A204892 -id:A204892 - cp's OEIS frontend

A205006 a(n) = s(k)-s(j), where (s(k),s(j)) is the least pair of distinct triangular numbers for which n divides their difference.

2, 2, 3, 4, 5, 12, 7, 8, 9, 20, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 56, 29, 30, 31, 32, 33, 34, 35, 72, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 132, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 156, 79, 80, 81

Offset: 1

Clark Kimberling, Jan 21 2012

nonn

For a guide to related sequences, see A204892.

Antti Karttunen, Table of n, a(n) for n = 1..23515

Cf. A000217, A205002, A205007, A204892.

Cf. A318894 (gives the positions where a(n) is not n).

Mathematica
```
(See the program at A205002.)
```

A205006(n) = for(k=2,oo,my(sk=binomial(k+1,2)); for(j=1,k-1,if(!((sk-binomial(j+1,2))%n),return(sk-binomial(j+1,2))))); \\ Antti Karttunen, Sep 27 2018

More terms from Antti Karttunen, Sep 27 2018

A205383 a(n) = (1/n)*A205382(n).

Original entry on oeis.org

8, 4, 8, 2, 8, 4, 8, 1, 8, 4, 8, 2, 8, 4, 8, 1, 8, 4, 8, 2, 8, 4, 8, 1, 8, 4, 8, 2, 8, 4, 8, 1, 8, 4, 8, 2, 8, 4, 8, 1, 8, 4, 8, 2, 8, 4, 8, 1, 8, 4, 8, 2, 8, 4, 8, 1, 8, 4, 8, 2

Offset: 1

Clark Kimberling, Jan 26 2012

nonn

Possibly a(n) = (1/n)*lcm(8,n); see A205382.

For a guide to related sequences, see A204892.

Cf. A205378, A204892.

Mathematica
```
(See the program at A205378.)
```

A205850 [s(k)-s(j)]/4, where the pairs (k,j) are given by A205847 and A205848, and s(k) denotes the (k+1)-st Fibonacci number.

Original entry on oeis.org

1, 3, 2, 5, 4, 2, 8, 13, 22, 21, 19, 17, 34, 58, 57, 55, 53, 36, 94, 93, 91, 89, 72, 36, 152, 144, 246, 233, 399, 398, 396, 394, 377, 341, 305, 644, 610, 1045, 1044, 1042, 1040, 1023, 987, 951, 646, 1691, 1690, 1688, 1686, 1669, 1633, 1597, 1292, 646

Offset: 1

Clark Kimberling, Feb 02 2012

nonn

For a guide to related sequences, see A205840.

			The first six terms match these differences:
s(4)-s(1) = 5-1 = 4 = 4*1
s(6)-s(1) = 13-1 = 12= 4*3
s(6)-s(4) = 13-5 = 8 = 4*2
s(7)-s(1) = 21-1 = 20 = 4*5
s(7)-s(4) = 21-5 = 16 = 4*4
s(7)-s(6) = 21-13 = 8 = 4*2

Cf. A204892, A205845, A205849.

s[n_] := s[n] = Fibonacci[n + 1]; z1 = 400; z2 = 60;
f[n_] := f[n] = Floor[(-1 + Sqrt[8 n - 7])/2];
Table[s[n], {n, 1, 30}]
u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
Table[u[m], {m, 1, z1}]  (* A204922 *)
v[n_, h_] := v[n, h] = If[IntegerQ[u[h]/n], h, 0]
w[n_] := w[n] = Table[v[n, h], {h, 1, z1}]
d[n_] := d[n] = Delete[w[n], Position[w[n], 0]]
c = 4; t = d[c]    (* A205846 *)
k[n_] := k[n] = Floor[(3 + Sqrt[8 t[[n]] - 1])/2]
j[n_] := j[n] = t[[n]] - f[t][[n]] (f[t[[n]]] + 1)/2
Table[k[n], {n, 1, z2}]    (* A205847 *)
Table[j[n], {n, 1, z2}]    (* A205848 *)
Table[s[k[n]] - s[j[n]], {n, 1, z2}] (* A205849 *)
Table[(s[k[n]] - s[j[n]])/c, {n, 1, z2}] (* A205850 *)

A204897 a(n) = (p(n)-q(n))/n, where (p(n), q(n)) is the least pair of primes for which n divides p(n)-q(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1

Offset: 1

Clark Kimberling, Jan 20 2012

nonn

For a guide to related sequences, see A204892.

It seems that a(A007921(n)) = 2 for all n. - Antti Karttunen, Oct 09 2018

			(3-2)/1=1
(5-3)/2=1
(5-2)/3=1
(7-3)/4=1
(7-2)/5=1
(11-5)/6=1
(17-3)/7=2

Antti Karttunen, Table of n, a(n) for n = 1..12000

Cf. A007921, A204892, A204896, A000040.

Mathematica
```
(See the program at A204892.)
```

A204897(n) = { my(d); forprime(p=3,oo, forprime(q=2,p-1,if(!((d=(p-q))%n),return(d/n),if(dAntti Karttunen, Oct 09 2018

More terms from Antti Karttunen, Oct 09 2018

A204914 Ordered differences of squared primes.

Original entry on oeis.org

5, 21, 16, 45, 40, 24, 117, 112, 96, 72, 165, 160, 144, 120, 48, 285, 280, 264, 240, 168, 120, 357, 352, 336, 312, 240, 192, 72, 525, 520, 504, 480, 408, 360, 240, 168, 837, 832, 816, 792, 720, 672, 552, 480, 312, 957, 952, 936, 912, 840, 792, 672

Offset: 1

Clark Kimberling, Jan 20 2012

nonn

For a guide to related sequences, see A204892.

			a(1) = s(2) - s(1) =  9 - 4 =  5;
a(2) = s(3) - s(1) = 25 - 4 = 21;
a(3) = s(3) - s(2) = 25 - 9 = 16;
a(4) = s(4) - s(1) = 49 - 4 = 45.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A204908, A204900, A204892.

