A025060 -id:A025060 - cp's OEIS frontend

A025062 Positions of even numbers in A025060 (numbers of form jk + ki + i*j, where 1 <= i < j < k).

2, 5, 7, 11, 12, 14, 15, 19, 20, 23, 25, 27, 29, 32, 34, 36, 38, 42, 44, 47, 49, 51, 52, 55, 57, 58, 60, 62, 64, 67, 68, 70, 72, 75, 77, 79, 82, 84, 86, 88, 91, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 123, 126, 128, 130, 132, 133, 135, 137, 139, 141

Offset: 1

Clark Kimberling

nonn

Cf. A025060, A025063.

a(46)-a(56), a(58)-a(61), a(63)-a(67) corrected by Gionata Neri, Sep 06 2016

A025063 Positions of odd numbers in A025060 (numbers of form jk + ki + i*j, where 1 <= i < j < k).

Original entry on oeis.org

1, 3, 4, 6, 8, 9, 10, 13, 16, 17, 18, 21, 22, 24, 26, 28, 30, 31, 33, 35, 37, 39, 40, 41, 43, 45, 46, 48, 50, 53, 54, 56, 59, 61, 63, 65, 66, 69, 71, 73, 74, 76, 78, 80, 81, 83, 85, 87, 89, 90, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 124, 125, 127, 129, 131

Offset: 1

Clark Kimberling

nonn

Cf. A025060, A025062.

a(55)-a(70) corrected by Gionata Neri, Sep 06 2016

A025061 Positions of primes in A025060 (numbers of form jk + ki + i*j, where 1 <=i < j < k).

Original entry on oeis.org

1, 3, 4, 6, 9, 10, 17, 18, 21, 26, 30, 31, 37, 40, 41, 46, 50, 54, 61, 65, 66, 69, 71, 74, 87, 90, 95, 97, 107, 109, 115, 121, 124, 129, 134, 136, 145, 147, 151, 153, 164, 176, 180, 182, 185, 191, 192, 202, 207, 213, 219, 221, 226, 229, 231, 241, 255, 259, 260, 264, 277, 283, 292, 294

Offset: 1

Clark Kimberling

nonn

More terms and a(29)-a(48) corrected by Gionata Neri, Sep 06 2016

A025064 Position of numbers of form 3n^2 in A025060 (numbers of form jk + ki + ij, where 1 <=i < j < k).

Original entry on oeis.org

8, 43, 70, 105, 146, 194, 248, 307, 374, 448, 528, 615, 707, 805, 910, 1021, 1138, 1260, 1388, 1523, 1664, 1810, 1963, 2122, 2287, 2458, 2635, 2818, 3007, 3202, 3403, 3610, 3823, 4042, 4267, 4498, 4735, 4978

Offset: 1

Clark Kimberling

nonn

N:= 10000: # to get positions of all 3*n^2 <= N
B:= sort(convert({seq(seq(seq(i*j + j*k + i*k, i=1..min(j-1, (N-j*k)/(j+k))),j=2..min(k-1,(N-k)/(1+k))),k=3..(N-2)/3)},list)):
count:= 1:
for n from 1 to floor(sqrt(N/3)) do
  if member(3*n^2,B,A[count]) then count:= count+1 fi
od:
seq(A[i],i=1..count-1); # Robert Israel, Sep 06 2016

It is conjectured that A000926 ends at 1848, in which case a(n) = 3*n^2+18*n-38 for all n >= 22. - Robert Israel, Sep 06 2016

More terms and a(4)-a(7) corrected by Gionata Neri, Sep 06 2016

A093669 Numbers having a unique representation as ab+ac+bc, with 0 < a < b < c.

Original entry on oeis.org

11, 14, 17, 19, 20, 27, 32, 34, 36, 43, 46, 49, 52, 64, 67, 73, 82, 97, 100, 142, 148, 163, 193

Offset: 1

T. D. Noe, Apr 08 2004

Are there more terms?

No more terms < 10^6. - Joerg Arndt, Oct 01 2017

			11 is on the list because 11 = 1*2+1*3+2*3.

See A025052.

Cf. A025052, A025058, A025060.

Cf. A000926 (numbers not of the form ab+ac+bc, 0

oneSol={}; Do[lim=Ceiling[(n-2)/3]; cnt=0; Do[If[n>a*b && Mod[n-a*b, a+b]==0 && Quotient[n-a*b, a+b]>b, cnt++; If[cnt>1, Break[]]], {a, 1, lim-1}, {b, a+1, lim}]; If[cnt==1, AppendTo[oneSol, n]], {n, 10000}]; oneSol

from collections import Counter
def aupto(N):
    acount = Counter()
    for i in range(1, N-1):
        for j in range(i+1, N//i + 1):
            p, s = i*j, i+j
            for k in range(j+1, (N-p)//s + 1):
                acount.update([p + s*k])
    return sorted([k for k in acount if acount[k] == 1])
print(aupto(10**5)) # Michael S. Branicky, Nov 14 2021

A025058 Numbers of form ij + jk + k*i, where 1 <=i < j < k, including repetitions.

Original entry on oeis.org

11, 14, 17, 19, 20, 23, 23, 26, 26, 27, 29, 29, 31, 31, 32, 34, 35, 35, 36, 38, 38, 39, 39, 41, 41, 41, 43, 44, 44, 44, 46, 47, 47, 47, 47, 49, 50, 50, 51, 51, 52, 53, 53, 54, 54, 55, 55, 56, 56, 56, 59, 59, 59, 59, 59, 61, 61, 62, 62, 62, 63, 63, 64, 65, 65

Offset: 1

Clark Kimberling

nonn

			11 is in the sequence because 11 = 1*2 + 2*3 + 3*1.
23 appears twice because 23 = 1*3 + 3*5 + 5*1 = 1*2 + 2*7 + 7*1.

Bo Gyu Jeong, Table of n, a(n) for n = 1..2000

Cf. A025060 (without repetitions), A093669.

def aupto(N):
    alst = []
    for i in range(1, N-1):
        for j in range(i+1, N//i + 1):
            p, s = i*j, i+j
            for k in range(j+1, (N-p)//s + 1):
                alst.append(p + s*k)
    return sorted(alst)
print(aupto(65)) # Michael S. Branicky, Nov 14 2021

Showing 1-6 of 6 results.

cp's OEIS Frontend

A025062 Positions of even numbers in A025060 (numbers of form j*k + k*i + i*j, where 1 <= i < j < k).

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A025063 Positions of odd numbers in A025060 (numbers of form j*k + k*i + i*j, where 1 <= i < j < k).

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Extensions

A025061 Positions of primes in A025060 (numbers of form j*k + k*i + i*j, where 1 <=i < j < k).

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A025064 Position of numbers of form 3*n^2 in A025060 (numbers of form j*k + k*i + i*j, where 1 <=i < j < k).

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Maple

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A093669 Numbers having a unique representation as ab+ac+bc, with 0 < a < b < c.

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Mathematica

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A025058 Numbers of form i*j + j*k + k*i, where 1 <=i < j < k, including repetitions.

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Python

A025062 Positions of even numbers in A025060 (numbers of form jk + ki + i*j, where 1 <= i < j < k).

A025063 Positions of odd numbers in A025060 (numbers of form jk + ki + i*j, where 1 <= i < j < k).

A025061 Positions of primes in A025060 (numbers of form jk + ki + i*j, where 1 <=i < j < k).

A025064 Position of numbers of form 3n^2 in A025060 (numbers of form jk + ki + ij, where 1 <=i < j < k).

A025058 Numbers of form ij + jk + k*i, where 1 <=i < j < k, including repetitions.