A066062 -id:A066062 - cp's OEIS frontend

A008929 Number of increasing sequences of Goldbach type with maximal element n.

1, 1, 1, 2, 3, 6, 10, 20, 37, 73, 139, 275, 533, 1059, 2075, 4126, 8134, 16194, 32058, 63910, 126932, 253252, 503933, 1006056, 2004838, 4004124, 7987149, 15957964, 31854676, 63660327, 127141415, 254136782, 507750109, 1015059238, 2028564292, 4055812657, 8107052520

Offset: 1

Mauro Torelli (torelli(AT)hermes.mc.dsi.unimi.it)

nonn

Equivalent to A066062 and A164047, except for initial term and offset, as shown by J. Marzuola and A. Miller in "Counting numerical sets with no small atoms" (2010). - Martin Fuller, Sep 13 2023

M. Torelli, Increasing integer sequences and Goldbach's conjecture, preprint, 1996.

Martin Fuller, Table of n, a(n) for n = 1..67
S. R. Finch, Monoids of natural numbers
S. R. Finch, Monoids of natural numbers, March 17, 2009. [Cached copy, with permission of the author]
J. Marzuola and A. Miller, Counting numerical sets with no small atoms, Journal of Combinatorial Theory A, Vol. 117, Issue 6 (2010), 650-667.
Mauro Torelli, Increasing integer sequences and Goldbach's conjecture, RAIRO Theoret. Informatics 40 (2) (2006) 107-121.
Index entries for sequences related to Goldbach conjecture

Cf. A066062, A164047.

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Sep 08 2010

a(34) onwards from Martin Fuller, Sep 13 2023

A158291 The number of numerical sets S with atom monoid A(S) equal to {0,n+1,n+2,n+3,n+4,...}.

Original entry on oeis.org

1, 2, 3, 6, 10, 20, 37, 74, 140, 280, 542, 1084, 2118, 4236, 8337, 16674

Offset: 1

Steven Finch, Mar 15 2009

S. R. Finch, Monoids of natural numbers
S. R. Finch, Monoids of natural numbers, March 17, 2009. [Cached copy, with permission of the author]
J. Marzuola and A. Miller, Counting Numerical Sets with No Small Atoms, arXiv:0805.3493 [math.CO], 2008.
J. Marzuola and A. Miller, Counting numerical sets with no small atoms, J. Combin. Theory A 117 (6) (2010) 650-667.

Cf. A066062.

A158449 The number of sigma-admissible subsets of {1,2,...,n} as defined by Marzuola-Miller.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 3, 1, 7, 3, 17, 7, 43, 24, 118, 74, 330, 206, 888, 612, 2571, 1810, 7274, 5552, 21099, 16334, 61252, 49025, 179239, 146048, 523455, 440980, 1554184, 1315927, 4572794, 3972193, 13569220, 11873290, 40263681, 35824869, 119901609, 107397585

Offset: 1

Steven Finch, Mar 19 2009

nonn

a(n), or Asigma(n), equals the number of sigma-admissible subsets of {1,2,...,n}.
Alternate description: (1) Asigma(k) is the same as the number of additive 2-bases for k which are not additive 2-bases for k+1. (2) Asigma(n) is the number of vertices at height n in the rooted tree in figure 5 of [Marzuola-Miller] which spawn only one vertex at height n+1. [Jeremy L. Marzuola (marzuola(AT)math.uni-bonn.de), Aug 08 2009]
The number of symmetric numerical sets S with atom monoid A(S) equal to {0,n+1,2n+2,2n+3,2n+4,2n+5,...}

			a(1)=a(3)=1 since {0,2,4,5,6,7,...} and {0,1,4,5,8,9,10,11,...} are the only sets satisfying the required conditions.

Martin Fuller, Table of n, a(n) for n = 1..65
S. R. Finch, Monoids of natural numbers [Broken link]
S. R. Finch, Monoids of natural numbers, March 17, 2009. [Cached copy, with permission of the author]
Martin Fuller, C program
J. Marzuola and A. Miller, Counting Numerical Sets with No Small Atoms, arXiv:0805.3493 [math.CO], 2008.
J. Marzuola and A. Miller, Counting numerical sets with no small atoms, J. Combin. Theory A 117 (6) (2010) 650-667.

Cf. A066062, A164047.

C
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See Martin Fuller link
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Recursively related to A164047 by the formula Asigma(2k+1)' = 2Asigma(2k)'-Asigma(k)

Definition rephrased by Jeremy L. Marzuola (marzuola(AT)math.uni-bonn.de), Aug 08 2009
Edited by R. J. Mathar, Aug 31 2009
a(33) onwards from Martin Fuller, Sep 13 2023

A265262 The tree of hemitropic sequences read by rows, arising from an Erdős-Turán conjecture.

