J. Lowell - cp's OEIS frontend

A374667 Triangle T(n,k) (1 <= k <= n) read by rows: T(n,k) = c_n * F(k)/F(k+2) where c_n = LCM of F(3), F(4), ... F(n+2) (and F() are the Fibonacci numbers).

1, 3, 2, 15, 10, 12, 60, 40, 48, 45, 780, 520, 624, 585, 600, 5460, 3640, 4368, 4095, 4200, 4160, 92820, 61880, 74256, 69615, 71400, 70720, 70980, 1021020, 680680, 816816, 765765, 785400, 777920, 780780, 779688, 90870780, 60580520, 72696624, 68153085, 69900600, 69234880, 69489420, 69392232, 69429360

Offset: 1

J. Lowell, Jul 15 2024

			Triangle begins:
      1;
      3,     2;
     15,    10,    12;
     60,    40,    48,    45;
    780,   520,   624,   585,   600;
   5460,  3640,  4368,  4095,  4200,  4160;
  92820, 61880, 74256, 69615, 71400, 70720, 70980;
  ...
Fifth row is 780, 520, 624, 585, 600. These are 1/2, 1/3, 2/5, 3/8, 5/13 of c_5 = 1560.

Cf. A000045, A035105.

row(n)={my(m=lcm(vector(n,k,fibonacci(k+2)))); vector(n, k, fibonacci(k)*m/fibonacci(k+2))}

T(n,k) = A035105(n+2) * A000045(k) / A000045(k+2).

A369562 Smallest positive n-digit number divisible by 7.

Original entry on oeis.org

7, 14, 105, 1001, 10003, 100002, 1000006, 10000004, 100000005, 1000000001, 10000000003, 100000000002, 1000000000006, 10000000000004, 100000000000005, 1000000000000001, 10000000000000003, 100000000000000002, 1000000000000000006, 10000000000000000004, 100000000000000000005

Offset: 1

J. Lowell, Jan 25 2024

The only semiprime terms are a(2) = 14 and a(n) such that (10^(n-1) + 3)/7 is a prime. - Jon E. Schoenfield, Jan 27 2024

			a(3) = 105 = 7*15.

Index entries for linear recurrences with constant coefficients, signature (11,-10,-1,11,-10).

Cf. A033940, A241217, A369613.

a[n_] := 10^(n - 1) + {6, 4, 5, 1, 3, 2}[[Mod[n, 6, 1]]]; Array[a, 30]
(* or *)
LinearRecurrence[{11, -10, -1, 11, -10}, {7, 14, 105, 1001, 10003, 100002}, 30] (* Amiram Eldar, Jan 27 2024 *)
Table[10^n+7-PowerMod[10,n,7],{n,0,20}] (* Harvey P. Dale, Jan 13 2025 *)

a(n) = (floor(10^(n-1)/7) + 1)*7.
a(n) = 10^(n-1) + A033940(n+2). - Amiram Eldar, Jan 27 2024
G.f.: 7*x*(1 - 9*x + 3*x^2 - x^3 - 3*x^4)/((1 - x)*(1 + x)*(1 - 10*x)*(1 - x + x^2)). - Stefano Spezia, Jan 28 2024

A367420 Numbers k with the property that if a Fibonacci number f is divisible by k then f is also divisible by 2*k.

Original entry on oeis.org

4, 6, 9, 12, 14, 17, 18, 19, 20, 22, 23, 24, 27, 28, 30, 31, 36, 38, 42, 44, 45, 46, 51, 52, 53, 54, 56, 57, 58, 60, 61, 62, 63, 66, 68, 69, 70, 72, 76, 78, 79, 81, 82, 83, 84, 85, 86, 90, 92, 93, 94, 95, 98, 99, 100, 102, 107, 108, 109, 110, 112, 114, 115, 116, 117

Offset: 1

J. Lowell, Nov 17 2023

nonn

If k is a term then so is b*k for any odd b.

			4 is in the sequence as any Fibonacci number divisible by 4 is divisible by 2*4.

