A164047 -id:A164047 - cp's OEIS frontend

A066062 Number of subsets S of T={0,1,2,...,n} such that each element of T is the sum of two (not necessarily distinct) elements of S.

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: 0

John W. Layman, Dec 01 2001

nonn

This sequence may be equivalent to A008929, but has a somewhat different definition. The size of the smallest subset counted by this sequence, for a given n, is given in A066063.
From Steven Finch, Mar 15 2009: (Start)
Such sets S are called additive 2-bases for {0,1,2,...,n}.
a(n) is also the number of symmetric numerical sets S with atom monoid A(S) equal to {0, 2n+2, 2n+3, 2n+4, 2n+5, ...}. (End)

			For n=2, the definition obviously requires that S contain both 0 and 1. The only subsets of {0,1,2} that do this are {0,1} and {0,1,2}. For both of these, we have 0=0+0, 1=0+1, 2=1+1, so a(2)=2.

Martin Fuller, Table of n, a(n) for n = 0..65
S. R. Finch, Monoids of natural numbers, March 17, 2009. [Cached copy, with permission of the author]
G. Grekos, L. Haddad, C. Helou, and J. Pihko, On the Erdos-Turán conjecture, J. Number Theory 102 (2003), no. 2, 339-352.
J. Marzuola and A. Miller, Counting numerical sets with no small atoms, arXiv:0805.3493 [math.CO], 2008. - _Steven Finch_, Mar 15 2009
J. Marzuola and A. Miller, Counting numerical sets with no small atoms, J. Combin. Theory A 117 (6) (2010) 650-667.

Cf. A008929, A066063, A158449, A164047.

Cf. A158291. - Steven Finch, Mar 15 2009

C
```
See Martin Fuller link in A158449
```

a[n_] := Module[{},
  T = Range[0, n];
  ST = Subsets[T, {Floor[n^(2/3)], n+1}];
  selQ[S_] := Intersection[T, Total /@ Tuples[S, {2}]] == T;
  SST = Select[ST, selQ]; min = Min[Length /@ SST];
  SST // Length
];
Table[an = a[n]; Print["a(", n, ") = ", an, " min = ", min]; an, {n, 0, 24}] (* Jean-François Alcover, Nov 05 2018 *)

def sums(s): return set((si+sj) for i, si in enumerate(s) for sj in s[i:])
def b(i, n, s):
    if sums(s) >= set(range(n+1)): return 2**(n+1-i)
    if i > n: return 0
    return b(i+1, n, s) + b(i+1, n, s+[i])
def a(n): return b(0, n, [])
print([a(n) for n in range(15)]) # Michael S. Branicky, Jan 15 2022

a(n) = 2*a(n-1) - A158449(n) [adapted from A164097]. - Martin Fuller, Sep 13 2023

a(n) >= A001405(n). - Michael Chu, Oct 15 2023

a(27)-a(30) from Michael S. Branicky, Jan 15 2022

a(31) onwards from Martin Fuller, Sep 13 2023

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

Original entry on oeis.org

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

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
```
See Martin Fuller link
```

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

cp's OEIS Frontend

A066062 Number of subsets S of T={0,1,2,...,n} such that each element of T is the sum of two (not necessarily distinct) elements of S.

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

C

Formula

Extensions