Original entry on oeis.org

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

Offset: 0

Labib Haddad and Michel Marcus, Dec 06 2015

Let A be a subset of the set N of nonnegative integers. Let pA(n) be the number of pairs (x, y) of elements of A such that n = x + y and x <= y. The sequence pA = [pA(0), pA(1), ... , pA(n), ... ] is called the profile of A. A Sidon set is a subset A whose profile has only 0's and 1's.
An [order 2 additive] basis of N is a subset A whose profile has no 0’s. Erdős and Turán conjectured that the profile of a basis is always unbounded (see the Erdős and Turán link). The conjecture is, up till now, still undecided.
The tree below displays the infinite sequences [1, pA(2), . . . ], associated to the profiles pA = [1, 1, pA(2), . . . ] of all the subsets A of N to which 0 and 1 belong. Those are the so-called hemitropic sequences. The Erdős-Turán conjecture would not hold if (and only if) the tree contained an infinite bounded branch with no 0's.
The length of the n-th row is 2^n. The right leaf of a node is equal to the left leaf + 1.

			First few levels of the tree:
                       1;
           1,                      2;
     0,          1,          1,         2;
  0,    1,    1,    2,    1,    2,    2,   3;
0, 1, 1, 2, 0, 1, 1, 2, 0, 1, 1, 2, 1, 2, 2, 3;
...
First few rows of the irregular array:
1;
1, 2;
0, 1, 1, 2;
0, 1, 1, 2, 1, 2, 2, 3;
0, 1, 1, 2, 0, 1, 1, 2, 0, 1, 1, 2, 1, 2, 2, 3;
...

Michel Marcus, Table of n, a(n) for n = 0..8190
P. Erdős and P. Turán, On a problem of Sidon in additive number theory, and on some related problems, J. Lond. Math. Soc. 16 (1941), 212-215.
Labib Haddad, Some peculiarities of order 2 bases of N and the Erdos-Turan conjecture, arXiv:1507.05849 [math.NT], 2015 (see The binary tree of hemitropic sequences chapter).
Wikipedia, Erdős-Turán conjecture on additive bases

Cf. A004137, A066062, A217793.

with(ListTools):
v:=n->Reverse( convert(n,base,2)):
m:=n->nops(v(n)):
c:=n-> v(n)[m(n)] + sum(v(n)[k]*v(n)[m(n)-k],k=1..floor(m(n)/2)):
d:= h->[seq(c(n),n=2^h..2^(h+1)-1)]: # the h-th row
f:= p->[seq(c(n),n=1..p)]: # the first p terms

f(t,n,va) = 1+ sum(k=1, n+1, va[k]*t^k);
row(n) = {if (n==0, vc = [1], vc = []; for (ni = 2^n, 2^(n+1)-1, b = binary(ni); ft = f(t, n, b); pt = (f(t, n, b)^2 + f(t^2, n, b))/2; vc = concat(vc, polcoeff(pt, n+1)););); vc;}
tabf(nn) = for (n=0, nn, vrow = row(n); for (k=1, #vrow, print1(vrow[k], ", ")); print());

For each k>=0, let u(k)=1 if k is in A, u(k)=0 is k is not in A. Then pA(n) = Sum_{k=0..floor(n/2)} u(k)*u(n-k). See formula (5) on p. 8 and p. 28 in Haddad link.

A066063 Size of the smallest subset S of T={0,1,2,...,n} such that each element of T is the sum of two elements of S.

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12

Offset: 0

John W. Layman, Dec 01 2001

If one counts all subsets S of T={0,1,2,...n} such that each number in T is the sum of two elements of S, sequence A066062 is obtained.
Since each k-subset of S covers at most binomial(k + 1, 2) members of T, we have binomial(a(n) + 1, 2) >= n + 1. It follows that A002024(n-1) is a lower bound. - Rob Pratt, May 14 2004
This is an instance of the <= 2-stamp postage problem with n denominations. For n > 0, a(n) = 1 + the smallest i such that A001212(i) >= n (adding one adjusts for the fact that A001212 has offset 1). - Tim Peters (tim.one(AT)comcast.net), Aug 25 2006

			For n=2, it is clear that S={0,1} is the unique subset of {0,1,2} that satisfies the definition, so a(2)=2.

Cf. A066062, A002024, A001212.

a(27)-a(50) from Rob Pratt, Aug 13 2020

A158448 a(n) equals the number of admissible pairs of subsets of {1,2,...,n} in the notation of Marzuola-Miller.

Original entry on oeis.org

1, 2, 3, 8, 18, 50, 135, 385, 1065, 3053, 8701, 25579, 73693, 217718, 635220, 1888802

Offset: 1

Steven Finch, Mar 19 2009

Alternate description: a(n) is the number of vertices at height n in the rooted tree in figure 4 of [Marzuola-Miller] which spawn only three vertices at height n+1.