Robin Visser, Table of n, a(n) for n = 1..10000

Cf. A000045, A001177.

def is_A367420(k):
    a, b = 1, 1
    while((a,b)!=(0,1)):
        if (a==k): return False
        a, b = b, (a+b)%(2*k)
    return True
print([k for k in range(1, 1000) if is_A367420(k)])  # Robin Visser, Jan 08 2024

More terms from David A. Corneth, Nov 17 2023

A365965 Numbers k such that A139315(k) = 0 but k is not in A138511.

Original entry on oeis.org

30, 50, 68, 76, 90, 92, 98, 116, 124, 132, 148, 150, 154, 160, 164, 165, 172, 174, 182

Offset: 1

J. Lowell, Sep 23 2023

			30 is not in A138511, but A139315(30)=0.

Cf. A139315, A138511, A337709.

A363882 Take 2 copies of Pascal's triangle. One copy has one inch between the terms of each row and the other copy has two inches between the terms of each row. Put one on top of the other so that the 1's at the very top of each copy coincide. Sequence is a triangle giving the differences between the overlapping terms.

Original entry on oeis.org

0, 1, 1, 0, 2, 0, 1, 3, 3, 1, 0, 4, 4, 4, 0, 1, 5, 10, 10, 5, 1, 0, 6, 12, 20, 12, 6, 0, 1, 7, 21, 35, 35, 21, 7, 1, 0, 8, 24, 56, 64, 56, 24, 8, 0, 1, 9, 36, 84, 126, 126, 84, 36, 9, 1, 0, 10, 40, 120, 200, 252, 200, 120, 40, 10, 0, 1, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1

Offset: 0

J. Lowell, Jun 25 2023

			Row n=8 is formed by taking row 8 of Pascal's triangle (1, 8, 28, 56, 70, 56, 28, 8, 1) and subtracting row 4 (1, 4, 6, 4, 1) spaced 2 apart. The numbers that overlap are 1, 28, 70, 28, 1 over 1, 4, 6, 4, 1, from which 1-1=0, 28-4=24, 70-6=64, 28-4=24, and 1-1=0. Thus, row 8 of the present triangle is 0, 8, 24, 56, 64, 56, 24, 8, 0.
Subtracting:
  1;                1;               0,
  1, 1;                              1, 1;
  1, 2, 1;       -  1,    1;      =  0, 2, 0;
  1, 3, 3, 1;                        1, 3, 3, 1;
  1, 4, 6, 4, 1;    1,    2,    1;   0, 4, 4, 4, 0;
  ...
Resulting triangle begins:
      k=0  1  2  3  4
  n=0:  0;
  n=1:  1, 1;
  n=2:  0, 2, 0;
  n=3:  1, 3, 3, 1;
  n=4:  0, 4, 4, 4, 0;
  ...

Cf. A007318.

T(n,k) = binomial(n,k) - [n mod 2 = k mod 2 = 0] * binomial(n/2,k/2).

A363299 a(n) is the sum of the n-th powers of the terms of row 4 of Pascal's triangle.

Original entry on oeis.org

5, 16, 70, 346, 1810, 9826, 54850, 312706, 1810690, 10601986, 62563330, 371185666, 2210336770, 13194911746, 78901035010, 472332468226, 2829699842050, 16961019183106, 101697395621890, 609909495824386, 3658357463318530, 21945746733400066, 131656888214355970, 789870960541958146

Offset: 0

J. Lowell, May 26 2023

			a(2) = 1^2 + 4^2 + 6^2 + 4^2 + 1^2 = 1 + 16 + 36 + 16 + 1 = 70.

Index entries for linear recurrences with constant coefficients, signature (11,-34,24).

Cf. A007318.

Cf. A000012 (row 0), A007395 (row 1), A052548 (row 2), A115099 (row 3).

Table[6^n + 2*(4^n + 1), {n, 0, 24}] (* Amiram Eldar, May 27 2023 *)

def A363299(n): return 2+(((1<Chai Wah Wu, Jun 27 2023

a(n) = 2 + 2*4^n + 6^n.
From Natalia L. Skirrow, Jun 25 2023: (Start)
G.f.: (5-39*x+64*x^2)/((1-x)*(1-4*x)*(1-6*x)).
E.g.f.: 2*e^x + 2*e^(4*x) + e^(6*x).
(End)

A357582 a(n) = A061300(n+1)/A061300(n).

