A094519 -id:A094519 - cp's OEIS frontend

A337315 Number of divisor pairs, (d1,d2), of A094519(n) such that (d1+d2) | A094519(n) and d1 < d2.

1, 3, 2, 1, 5, 4, 6, 2, 3, 7, 3, 1, 12, 2, 1, 11, 2, 3, 9, 9, 9, 2, 2, 9, 1, 2, 2, 21, 7, 7, 2, 4, 16, 7, 7, 4, 4, 17, 2, 26, 1, 2, 11, 5, 4, 6, 14, 17, 4, 2, 3, 6, 4, 31, 2, 20, 2, 2, 13, 14, 1, 6, 9, 2, 1, 21, 5, 21, 5, 1, 12, 2, 5, 12, 11, 25, 2, 5, 6, 2, 2, 47, 2, 2, 6, 11, 3, 13

Offset: 1

Wesley Ivan Hurt, Aug 22 2020

nonn

a(n) >= 1.

			a(2) = 3; A094519(2) = 12 has divisors {1,2,3,4,6,12}. There are 3 divisor pairs, (d1,d2) such that (d1+d2) | 12 and where d1 < d2: (1,2), (1,3) and (2,4); so a(2) = 3.
a(3) = 2; A094519(3) = 18 has divisors {1,2,3,6,9,18}. There are 2 divisor pairs, (d1,d2) such that (d1+d2) | 18 and where d1 < d2: (1,2) and (3,6); so a(3) = 2.
a(4) = 1; A094519(4) = 20 has divisors {1,2,4,5,10,20}. There is 1 divisor pair, (d1,d2) such that (d1+d2) | 20 and where d1 < d2: (1,4). So a(4) = 1.
a(5) = 5; A094519(5) = 24 has divisors {1,2,3,4,6,8,12,24}. There are 5 divisor pairs, (d1,d2) such that (d1+d2) | 24 and where d1 < d2: (1,2), (1,3), (2,4), (2,6) and (4,8). So a(5) = 5.

Cf. A094519.

A094518 Number of pairs (x,y) of divisors of n with x

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 5, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 6, 0, 0, 0, 2, 0, 3, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 3, 0, 1, 0, 0, 0, 12, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 11, 0, 0, 0, 0, 0, 2, 0, 3, 0, 0, 0, 9, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 9, 0, 0, 0, 2, 0, 2

Offset: 1

Reinhard Zumkeller, May 06 2004

nonn

a(A094519(n)) > 0, a(A094520(n)) = 0.

a(A097370(n)) = n and a(m) <> n for m < A097370(n).

			n=30 with divisor set {1,2,3,5,6,10,15,30}: a(30)=4, as 1<2 & 3=1+2, 1<5 & 6=1+5, 2<3 & 5=2+3 and 5<10 & 15=5+10.

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

Cf. A091009.

Table[Count[Subsets[#, {2}], ?(Mod[n, Total@ #] == 0 &)] &@ Divisors@ n, {n, 102}] (* _Michael De Vlieger, Sep 10 2018 *)

A094518(n) = if(1==n,0,my(d=divisors(n),c=0); for(i=1,(#d-1),for(j=(i+1),#d,if(!(n%(d[i]+d[j])),c++))); (c)); \\ Antti Karttunen, Sep 10 2018

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.

A096472 Numbers containing squares of Pythagorean triples in their divisor set.

Original entry on oeis.org

3600, 7200, 10800, 14400, 18000, 21600, 25200, 28800, 32400, 36000, 39600, 43200, 46800, 50400, 54000, 57600, 61200, 64800, 68400, 72000, 75600, 79200, 82800, 86400, 90000, 93600, 97200, 100800, 104400, 108000, 111600, 115200, 118800, 122400, 126000, 129600, 133200

Offset: 1

Reinhard Zumkeller, Aug 13 2004

a(n) = m * (A046083(k)*A046084(k)*A009000(k))^2 for appropriate, not necessarily unique m and k.

			5^2 + 12^2 = 13^2: 5^2, 12^2 and 13^2 are divisors of 608400 = (13*5*3*2^2)^2, therefore 608400 is a term.

