Ken Levasseur - cp's OEIS frontend

User: Ken Levasseur

Ken Levasseur has authored 4 sequences.

A238729 Maximum flow of a number.

2, 3, 2, 5, 4, 7, 2, 3, 4, 11, 4, 13, 4, 6, 2, 17, 5, 19, 4, 6, 4, 23, 4, 5, 4, 3, 4, 29, 8, 31, 2, 6, 4, 10, 5, 37, 4, 6, 4, 41, 8, 43, 4, 6, 4, 47, 4, 7, 6, 6, 4, 53, 5, 10, 4, 6, 4, 59, 8, 61, 4, 6, 2, 10, 8, 67, 4, 6, 8, 71, 5, 73, 4, 8, 4, 14, 8, 79, 4, 3

Offset: 2

Ken Levasseur, Mar 03 2014

nonn

F(n) is the maximal flow in a network whose nodes are the divisors of n, with an edge from a to b if and only if b/a is a prime factor of n, in which case the capacity of the edge is b/a. The source of the network is 1 and the sink is n.

Alois P. Heinz, Table of n, a(n) for n = 2..10000
Wikipedia, Maximum flow problem
Wikipedia, Max-flow min-cut theorem

with(combinat):
a:= proc(n) local S, s, f, l, m;
      f:= infinity; l:= ifactors(n)[2]; m:= nops(l);
      S:= subsets({$1..m}):
      while not S[finished] do s:= S[nextvalue]();
        if s={} then next fi:
        f:= min(f, mul(1+l[i][2], i=({$1..m} minus s))*add(l[i][1], i=s))
      od; f
    end:
seq(a(n), n=2..100);  # Alois P. Heinz, Mar 04 2014

F[n_] := F[n] = Module[{v, e, flowgraph, flow},
   v = Divisors[n];
   e = Apply[DirectedEdge,
     Select[Subsets[v, {2}], PrimeQ[Last[#]/First[#]] &], {1}];
   flowgraph =
    Graph[e, EdgeCapacity -> Map[Rule[#, (Divide @@ Reverse[#])] &, e]];
   flow = FindMaximumFlow[flowgraph, 1, n, "OptimumFlowData"];
   flow["FlowValue"]]

a(n) = min_{S:P([m])\{}} Product_{i:[m]\S} (e_i+1) * Sum_{i:S} p_i, where n = Product_{i=1..m} p_i^e_i and P([m]) is the powerset of {1,...,m}.

A145021 a(n) = number of different positive integers that can be formed from different groupings of expressions of the form n op1 n op2 n op3 n, where each of op1, op2 and op3 are addition, subtraction, multiplication or division.

Original entry on oeis.org

4, 10, 20, 25, 27, 29, 30, 31, 30, 31, 30, 31, 30, 31, 30, 31, 30, 31, 30, 31, 30, 31, 30, 31, 30, 31, 30, 31, 30, 31, 30, 31, 30, 31, 30, 31, 30, 31, 30, 31, 30, 31, 30, 31, 30, 31, 30, 31, 30, 31, 30, 31, 30, 31, 30, 31, 30, 31, 30, 31, 30, 31, 30, 31, 30

Offset: 1

Ken Levasseur, Sep 29 2008

If one uses all 4^3=64 forms of this type but no parentheses, the sequence starts 4,9,15,13,15,14... In this case 4/4/4/4=1/4/4=1/16 is not an integer (association left-to-right), whereas with parenthesis one could write (4/4)/(4/4)=1, an integer, for example. The definition need clarification in this respect. [From R. J. Mathar, Jan 22 2009]

			You can form the numbers 1, 2, 3, 4 with 4 ones; hence the first term is 4.

If k >3, a(2k-1)=30 and a(2k)=31. - Ken Levasseur, Oct 01 2008

A054736 Smallest losing position after your opponent has taken k stones in a variation of "Fibonacci Nim".

Original entry on oeis.org

4, 8, 11, 15, 21, 29, 40, 55, 76, 105, 145, 200, 276, 381, 526, 726, 1002, 1383, 1909, 2635, 3637, 5020, 6929, 9564, 13201, 18221, 25150, 34714, 47915, 66136, 91286, 126000, 173915, 240051, 331337, 457337, 631252, 871303, 1202640, 1659977, 2291229, 3162532, 4365172

Offset: 1

Ken Levasseur, Apr 22 2000

nonn

In Fibonacci Nim, the first player takes any number of stones (except all) and then each player takes no more than twice the number taken in the previous move. This sequence concerns the game where 2 is replaced by 3.

