A094518 -id:A094518 - cp's OEIS frontend

A091009 Number of triples (u,v,w) of divisors of n with v-u = w-v, and u < v < w.

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 3, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 0, 5, 0, 0, 0, 1, 0, 4, 0, 0, 0, 0, 0, 6, 0, 0, 0, 1, 0, 2, 0, 0, 3, 0, 0, 7, 0, 0, 0, 0, 0, 3, 0, 2, 0, 0, 0, 11, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 10, 0, 0, 2, 0, 0, 2, 0, 2, 0, 0, 0, 9, 0, 0, 0, 0, 0, 10, 1, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0

Offset: 1

Reinhard Zumkeller, Dec 13 2003

nonn

a(A091014(n))=n and a(m)<>n for m<=A091014(n);
a(A091010(n))=0; a(A091011(n))>0; a(A091012(n))=1; a(A091013(n))>1.
Number of pairs (x,y) of divisors of n with xAntti Karttunen, Sep 10 2018

			a(30)=4, as there are exactly 4 triples of divisors with the defining property: (1,2,3), (1,3,5), (2,6,10) and (5,10,15).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. also A094518.

Array[Count[Subsets[#, {3}], ?(#2 - #1 == #3 - #2 & @@ # &)] &@ Divisors@ # &, 105] (* _Michael De Vlieger, Sep 10 2018 *)

A091009(n) = if(1==n,0,my(d=divisors(n),c=0); for(i=1,(#d-1),for(j=(i+1),#d,if(!(n%(d[j]+(d[j]-d[i]))),c++))); (c)); \\ Antti Karttunen, Sep 10 2018

Definition clarified by Antti Karttunen, Sep 10 2018

A094519 Numbers having at least one pair (x,y) of divisors with x

Original entry on oeis.org

6, 12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 90, 96, 100, 102, 108, 110, 112, 114, 120, 126, 132, 138, 140, 144, 150, 156, 160, 162, 168, 174, 180, 182, 186, 192, 198, 200, 204, 210, 216, 220, 222, 224, 228, 234, 240, 246

Offset: 1

Reinhard Zumkeller, May 06 2004

nonn

If m is in the sequence then so is k*m for k > 0. Furthermore, all terms are even. - David A. Corneth, Aug 31 2019
If (x,y) = (1,m) with m > 1, then oblong numbers m*(m+1) >= 6 belong to this sequence, and each oblong number >= 6 is a primitive term of the subsequence {k*m*(m+1), k >= 1}. Examples: with pair (1,2), we get multiples of 6 (see A008588); with (1,3) we get multiples of 12 (see A008594); with (1,4) we get multiples of 20 (see A008602); with (1,7) we get multiples of 56. - Bernard Schott, Aug 31 2019
The numbers of terms that do not exceed 10^k, for k = 1, 2, ..., are 1, 22, 230, 2317, 23201, 232209, 2322920, 23232166, 232332309, 2323370184, ... . Apparently, the asymptotic density of this sequence exists and equals 0.23233... . - Amiram Eldar, Apr 20 2025

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A094518.
Complement of A094520.
A superset of A088723. - R. J. Mathar, Sep 16 2007
Subsequences: A002378 \ {0, 2}, A008588 \ {0}, A008602 \ {0}.

aQ[n_] := AnyTrue[Total /@ Subsets[Divisors[n], {2}], Divisible[n, #] &]; Select[Range[250], aQ] (* Amiram Eldar, Aug 31 2019 *)

is(n) = {my(d = divisors(n)); for(i = 1, #d - 2, for(j = i + 1, #d - 1, if(n % (d[i] + d[j]) == 0, return(1) ) ) ); 0 } \\ David A. Corneth, Aug 31 2019

from itertools import count, islice
from sympy import divisors
def A094519_gen(): # generator of terms
    for n in count(1):
        for i in range(1,len(d:=divisors(n))):
            di = d[i]
            for j in range(i):
                if n % (di+d[j]) == 0:
                    yield n
                    break
            else:
                continue
            break
A094519_list = list(islice(A094519_gen(),20)) # Chai Wah Wu, Dec 26 2021

A094518(a(n)) > 0.

A094520 Numbers such that all sums of two distinct divisors are not divisors.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 64, 65, 67, 68, 69, 71, 73, 74, 75, 76, 77, 79, 81, 82, 83, 85, 86, 87, 88, 89, 91

Offset: 1

Reinhard Zumkeller, May 06 2004

nonn

The numbers of terms that do not exceed 10^k, for k = 1, 2, ..., are 9, 78, 770, 7683, 76799, 767791, 7677080, 76767834, 767667691, 7676629816, ... . Apparently, the asymptotic density of this sequence exists and equals 0.76766... . - Amiram Eldar, Apr 20 2025

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Sum-Free Set.

Cf. A094518.

Complement of A094519.

aQ[n_] := AllTrue[Total /@ Subsets[Divisors[n], {2}], ! Divisible[n, #] &]; Select[Range[91], aQ] (* Amiram Eldar, Aug 31 2019 *)

isok(k) = {my(d = divisors(k)); for(i = 1, #d, for(j = 1, i-1, if(!(k % (d[i] + d[j])), return(0)))); 1;} \\ Amiram Eldar, Apr 20 2025

A094518(a(n)) = 0.

A097370 Smallest number such that there exist exactly n triples (x,y,z) of distinct divisors with z=x+y.

Original entry on oeis.org

1, 6, 18, 12, 30, 24, 36, 48, 462, 84, 390, 72, 60, 264, 210, 468, 144, 168, 624, 528, 252, 120, 1530, 648, 1248, 336, 180, 1140, 936, 1176, 780, 240, 630, 660, 1296, 4212, 2304, 3432, 504, 2730, 420, 480, 3672, 1872, 2268, 2592, 900, 360, 2184, 2040, 1848, 960

Offset: 1

Reinhard Zumkeller, Aug 10 2004

nonn

A094518(a(n)) = n and A094518(m) <> n for m < a(n).

cp's OEIS Frontend

A091009 Number of triples (u,v,w) of divisors of n with v-u = w-v, and u < v < w.

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A094519 Numbers having at least one pair (x,y) of divisors with x

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A094520 Numbers such that all sums of two distinct divisors are not divisors.

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A097370 Smallest number such that there exist exactly n triples (x,y,z) of distinct divisors with z=x+y.

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