Mamuka Jibladze - cp's OEIS frontend

User: Mamuka Jibladze

Mamuka Jibladze has authored 5 sequences.

A343264 Cardinalities of the sets of fusible numbers obtained at the consecutive steps of their construction as follows. We set S(0) = {0}. S(n+1) is obtained by adding to S(n) the sums (x+y+1)/2 for all x,y from S(n) with the property |x-y| < 1. Then, a(n) is the number of elements in S(n).

1, 2, 4, 9, 21, 50, 119, 281, 656, 1513, 3449, 7777, 17363, 38422, 84355, 183915, 398526, 858901

Offset: 0

Mamuka Jibladze, Apr 09 2021

			a(1) = 2 because S(1) = {0, 1/2};
a(2) = 4 because S(2) = {0, 1/2, 3/4, 1};
a(3) = 9 because S(3) = {0, 1/2, 3/4, 7/8, 1, 9/8, 5/4, 11/8, 3/2}.

Jeff Erickson, Gabriel Nivasch and Junyan Xu, Fusible numbers and Peano Arithmetic, arXiv:2003.14342 [cs.LO], 2020.
David A. Corneth, PARI program

Cf. A283075.

s:= proc(n) option remember; `if`(n=0, {0}, (l-> (m-> {seq([2*x, seq(
     `if`(abs(x-y) nops(s(n)):
seq(a(n), n=0..10);  # Alois P. Heinz, Apr 09 2021

S[n_]:=S[n]=If[n==0,{0},S[n-1]\[Union]Map[(#[[1]]+#[[2]]+1)/2&,Select[Tuples[S[n-1],{2}],Abs[#[[1]]-#[[2]]]<1&]]]; Table[Length[S[n]],{n,0,12}]

\\ See Corneth link. David A. Corneth, Apr 09 2021

a(13) from Alois P. Heinz, Apr 09 2021

a(14)-a(17) from David A. Corneth, Apr 10 2021

A279170 a(n) is the smallest among the natural numbers m with the property that there exists a non-constant quadratic map S^n -> S^m from the n-dimensional sphere to the m-dimensional sphere.

Original entry on oeis.org

1, 2, 2, 4, 4, 4, 4, 8, 8, 8, 8, 8, 8, 8, 8, 16, 16, 16, 16, 16, 16, 16, 16, 16, 24, 24, 24, 24, 24, 24, 24, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 40, 40, 40, 40, 40, 40, 48, 48, 48, 48, 48, 48, 48, 48, 48, 56, 56, 56, 56, 56, 56, 56, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 72, 72, 72, 72, 80, 80, 80, 80, 80, 80, 80, 80, 80, 88, 88, 88, 88, 88, 88, 88, 96, 96, 96, 96, 96

Offset: 1

Mamuka Jibladze, Dec 07 2016

nonn

Coincides with A053644 until n=24.

P. Yiu, Quadratic Forms between Euclidean Spheres, Manuscripta Math. 83, pp. 171-181 (1994).

A003484 used in the definition. Cf. A053644.

Uniquely determined by the following: a(2^t + m) = 2^t if 0 <= m < A003484(2^t); a(2^t + m) = 2^t + a(m) if A003484(2^t) <= m < 2^t.

A254033 Number of primes dividing exactly one number in the next largest gap between primes.

Original entry on oeis.org

0, 1, 2, 3, 6, 10, 15, 20, 21, 28, 37, 44, 53, 76, 96, 113, 123, 135, 142, 150, 181, 191, 235, 270, 291, 294, 313, 327, 334, 395, 403, 411, 445, 478, 496, 539, 582, 587, 654, 693, 722, 732, 757, 754, 772, 778, 791, 832, 830, 848, 920, 930, 955, 1004, 1053, 1151, 1240

Offset: 1

Mamuka Jibladze, Jan 23 2015

nonn

			The 5th largest prime gap (after 2-3, 3-5, 7-11 and 23-29) occurs between 89 and 97, and there are 6 primes which occur exactly once in this gap, namely 7 (dividing 91), 13 (dividing 91), 19 (dividing 95), 23 (dividing 92), 31 (dividing 93) and 47 (dividing 94), so a(5)=6.

