar, Oct 08 2006. Comment from Martin Fuller, Mar 18 2009: I agree with this conjecture.

lim inf a(n) / n^2 > 0.2857 lim sup a(n) / n^2 < 0.4789 - Charles R Greathouse IV, Aug 11 2007

More terms (from Smith's web site) from R. J. Mathar, Oct 08 2006
Entry revised by N. J. A. Sloane, Aug 06 2017
a(13)-a(25) from Herzog et al. added by Stefano Spezia, Jul 05 2024

cp's OEIS Frontend

A005342 a(n) = solution to the postage stamp problem with n denominations and 7 stamps.

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A005343 a(n) = solution to the postage stamp problem with n denominations and 8 stamps.

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A005344 a(n) = solution to the postage stamp problem with n denominations and 9 stamps.

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A053346 a(n) = solution to the postage stamp problem with 7 denominations and n stamps.

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A053348 a(n) = solution to the postage stamp problem with 8 denominations and n stamps.

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A075060 a(n) = solution to the postage stamp problem with n denominations and 10 stamps.

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A084192 Array read by antidiagonals: T(n,k) = solution to postage stamp problem with n stamps and k denominations (n >= 1, k >= 1).

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A084193 Array read by antidiagonals: T(k,n) = solution to postage stamp problem with n stamps and k denominations (n >= 1, k >= 1).

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A196416 Table read by antidiagonals: V(n,m) = solution to postage stamp problem with n stamps in set, m stamps on letter.

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