The number of numerical sets S with atom monoid A(S) equal to {0,n+1, 2n+2,2n+3,2n+4,2n+5,...}

			a(3)=3 since {0,4,8,9,10,11,...}, {0,1,4,5,8,9,10,11,...} and {0,1,2, 4,5,6,8,9,10,11,...} are the only three sets satisfying the required conditions.

S. R. Finch, Monoids of natural numbers
S. R. Finch, Monoids of natural numbers, March 17, 2009. [Cached copy, with permission of the author]
J. Marzuola and A. Miller, Counting numerical sets with no small atoms, arXiv:0805.3493 [math.CO], 2008.
J. Marzuola and A. Miller, Counting numerical sets with no small atoms, J. Combin. Theory A 117 (6) (2010) 650-667.

Cf. A158291, A164048.

Recursively related to A164048 (call it A'()) by the formula A(2k+1)' = 2A(2k)'-a(k).

Definition rephrased by Jeremy L. Marzuola (marzuola(AT)math.uni-bonn.de), Aug 08 2009

Edited by R. J. Mathar, Aug 31 2009

A158278 Number of symmetric numerical semigroups with Frobenius number 2n-1; that is, symmetric numerical semigroups for which the largest integer not belonging to them is 2n-1.

Original entry on oeis.org

1, 1, 2, 3, 3, 6, 8, 7, 15, 20, 18, 36, 44, 45, 83, 109, 101, 174, 246, 227

Offset: 1

Steven Finch, Mar 15 2009

			a(3)=2: the only 2 symmetric semigroups with Frobenius number 5=2*3-1 are generated by {3, 4} and {2, 7}.

CombOS - Combinatorial Object Server, Generate numerical semigroups
S. R. Finch, Monoids of natural numbers
S. R. Finch, Monoids of natural numbers, March 17, 2009. [Cached copy, with permission of the author]
J. C. Rosales, P. A. Garcia-Sanchez, J. I. Garcia-Garcia and J. A. Jimenez-Madrid, Fundamental gaps in numerical semigroups, Journal of Pure and Applied Algebra 189 (2004) 301-313.
Clayton Cristiano Silva, Irreducible Numerical Semigroups, University of Campinas, São Paulo, Brazil (2019).

Cf. A124506, A158206.

a(n) = A158206(2*n-1).

A158279 Number of pseudo-symmetric numerical semigroups with Frobenius number 2n; that is, pseudo-symmetric numerical semigroups for which the largest integer not belonging to them is 2n.

Original entry on oeis.org

1, 1, 1, 2, 3, 2, 6, 7, 7, 11, 20, 14, 35, 37, 36, 70, 106, 77, 182

Offset: 1

Steven Finch, Mar 15 2009

			a(3)=1: the unique pseudo-symmetric semigroup with Frobenius number 6=2*3 is generated by {4, 5, 7}.

S. R. Finch, Monoids of natural numbers
S. R. Finch, Monoids of natural numbers, March 17, 2009. [Cached copy, with permission of the author]
J. C. Rosales, P. A. Garcia-Sanchez, J. I. Garcia-Garcia and J. A. Jimenez-Madrid, Fundamental gaps in numerical semigroups, Journal of Pure and Applied Algebra 189 (2004) 301-313.
Clayton Cristiano Silva, Irreducible Numerical Semigroups, University of Campinas, São Paulo, Brazil (2019).

Cf. A124506, A158206.

a(n) = A158206(2*n).

cp's OEIS Frontend

A008929 Number of increasing sequences of Goldbach type with maximal element n.

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A158291 The number of numerical sets S with atom monoid A(S) equal to {0,n+1,n+2,n+3,n+4,...}.

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A158449 The number of sigma-admissible subsets of {1,2,...,n} as defined by Marzuola-Miller.

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C

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A265262 The tree of hemitropic sequences read by rows, arising from an Erdős-Turán conjecture.

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Maple

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A066063 Size of the smallest subset S of T={0,1,2,...,n} such that each element of T is the sum of two elements of S.

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A158448 a(n) equals the number of admissible pairs of subsets of {1,2,...,n} in the notation of Marzuola-Miller.

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Extensions

A158278 Number of symmetric numerical semigroups with Frobenius number 2*n-1; that is, symmetric numerical semigroups for which the largest integer not belonging to them is 2*n-1.

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A158279 Number of pseudo-symmetric numerical semigroups with Frobenius number 2*n; that is, pseudo-symmetric numerical semigroups for which the largest integer not belonging to them is 2*n.

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A158278 Number of symmetric numerical semigroups with Frobenius number 2n-1; that is, symmetric numerical semigroups for which the largest integer not belonging to them is 2n-1.

A158279 Number of pseudo-symmetric numerical semigroups with Frobenius number 2n; that is, pseudo-symmetric numerical semigroups for which the largest integer not belonging to them is 2n.