Robert G. Wilson v, Table of n, a(n) for n = 1..75

Sequences related to increasing prime gaps: A005250, A002386, A000101, A005669.

gp = (* the list of primes in A002386 *); f[n_] := Block[{p = gp[[n]], q = NextPrime[ gp[[n]]]}, r = Range[p + 1, q - 1]; lng = Length@ r; t = Split@ Sort@ Flatten@ Table[ First@# & /@ FactorInteger[ r[[i]]], {i, lng}]; Length@ Select[t, Length@# == 1 &]]; Array[f, 75] (* Robert G. Wilson v, Jan 23 2015 *)

a(43)-a(57) from Robert G. Wilson v, Jan 23 2015

A230439 Number of contractible "tight" meanders of width n.

Original entry on oeis.org

1, 2, 6, 14, 34, 68, 150, 296, 586, 1140, 2182, 4130, 7678, 14368, 26068, 48248, 86572, 158146, 281410, 509442, 901014, 1618544, 2852464, 5089580, 8948694, 15884762, 27882762, 49291952, 86435358, 152316976, 266907560, 469232204, 821844316

Offset: 1

Mamuka Jibladze, Nov 04 2013

nonn

A tight meander of width n is a special kind of meander defined as follows.

For any pair (S={s_1,...,s_k},T={t_1,...,t_l}) of subsets of {1,...,n-1} (k or l might be 0), the tight meander M(S,T) defined by (S,T) is the following subset of R^2:

assuming S and T ordered so that 0=s_0

semicircles in the upper half-plane with endpoints (s_{i-1}+j,0) and (s_i+1-j,0), for i=1,...,k+1, and j positive integer with s_{i-1}+j

and semicircles in the lower half-plane with endpoints (t_{i-1}+j,0) and (t_i+1-j,0), for i=1,...,l+1, and j positive integer with t_{i-1}+j

The tight meander M(S,T) is called contractible if it is a contractible subspace of R^2, i.e., is either a single point or homeomorphic to an interval.

Then, a(n) is the number of pairs (S,T) as above such that the tight meander M(S,T) is contractible.

From Roger Ford, Jul 05 2023: (Start)

The following is a definition for closed meanders that yield the same sequence as tight meanders. T(n,k) = the number of closed meanders with n top arches and with k exterior arches and k arches of length 1.

e = exterior arch (arch with no covering arch), 1 = arch with length 1, e1 = arch that is exterior with a length of 1:

e exterior length 1

__________ arches arches

/ ____ \

e1 / / \ \ top = 2 top = 2

/\ / / /\1 \ \

/ \ / / / \ \ \

\ \ / / \ \ / / bottom = 2 bottom = 2

\ \/1 / \ \/1 / total = 4 total = 4

\____/ \____/

e e Example T(4,4).

(End)

			For n=3 the a(3)=6 contractible tight meanders of width 3 correspond to the following pairs of subsets of {1,2}: ({},{1}), ({},{2}), ({1},{}), ({2},{}), ({1},{2}), ({2},{1}).

Mamuka Jibladze, Table of n, a(n) for n = 1..100 (first 64 terms by Martin Plechsmid)
Vincent Coll, Colton Magnant, and Hua Wang, The Signature of a Meander, arXiv:1206.2705 [math.QA], 2012.
Vladimir Dergachev and Alexandre Kirillov, Index of Lie algebras of Seaweed Type, J. Lie Theory, 10 (2000), 331-343
Mathoverflow, "Special" meanders
Dmitri I. Panyushev, Inductive Formulas for the Index of Seaweed Lie Algebras, Moscow Math. J., 1 (2001), 221-241.

For various kinds of meandric numbers see A005315, A005316, A060066, A060089, A060206.

