A053348 -id:A053348 - cp's OEIS frontend

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

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

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.
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

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

A114139 Changes in United States postal rates per ounce since 1863.

Original entry on oeis.org

-2, -2, 1, -1, 1, 1, 1, 1, 2, 2, 0, 3, 2, 3, 2, 2, 3, 4, 3, 3, 1, 1, 3, 2

Offset: 1

Jonathan Vos Post, Feb 03 2006

Benjamin Franklin, first Postmaster General of the United States, applied computational complexity theory to Economics by changing the business plan for American mail by changing from payment by distance to payment by weight. "Before stamps were used a person had to collect his mail at the post office and pay for it. Franklin stopped the money loss on unclaimed mail in Philadelphia by printing in his paper the names of persons who had mail awaiting them. He also developed a simple, accurate way of keeping post-office accounts. In 1753 Franklin was made deputy postmaster general for all the colonies." [Encyclopedia Britannica]

			a(1) = -2 because the rate per half ounce was lowered effective 3 March 1863 from 3 cents to 2 cents; thereafter rates were per ounce.

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

R. Alter and J. A. Barnett, A postage stamp problem, Amer. Math. Monthly, Vol. 87, No. 3 (1980), 206-210.
A. K. Dart, The History of Postage Rates in the United States
W. F. Lunnon, A postage stamp problem, Comput. J., Vol. 12, No. 4 (1969) 377-380.
USPS, Benjamin Franklin Commemorative Stamps
1847usa.com, United States Postage Stamps By Year.

First differences of A114062.

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

A140571 Decimal expansion of the universal constant in E(h), the maximum number of essential elements of order h.

Original entry on oeis.org

2, 0, 5, 7, 2, 8, 4, 1, 2, 8, 4, 7, 8, 7, 9, 3, 4, 1, 2, 8, 5, 8, 2, 2, 3, 9, 6, 4, 4, 8, 3, 7, 6, 9, 0, 9, 1, 0, 0, 4, 3, 4, 7, 8, 2, 7, 4, 9, 4, 2, 1, 2, 6, 8, 0, 7, 4, 1, 5, 3, 8, 1, 9, 6, 6, 2, 4, 2, 3, 6, 9, 2, 9, 5, 4, 2, 7, 6, 3, 5, 1, 3, 3, 4, 9, 8, 5, 1, 9, 0, 8, 0, 7, 8, 9, 0, 1, 6, 5, 3, 6, 5, 5, 9, 7, 7

Offset: 1

Jonathan Vos Post, Jul 05 2008

A fundamental result of Erdos and Graham is that every integer basis possesses only finitely many essential elements. Grekos refined this, showing that the number of essential elements in a basis or order h is bounded by a function of h only. Deschamps and Farhi (2007) proved a best possible upper bound on this function, which contains a constant whose digits are this sequence.
Abstract: Plagne recently determined the asymptotic behavior of the function E(h), which counts the maximum possible number of essential elements in an additive basis for N of order h. Here we extend his investigations by studying asymptotic behavior of the function E(h,k), which counts the maximum possible number of essential subsets of size k, in a basis of order h. For a fixed k and with h going to infinity, we show that
E(h,k) = Theta_{k} ([h^{k}/log h]^{1/(k+1)}). The determination of a more precise asymptotic formula is shown to depend on the solution of the well-known "postage stamp problem" in finite cyclic groups. On the other hand, with h fixed and k going to infinity, we show that E(h,k) ~ (h-1) (log k)/(log log k).

			2.0572841284787934...

Bruno Deschamps, Bakir Farhi, Essentialité dans les bases additives, J. Number Theory, 123 (2007), p. 170-192.
P. Erdos and R. L. Graham, On Bases with an Exact Order, Acta Arith. 37(1980)201-207.
G. Grekos, Sur l'ordre d'une base additive, (French) Séminaire de Théorie des nombres de Bordeaux, 1987/1988, exposé 31.
Peter Hegarty, The Postage Stamp Problem and Essential Subsets in Integer Bases, arXiv:0807.0463 [math.NT], 2008.

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

RealDigits[(30*Sqrt[Log[1564]/1564]),10,120][[1]] (* Harvey P. Dale, Sep 27 2023 *)

30*sqrt(log(1564)/1564) \\ Michel Marcus, Oct 18 2018

Equals 30*sqrt(log(1564)/1564).

a(100) corrected by Georg Fischer, Jul 12 2021

cp's OEIS Frontend

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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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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A114139 Changes in United States postal rates per ounce since 1863.

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A140571 Decimal expansion of the universal constant in E(h), the maximum number of essential elements of order h.

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