A044102 -id:A044102 - cp's OEIS frontend

A383488 Numbers k that have at least one divisor d_i(k) for which a divisor d_j(k) exists such that d_i(k) < d_j(k) < sigma(d_i(k)).

12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96, 99, 100, 102, 104, 105, 108, 110, 112, 114, 120, 126, 130, 132, 138, 140, 144, 150, 154, 156, 160, 162, 168, 170, 174, 176, 180, 186, 189, 192, 196, 198, 200, 204, 208, 210, 216

Offset: 1

Felix Huber, May 03 2025

nonn

Numbers k (without multiplicity) that are multiples of lcm(c,i), where c is any composite and i is any integer from [c + 1, sigma(c) - 1].

			All multiples of 12 (A008594) are terms because 12 has the divisors 4 and 6 where sigma(4) = 7 > 6.
All multiples of 18 (A008600) are terms because 18 has the divisors 6 and 9 where sigma(6) = 12 > 9.
All multiples of 20 (A008602) are terms because 20 has the divisors 4 and 5 where sigma(4) = 7 > 5.

Felix Huber, Table of n, a(n) for n = 1..10000

Supersequence of A008594, A008600, A008602, A008606, A044102, A135628.

Cf. A000203, A002808, A027750, A109042, A383360, A383489.

with(NumberTheory):
A383488:=proc(n)
    option remember;
    local k,i,L;
    if n=1 then
        12
    else
        for k from procname(n-1)+1 do
            L:=Divisors(k);
            for i to nops(L)-1 do
                if sigma(L[i])>L[i+1] then
                    return k
                fi
            od
        od
    fi;
end proc;
seq(A383488(n),n=1..57);

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

A158062 a(n) = 36n^2 - 2n.

Original entry on oeis.org

34, 140, 318, 568, 890, 1284, 1750, 2288, 2898, 3580, 4334, 5160, 6058, 7028, 8070, 9184, 10370, 11628, 12958, 14360, 15834, 17380, 18998, 20688, 22450, 24284, 26190, 28168, 30218, 32340, 34534, 36800, 39138, 41548, 44030, 46584, 49210, 51908

Offset: 1

Vincenzo Librandi, Mar 12 2009

The identity (36*n - 1)^2 - (36*n^2 - 2*n)*6^2 = 1 can be written as (A044102(n+1) - 1)^2 - a(n)*6^2 = 1. - Vincenzo Librandi, Feb 11 2012

The continued fraction expansion of sqrt(a(n)) is [6n-1; {1, 4, 1, 12n-2}]. - Magus K. Chu, Nov 08 2022

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
E. J. Barbeau, Polynomial Excursions, Chapter 10: Diophantine equations (2010), pages 84-85 (row 15 in the first table at p. 85, case d(t) = t*(6^2*t-2)).
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A044102.

Magma
```
[36*n^2 - 2*n: n in [1..50]]
```

LinearRecurrence[{3, -3, 1}, {34, 140, 318}, 50] (* Vincenzo Librandi, Feb 11 2012 *)

for(n=1, 50, print1(36*n^2 - 2*n ", ")); \\ Vincenzo Librandi, Feb 11 2012

G.f.: x*(-34 - 38*x)/(x-1)^3. - Vincenzo Librandi, Feb 11 2012

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Vincenzo Librandi, Feb 11 2012

A214394 If n mod 6 = 0 then n/6 else n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 1, 7, 8, 9, 10, 11, 2, 13, 14, 15, 16, 17, 3, 19, 20, 21, 22, 23, 4, 25, 26, 27, 28, 29, 5, 31, 32, 33, 34, 35, 6, 37, 38, 39, 40, 41, 7, 43, 44, 45, 46, 47, 8, 49, 50, 51, 52, 53, 9, 55, 56, 57, 58, 59, 10, 61

Offset: 0

Jeremy Gardiner, Jul 15 2012

			a(36) = 36/6 = 6.

Jeremy Gardiner, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,2,0,0,0,0,0,-1).

Cf. A026741, A051176, A214392.

Cf. A044102, A244414.

Table[If[Mod[n, 6] == 0, n/6, n], {n, 0, 50}] (* G. C. Greubel, Oct 26 2017 *)

a(n)=if(n%6,n,n/6) \\ G. C. Greubel, Oct 26 2017

first(n) = my(res = vector(n, i, i-1)); forstep(i = 1, n, 6, res[i] \= 6); res \\ David A. Corneth, Oct 28 2017

a(n) = 2*a(n-6) - a(n-12). - G. C. Greubel, Oct 26 2017 [corrected by Georg Fischer, Mar 12 2020]
a(n) = A244414(n) when n is not a multiple of 36 (A044102). - Michel Marcus, Oct 28 2017
a(n) = floor(n/6) + sign(n mod 6) * (n - floor(n/6)). - Wesley Ivan Hurt, Oct 28 2017

A242570 a(n) = 252 * n.