# program based on the C code by Martin Plechsmid:
proc()
local n,a,b,d,r;
option remember;
  if args[1]=1 then
   1
  elif nargs=1 then
   2*`+`(''procname(args,[i],[j])'$'j'=1..i-1'$'i'=2..args)
  else
   n:=args[1]; a:=args[2]; b:=args[3];
   if b=[] then
    `+`('procname(n,a,[k])'$'k'=1..n)
   elif a[1]=b[1] then
    0
   elif a[1]0 then
     procname(n-b[1],[d-r,op(subsop(1=r,a))],subsop(1=NULL,b))
    else
     procname(n-b[1],subsop(1=d,a),subsop(1=NULL,b))
    fi
   fi
  fi
end;

(* program based on the C code by Martin Plechsmid: *)
f[n_,a_,b_]:=Which[
n==1, 1,
b=={}, f[n,a,b]=Sum[f[n,a,{i}],{i,n}],
a=={} || First[a]

A096337 Number of those nonnegative integer solutions of the congruence x_1+2x_2+...+(n-1)x_{n-1} = 0 (mod n) which are indecomposable, that is, are not nonnegative linear combinations of other nonnegative integer solutions.

Original entry on oeis.org

0, 1, 3, 6, 14, 19, 47, 64, 118, 165, 347, 366, 826, 973, 1493, 2134, 3912, 4037, 7935, 8246, 12966, 17475, 29161, 28064, 49608, 59357, 83419, 97242, 164966, 152547, 280351, 295290, 405918, 508161, 674629, 708818, 1230258, 1325731, 1709229, 1868564, 3045108

Offset: 1

Mamuka Jibladze, Jun 28 2004

nonn

a(n) is a lower bound for the number of fundamental invariants of binary forms of degree n+2 - see Kac. A lower estimate for a(n) is given by Dixmier et al.

a(n) is the number of nonempty multisets of positive integers < n such that their sum modulo n is zero and that no proper nonempty subset has this property. - George B. Salomon, Sep 29 2019

			a(3)=3 since 3+2*0=3, 1+2*1=3 and 0+2*3=6 are the only indecomposable nonnegative integer solutions to x_1+2x_2=0 (mod 3): all other nonnegative integer solutions have form x_1=p*3+q*1+r*0, x_2=p*0+q*1+r*3 for nonnegative integers p, q, r.

Vakhtang Tsiskaridze, Table of n, a(n) for n = 1..64, computed by a Pascal code (1994, unpublished)
J. Dixmier, P. Erdős and J.-L. Nicolas, Sur le nombre d'invariants fondamentaux des formes binaires, C. R. Acad. Sci. Paris Ser. I Math. 305 (1987), no. 8, 319-322.
John C. Harris and David L. Wehlau, Non-negative Integer Linear Congruences, Indag. Math. 17 (2006) 37-44.
V. Kac, Root systems, representations of quivers and invariant theory, Invariant theory (Montecatini, 1982), 74-108, Lecture Notes in Math., 996, Springer, Berlin, 1983.
Klaus Pommerening, The Indecomposable Solutions of Linear Congruences, arXiv:1703.03708 [math.NT], 2017.

cp's OEIS Frontend

User: Mamuka Jibladze

A343264 Cardinalities of the sets of fusible numbers obtained at the consecutive steps of their construction as follows. We set S(0) = {0}. S(n+1) is obtained by adding to S(n) the sums (x+y+1)/2 for all x,y from S(n) with the property |x-y| < 1. Then, a(n) is the number of elements in S(n).

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A279170 a(n) is the smallest among the natural numbers m with the property that there exists a non-constant quadratic map S^n -> S^m from the n-dimensional sphere to the m-dimensional sphere.

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A254033 Number of primes dividing exactly one number in the next largest gap between primes.

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Examples

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Programs

Mathematica

Extensions

A230439 Number of contractible "tight" meanders of width n.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A096337 Number of those nonnegative integer solutions of the congruence x_1+2x_2+...+(n-1)x_{n-1} = 0 (mod n) which are indecomposable, that is, are not nonnegative linear combinations of other nonnegative integer solutions.

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