A060275 -id:A060275 - cp's OEIS frontend

A060277 Number of m for which a+b+c = n; abc = m has at least two distinct solutions (a,b,c) with 1 <= a <= b <= c.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 1, 1, 0, 1, 1, 3, 1, 1, 1, 1, 3, 2, 7, 3, 2, 5, 4, 3, 5, 9, 2, 5, 6, 9, 5, 9, 14, 9, 7, 5, 10, 10, 11, 18, 7, 11, 16, 14, 12, 12, 23, 19, 13, 18, 11, 20, 19, 32, 17, 21, 18, 25, 19, 21, 27, 22, 21, 31, 27, 24, 28, 42, 34, 33, 21, 28, 31, 35, 47

Offset: 1

Naohiro Nomoto, Mar 23 2001

nonn

A triple (a,b,c) as described in the name cannot have c prime. - David A. Corneth, Aug 01 2018

			(14 = 6+6+2 = 8+3+3, 72 = 6*6*2 = 8*3*3); (14 = 8+5+1 = 10+2+2, 40 = 8*5*1 = 10*2*2); 14 has two "m" variables. so a(14)=2.

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1500 terms from Hugo Pfoertner)
IBM Research, Ponder This Challenge - July 2018.
John B. Kelly, Partitions with equal products, Proc. Amer. Math. Soc. 15 (1964), 987-990
John B. Kelly, Partitions with equal products. II, Proc. Amer. Math. Soc. 107 (1989), 887-893

Cf. A001399, A060275, A069905, A103277, A103278, A317388, A317578.

a[n_] := Count[ Tally[ Times @@@ IntegerPartitions[n, {3}]], {m_,c_} /; c>1]; Array[a, 84] (* Giovanni Resta, Jul 27 2018 *)

a(n)={my(M=Map()); for(i=n\3, n, for(j=(n-i+1)\2, min(n-1-i, i), my(k=n-i-j); my(m=i*j*k); my(z); mapput(M, m, if(mapisdefined(M, m, &z), z + 1, 1)))); #select(z->z>=2, if(#M, Mat(M)[, 2], []))} \\ Andrew Howroyd, Jul 27 2018

a(n) = Sum_{k>=2} A317578(n,k). - Alois P. Heinz, Aug 01 2018

Description revised by David W. Wilson and Don Reble, Jun 04 2002

A060292 At least two unordered triples of positive numbers have product n and equal sums.

Original entry on oeis.org

36, 40, 72, 90, 96, 126, 144, 168, 176, 200, 225, 234, 240, 252, 270, 280, 288, 297, 320, 360, 396, 408, 420, 432, 450, 480, 504, 520, 540, 546, 550, 560, 576, 588, 600, 630, 648, 672, 675, 690, 714, 720, 735, 736, 768, 780, 784, 800, 816, 840, 850, 855

Offset: 1

Naohiro Nomoto, Mar 24 2001

			36=6*6*1=9*2*2. 6+6+1=9+2+2. so 36 is in the sequence.

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 160 terms from Carmine Suriano)

Cf. A060275.

N:= 1000: # to get all entries <= N
for i from 1 to N do R[i]:= {} od:
A:= {}:
for a from 1 to floor(N^(1/3)) do
  for b from a to floor((N/a)^(1/2)) do
    for c from b to floor(N/(a*b)) do
       p:= a*b*c;
       s:= a+b+c;
       if member(s,R[p]) then A:= A union {p}
       else R[p]:= R[p] union {s}
       fi;
od od od:
A;
# if using Maple 11 or earlier, uncomment the next line
# sort(convert(A,list)); # Robert Israel, Feb 09 2015
# second Maple program:
b:= proc(n, k, t) option remember; expand(`if`(t=0, `if`(kk, 0, b(n/d, d, t-1)*x^d), d=numtheory[divisors](n))))
    end:
a:= proc(n) option remember; local k; for k from 1+
      `if`(n=1, 0, a(n-1)) while max(coeffs(b(k$2, 2)))<2 do od; k
    end:
seq(a(n), n=1..50);  # Alois P. Heinz, May 16 2020

b[n_, k_, t_] := b[n, k, t] = Expand[If[t == 0, If[k < n, 0, x^n], Sum[If[d > k, 0, b[n/d, d, t - 1] x^d], {d, Divisors[n]}]]];
a[n_] := a[n] = Module[{k}, For[k = 1 + If[n == 1, 0, a[n - 1]], Max[ CoefficientList[b[k, k, 2], x]] < 2, k++]; k];
Array[a, 52] (* Jean-François Alcover, May 30 2020, after Alois P. Heinz *)

Name changed by Robert Israel, Feb 09 2015

cp's OEIS Frontend

A060277 Number of m for which a+b+c = n; abc = m has at least two distinct solutions (a,b,c) with 1 <= a <= b <= c.

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