Paolo Xausa, Table of n, a(n) for n = 1..10000
Tanya Khovanova, Recursive Sequences.
Eric Weisstein's World of Mathematics, Pythagorean Triple.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. Pythagorean triples: A046083, A046084, A009000.

Cf. A044102, A094519, A169823.

Range[50]*3600 (* Paolo Xausa, Jul 01 2025 *)

my(x='x+O('x^38)); Vec(3600*x/(1-x)^2) \\ Elmo R. Oliveira, Jun 30 2025

a(n) = n*60^2.
From Elmo R. Oliveira, Jun 30 2025: (Start)
G.f.: 3600*x/(1-x)^2.
E.g.f.: 3600*x*exp(x).
a(n) = 60*A169823(n) = 100*A044102(n).
a(n) = 2*a(n-1) - a(n-2) for n > 2. (End)

Name clarified by Tanya Khovanova, Jul 05 2021

More terms from Elmo R. Oliveira, Jun 30 2025

A323640 Numbers m having at least one pair (x,y) of divisors with x

Original entry on oeis.org

6, 20, 56, 70, 110, 182, 272, 286, 308, 506, 646, 650, 812, 884, 992, 1150, 1406, 1672, 1748, 1798, 1892, 2162, 2756, 2990, 3422, 3526, 3782, 4030, 4466, 4556, 4606, 4930, 5402, 5510, 5704, 6032, 6068, 6806, 7198, 7310, 7378, 7832, 7904, 8084, 8170, 8246, 8584, 8710

Offset: 1

David A. Corneth, Aug 31 2019

nonn

Primitive terms of A094519.
From Bernard Schott, Aug 31 2019: (Start)
Some subsequences (this list is not exhaustive):
1) Oblong numbers of the form (3*k+1)*(3*k+2). These are in A001504 and the pair (x,y) = (1,3*k+1). Only 6 is oblong and not of this form. The first few terms are 20, 56, 110, 182, 272, ...
2) Numbers of the form 2*p*q  where (p, q) is a twin prime pair. These terms are precisely A071142 \ {30} and the pair (x,y) = (2,p). The first few terms are 70, 286, 646, ...
3) Numbers of the form 2^2 * p * q where p and q = p+4 are primes and p > 3. These primes p are in A023200 \ {3} and the pair (x,y) = (4,p). The first few terms are 308, 884, ...
4) More generally, numbers of the form 2^k * p * q where p and q = p+2^k are primes and the pair (x,y) = (2^k,p). For k = 3, the smallest such term is 1672 with p = 11. (End)

			56 is in the sequence as 1, 7 and 1 + 7 = 8 are divisors of 56 and no divisor of 56 is in the sequence.

David A. Corneth, Table of n, a(n) for n = 1..11106 (terms <= 5*10^7)

Cf. A094519.

filter:= proc(n) local D,i,j,nD;
  D:= numtheory:-divisors(n);
  nD:= nops(D);
  for i from 1 to nD-1 do
    for j from i+1 to nD do
      if (n/(D[i]+D[j]))::integer then return true fi
  od od;
  false
end proc:
N:= 10000: # for terms <= N
C:= Vector(N):
R:= NULL:
for i from 1 to N do
  if C[i]=0 and filter(i) then
    R:= R, i;
    C[[seq(i*j,j=2..N/i)]]:= 1
  fi
od:
R; # Robert Israel, Sep 02 2019

upto(n) = {my(charprim = vector(n, i, 1), res = List()); for(i = 1, n, if(charprim[i] == 1, if(isA094519(i), listput(res, i); for(k = 2, n \ i, charprim[i*k] = 0 ) , charprim[i] = 0; ) ) ); res }
isA094519(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 }

cp's OEIS Frontend

A337315 Number of divisor pairs, (d1,d2), of A094519(n) such that (d1+d2) | A094519(n) and d1 < d2.

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A094518 Number of pairs (x,y) of divisors of n with x

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

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A096472 Numbers containing squares of Pythagorean triples in their divisor set.

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A323640 Numbers m having at least one pair (x,y) of divisors with x

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