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A193101 Minimal number of numbers of the form (m^3+5m)/6 (see A004006) needed to sum to n. 3
1, 2, 1, 2, 3, 2, 1, 2, 3, 2, 3, 4, 3, 1, 2, 3, 2, 3, 4, 3, 2, 3, 4, 3, 1, 2, 3, 2, 3, 4, 3, 2, 3, 4, 3, 4, 5, 4, 2, 3, 1, 2, 3, 2, 3, 3, 3, 2, 3, 2, 3, 4, 3, 4, 2, 3, 3, 3, 4, 4, 4, 3, 1, 2, 3, 2, 3, 4, 3, 2, 3, 4, 3, 4, 3, 4, 2, 3, 4, 3, 4, 2, 3, 3, 3, 4, 4, 2, 3, 4, 3, 1, 2, 3, 2, 3, 4, 3, 2, 3, 4, 3, 4, 2, 3, 2, 3, 4, 3, 4, 3, 4, 3, 4, 5, 4, 2, 3, 4, 3 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Watson showed that a(n) <= 8 for all n.

It is conjectured that a(n) <= 5 for all n.

REFERENCES

Salzer, H. E. and Levine, N., Proof that every integer <= 452,479,659 is a sum of five numbers of the form Q_x = (x^3+5x)/6, x>= 0, Math. Comp., (1968), 191-192.

G. L. Watson, Sums of eight values of a cubic polynomial, J. London Math. Soc., 27 (1952), 217-224.

LINKS

Table of n, a(n) for n=1..120.

N. J. A. Sloane, Transforms

MAPLE

# LAGRANGE transform of a sequence {a(n)}

# Suggested by Lagrange's theorem that at most 4 squares are needed to sum to n.

# Returns b(n) = minimal number of terms of {a} needed to sum to n for 1 <= n <= M.

# C = maximal number of terms of {a} to try to build n

# M = upper limit on n

# Internally, the initial terms of both a and b are taken to be 0, but since this is a number-theoretic function, the output starts at n=1

LAGRANGE:=proc(a, C, M)

local t1, ip, i, j, a1, a2, b, c, N1, N2, Nc;

if whattype(a) <> list then RETURN([]); fi:

# sort a, remove duplicates, include 0

t1:=sort(a);

a1:=sort(convert(convert(a, set), list));

if not member(0, a1) then a1:=[0, op(a1)]; fi;

N1:=nops(a1);

b:=Array(1..M+1, -1);

for i from 1 to N1 while a1[i]<=M do b[a1[i]+1]:=1; od;

a2:=a1; N2:=N1;

for ip from 2 to C do

c:={}:

   for i from 1 to N1 while a1[i] <= M do

      for j from 1 to N2 while a1[i]+a2[j] <= M do

c:={op(c), a1[i]+a2[j]};

                                                od;

                                       od;

c:=sort(convert(c, list));

Nc:=nops(c);

   for i from 1 to Nc do

      if b[c[i]+1] = -1 then b[c[i]+1]:= ip; fi;

                      od;

a2:=c; N2:=Nc;

                   od;

[seq(b[i], i=2..M+1)];

end;

Q:=[seq((m^3+5*m)/6, m=0..20)];

LAGRANGE(Q, 8, 120);

CROSSREFS

Cf. A004006. A002828, A104246, A193105.

Sequence in context: A066856 A089280 A192099 * A100661 A088696 A004738

Adjacent sequences:  A193098 A193099 A193100 * A193102 A193103 A193104

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Jul 15 2011

STATUS

approved

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Last modified April 25 04:22 EDT 2014. Contains 240994 sequences.