			If your opponent has just removed 1 or 2 stones from the pile leaving you with 8, then you lose. Any fewer stones after your opponent has taken 2 will be a win for you.

R. K. Guy, Fair Game: How to play impartial combinatorial games, COMAP's Mathematical Exploration Series, 1989; see p. 22.

MAXTERM=10**9
cache, oldk = [MAXTERM], 1
for nleft in range(1,MAXTERM+1):
  for k in range(1,nleft+1):
    if koldk:
    print(nleft)
    oldk=mk
# Bert Dobbelaere, Apr 07 2024

More terms from Bert Dobbelaere, Apr 07 2024

A008837 a(n) = p*(p-1)/2 for p = prime(n).

Original entry on oeis.org

1, 3, 10, 21, 55, 78, 136, 171, 253, 406, 465, 666, 820, 903, 1081, 1378, 1711, 1830, 2211, 2485, 2628, 3081, 3403, 3916, 4656, 5050, 5253, 5671, 5886, 6328, 8001, 8515, 9316, 9591, 11026, 11325, 12246, 13203, 13861, 14878, 15931, 16290, 18145, 18528, 19306

Offset: 1

N. J. A. Sloane, Ken Levasseur

Whereas A034953 is the sequence of triangular numbers with prime indices, this is the sequence of triangular numbers with numbers one less than primes for indices. - Alonso del Arte, Aug 17 2014
From Jianing Song, Apr 13 2019: (Start)
a(n) is both the number of quadratic residues and the number of nonresidues modulo prime(n)^2 that are coprime to prime(n).
For k coprime to prime(n), k^a(n) == +-1 (mod prime(n)^2). (End)

Harry J. Smith, Table of n, a(n) for n = 1..1000

Half the terms of A036689.
Cf. A000217 (triangular numbers), A112456 (least triangular number divisible by n-th prime). - Klaus Brockhaus, Nov 18 2008
Cf. A000010, A000040, A072230, A034953, A271780.
Column 1 of A257253. (Row 1 of A257254).

[ (k-1)*k/2 where k is NthPrime(n): n in [1..44] ]; // Klaus Brockhaus, Nov 18 2008

a:= n-> (p-> p*(p-1)/2)(ithprime(n)):
seq(a(n), n=1..65);  # Alois P. Heinz, Apr 20 2022

Table[Prime[n] * (Prime[n] - 1)/2, {n, 22}] (* Vladimir Joseph Stephan Orlovsky, Apr 29 2008 *)
Table[Binomial[Prime[n], 2], {n, 40}] (* Alonso del Arte, Aug 22 2014, based on the formula from Enrique Pérez Herrero *)
(#(#-1))/2&/@Prime[Range[50]] (* Harvey P. Dale, Oct 02 2019 *)

{ n=0; forprime (p=2, prime(1000), write("b008837.txt", n++, " ", p*(p - 1)/2) ) } \\ Harry J. Smith, Jul 25 2009

(define (A008837 n) (/ (A036689 n) 2)) ;; Antti Karttunen, May 01 2015

a(n) = binomial(prime(n), 2) = A000217(A000040(n)-1). - Enrique Pérez Herrero, Dec 10 2011
a(n) = (1/2)*A072230(A000040(n)). - L. Edson Jeffery, Apr 07 2012
a(n) = (phi(prime(n))^2 + phi(prime(n)))/2, where phi(n) is Euler's totient function, A000010. - Alonso del Arte, Aug 22 2014
a(n) = A036689(n)/2. - Antti Karttunen, May 01 2015
Product_{n>=2} (1 - 1/a(n)) = A271780. - Amiram Eldar, Nov 22 2022

Offset changed from 2 to 1 by Harry J. Smith, Jul 25 2009

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User: Ken Levasseur

A238729 Maximum flow of a number.

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A145021 a(n) = number of different positive integers that can be formed from different groupings of expressions of the form n op1 n op2 n op3 n, where each of op1, op2 and op3 are addition, subtraction, multiplication or division.

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A054736 Smallest losing position after your opponent has taken k stones in a variation of "Fibonacci Nim".

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A008837 a(n) = p*(p-1)/2 for p = prime(n).

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