Original entry on oeis.org

0, 252, 504, 756, 1008, 1260, 1512, 1764, 2016, 2268, 2520, 2772, 3024, 3276, 3528, 3780, 4032, 4284, 4536, 4788, 5040, 5292, 5544, 5796, 6048, 6300, 6552, 6804, 7056, 7308, 7560, 7812, 8064, 8316, 8568, 8820, 9072, 9324, 9576, 9828, 10080, 10332, 10584, 10836, 11088, 11340

Offset: 0

Derek Orr, May 17 2014

As lcm(1,2,3,...,9) = 2520, 10*a(n) + k is divisible by each k from 1 through 9.

Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A001477, A008600, A008603, A008589, A008591, A008594, A008596.

Cf. A044102, A135628.

252*Range[0, 49] (* Alonso del Arte, May 17 2014 *)
LinearRecurrence[{2,-1},{0,252},50] (* Harvey P. Dale, Mar 25 2025 *)

PARI
```
for(n=0,50,print(252*n))
```

From Elmo R. Oliveira, Apr 16 2024: (Start)
G.f.: 252*x/(x-1)^2.
E.g.f.: 252*x*exp(x).
a(n) = 2*a(n-1) - a(n-2) for n >= 2.
a(n) = 7*A044102(n) = 9*A135628(n) = 12*A008603(n) = 14*A008600(n) = 18*A008596(n) = 21*A008594(n) = 28*A008591(n) = 36*A008589(n) = 252*A001477(n). (End)

A363436 Array read by ascending antidiagonals: A(n, k) = k*n^2, with k >= 0.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 4, 2, 0, 0, 9, 8, 3, 0, 0, 16, 18, 12, 4, 0, 0, 25, 32, 27, 16, 5, 0, 0, 36, 50, 48, 36, 20, 6, 0, 0, 49, 72, 75, 64, 45, 24, 7, 0, 0, 64, 98, 108, 100, 80, 54, 28, 8, 0, 0, 81, 128, 147, 144, 125, 96, 63, 32, 9, 0, 0, 100, 162, 192, 196, 180, 150, 112, 72, 36, 10, 0

Offset: 0

Stefano Spezia, Jul 08 2023

			The array begins:
  0,  0,  0,   0,   0,   0,   0, ...
  0,  1,  2,   3,   4,   5,   6, ...
  0,  4,  8,  12,  16,  20,  24, ...
  0,  9, 18,  27,  36,  45,  54, ...
  0, 16, 32,  48,  64,  80,  96, ...
  0, 25, 50,  75, 100, 125, 150, ...
  0, 36, 72, 108, 144, 180, 216, ...
  ...

Cf. A000290 (k = 1), A001105 (k = 2), A033428 (k = 3), A016742 (k = 4), A033429 (k = 5), A033581 (k = 6), A033582 (k = 7), A139098 (k = 8), A016766 (k = 9), A033583 (k = 10), A033584 (k = 11), A135453 (k = 12),  A152742 (k = 13), A144555 (k = 14), A064761 (k = 15), A016802 (k = 16), A244630 (k = 17), A195321 (k = 18), A244631 (k = 19), A195322 (k = 20), A064762 (k = 21), A195323 (k = 22), A244632 (k = 23), A195824 (k = 24), A016850 (k = 25), A244633 (k = 26), A244634 (k = 27), A064763 (k = 28), A244635 (k = 29), A244636 (k = 30).
Cf. A001477 (n = 1), A008586 (n = 2), A008591 (n = 3), A008598 (n = 4), A008607 (n = 5), A044102 (n = 6), A152691 (n = 8).
Cf. A000007 (n = 0 or k = 0), A000578 (main diagonal), A002415 (antidiagonal sums), A004247.

A[n_,k_]:=k n^2; Table[A[n-k,k],{n,0,11},{k,0,n}]//Flatten

O.g.f.: x*y*(1 + x)/((1 - x)^3*(1 - y)^2).
E.g.f.: x*y*(1 + x)*exp(x + y).
A(n, k) = n*A004247(n, k).

cp's OEIS Frontend

A383488 Numbers k that have at least one divisor d_i(k) for which a divisor d_j(k) exists such that d_i(k) < d_j(k) < sigma(d_i(k)).

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

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A158062 a(n) = 36*n^2 - 2*n.

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Magma

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A214394 If n mod 6 = 0 then n/6 else n.

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A242570 a(n) = 252 * n.

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A363436 Array read by ascending antidiagonals: A(n, k) = k*n^2, with k >= 0.

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Programs

Mathematica

Formula

A158062 a(n) = 36n^2 - 2n.