Original entry on oeis.org

1, 2, 6, 30, 154, 1105, 4788, 20677, 216931, 858925, 7105392, 5546059, 2018025900, 1480452337, 3238556831, 107972737, 18425956230000, 4683032671, 14053747110612300, 160436746661, 33809725025123, 15260431896321667, 1583855315457687090000

Offset: 0

J. Lowell, Oct 04 2022

It is not known if the ratios A061300(n+1)/A061300(n) are always integer, but so far (for the listed terms) they are. - Max Alekseyev, Sep 05 2023

			a(5) = 1105 as A061300(5+1) / A061300(5) = 61261200 / 55440 = 1105.

Max Alekseyev, Table of n, a(n) for n = 0..29

Cf. A061300.

A061300[n_Integer?NonNegative] := A061300[n] = Module[{fact = n!, num = 1}, Monitor[While[Length@Divisors@num != fact, num++]; num, {n, num}]]; a[n_] := A061300[n + 1]/A061300[n]; Table[a[n], {n, 0, 4}] (* Robert P. P. McKone, Sep 07 2023 *)

a(11)-a(21) from David A. Corneth, Oct 05 2022

a(22)-a(29) from Max Alekseyev, Sep 05 2023

A357249 a(n) = A139315(n)*n.

Original entry on oeis.org

2, 6, 24, 60, 360, 840, 10080, 7560, 0, 27720, 332640, 720720, 0, 10810800, 17297280, 36756720, 1102701600, 698377680, 27935107200, 48886437600, 0, 16062686640, 385504479360, 1204701498000, 0, 20238985166400, 4497552259200, 6987268688400, 0, 216605329340400

Offset: 2

J. Lowell, Sep 19 2022

nonn

			a(8) = A139315(8)*8 = 1260*8 = 10080.

Cf. A129902, A139315.

a(n) = A129902(A139315(n)).

More terms from Michel Marcus, Sep 20 2022

A356078 Highly composite numbers that are multiples of their number of divisors.

Original entry on oeis.org

1, 2, 12, 24, 36, 60, 180, 240, 360, 720, 1260, 1680, 5040, 10080, 15120, 20160, 25200, 55440, 110880, 221760, 277200, 665280, 720720, 1441440, 2882880, 3603600, 8648640, 14414400, 32432400, 43243200, 61261200, 245044800, 551350800, 735134400, 1102701600, 2205403200

Offset: 1

J. Lowell, Jul 25 2022

nonn

Conjecture: this sequence is finite. The prime factorization of the number of divisors of a large highly composite number usually has a high power of 2; this is not true of the number of divisors of a highly composite number itself.

			180 (in A002182) has 18 divisors, and 180/18 equals 10; so, 180 is in this sequence.

Intersection of A002182 and A033950.

Cf. A002183.

More terms from Amiram Eldar, Jul 25 2022

A355286 Highly composite numbers that are not a product of two highly composite numbers greater than 1.

Original entry on oeis.org

1, 2, 6, 60, 180, 840, 1260, 25200, 27720, 83160, 277200, 720720, 1081080, 3603600, 10810800, 32432400, 36756720, 61261200, 110270160, 183783600, 551350800, 698377680, 2095133040, 2327925600, 3491888400, 10475665200, 48886437600, 64250746560, 73329656400, 80313433200

Offset: 1

J. Lowell, Jun 26 2022

nonn

Cf. A002182, A307763.

More terms from Amiram Eldar, Jun 27 2022

cp's OEIS Frontend

User: J. Lowell

A374667 Triangle T(n,k) (1 <= k <= n) read by rows: T(n,k) = c_n * F(k)/F(k+2) where c_n = LCM of F(3), F(4), ... F(n+2) (and F() are the Fibonacci numbers).

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A369562 Smallest positive n-digit number divisible by 7.

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A367420 Numbers k with the property that if a Fibonacci number f is divisible by k then f is also divisible by 2*k.

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A365965 Numbers k such that A139315(k) = 0 but k is not in A138511.

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A363299 a(n) is the sum of the n-th powers of the terms of row 4 of Pascal's triangle.

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A357582 a(n) = A061300(n+1)/A061300(n).

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A357249 a(n) = A139315(n)*n.

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A356078 Highly composite numbers that are multiples of their number of divisors.

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