Mathematica
```
(See the program at A204916.)
```

from math import isqrt
from sympy import prime, primerange
def aupton(terms):
  sqps = [p*p for p in primerange(1, prime(isqrt(2*terms)+1)+1)]
  return [b-a for i, b in enumerate(sqps) for a in sqps[:i]][:terms]
print(aupton(52)) # Michael S. Branicky, May 21 2021

A204925 a(n) is the index j < k such that n divides s(k) - s(j), where k is the least index (A204924) for which such j exists, and s=(1,2,3,5,8,13,...), the Fibonacci numbers.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 1, 4, 3, 3, 2, 1, 5, 6, 5, 4, 7, 3, 2, 1, 6, 1, 3, 10, 4, 5, 1, 4, 4, 8, 3, 2, 1, 7, 1, 10, 9, 6, 5, 11, 7, 6, 3, 1, 5, 3, 5, 10, 8, 4, 8, 3, 2, 1, 8, 5, 4, 1, 10, 16

Offset: 1

Clark Kimberling, Jan 21 2012

nonn

For a guide to related sequences, see A204892.

			(See the Example section at A204924.)

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A204924, A204892, A000045.

Mathematica
```
(See the program at A204924.)
```

A204928 s(k(n)) - s(j(n)), where (s(k(n)), s(j(n))) is the least pair of distinct Fibonacci numbers for which n divides s(k(n)) - s(j(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 18, 10, 11, 12, 13, 42, 225, 16, 34, 18, 19, 20, 21, 88, 230, 144, 50, 26, 54, 84, 29, 2550, 31, 32, 33, 34, 17710, 144, 555, 76, 4173, 2440, 123, 42, 86, 88, 225, 230, 47, 144, 343, 50, 2550, 52, 53, 54, 55, 2576, 228, 232, 121304

Offset: 1

Clark Kimberling, Jan 21 2012

nonn

The ordering of the pairs (s(k(n)), s(j(n))) is given by A204922. For a guide to related sequences, see A204892.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A204900, A204892, A204929.

Mathematica
```
(See the program at A204924.)
```

A204929 (s(k(n)) - s(j(n)))/n, where (s(k(n)), s(j(n))) is the least pair of distinct Fibonacci numbers for which n divides s(k(n)) - s(j(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 15, 1, 2, 1, 1, 1, 1, 4, 10, 6, 2, 1, 2, 3, 1, 85, 1, 1, 1, 1, 506, 4, 15, 2, 107, 61, 3, 1, 2, 2, 5, 5, 1, 3, 7, 1, 50, 1, 1, 1, 1, 46, 4, 4, 2056, 451

Offset: 1

Clark Kimberling, Jan 21 2012

nonn

For a guide to related sequences, see A204892.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A204924, A204892, A204928.

Mathematica
```
(See the program at A204924.)
```

A204935 The number j! such that n divides k!-j!>0, where k is the least positive integer for which such a j exists.

Original entry on oeis.org

1, 2, 6, 2, 1, 6, 1, 24, 6, 120, 2, 24, 24, 6, 120, 24, 1, 6, 6, 120, 6, 2, 1, 24, 120, 24, 720, 5040, 24, 120, 2, 24, 24, 6, 5040, 720, 720, 6, 24, 120, 120, 6, 40320, 24, 720, 24, 24, 24, 5040, 120, 6, 24, 2, 720, 720, 5040, 6, 24, 2, 120, 40320, 2, 5040

Offset: 1

Clark Kimberling, Jan 21 2012

nonn

For a guide to related sequences, see A204892.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A204932, A204892, A204933.

f:= proc(n) local t,k,V;
  t:= 1:
  for k from 1 do
    t:= t*k mod n;
    if assigned(V[t]) then return V[t]! else V[t]:= k fi
  od
end proc:
map(f, [$1..100]); # Robert Israel, Nov 10 2024

Mathematica
```
(See the program at A204932.)
```

A204988 The index j < k such that n divides 2^k - 2^j, where k is the least index (A204987) for which such j exists.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 3, 1

Offset: 1

Clark Kimberling, Jan 21 2012

For a guide to related sequences, see A204892.

			(See example at A204987.)

Antti Karttunen, Table of n, a(n) for n = 1..6556

Cf. A007814, A204892, A204987.

(See the program at A204987.)
a[n_] := Max[1, IntegerExponent[n, 2]]; Array[a, 100] (* Amiram Eldar, Oct 22 2022 *)

\\ Use the program at A204987. - Antti Karttunen, Nov 19 2017

a(n)=max(1, valuation(n,2)); \\ Andrew Howroyd, Aug 08 2018

a(n) = A007814(n) + (1-(-1)^n)/2 (conjecture). - Velin Yanev, Nov 14 2016.
From Andrew Howroyd, Aug 08 2018: (Start)
The above conjecture is true because the definition of this sequence and A204987 requires j to be at least 1 and 2^k - 2^j can be written 2^j*(2^(k-j) - 1).
a(n) = max(1, A007814(n)). (End)
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 3/2. - Amiram Eldar, Oct 22 2022

More terms from Antti Karttunen, Nov 19 2017

Keyword:mult added by Andrew Howroyd, Aug 08 2018

cp's OEIS Frontend

A205006 a(n) = s(k)-s(j), where (s(k),s(j)) is the least pair of distinct triangular numbers for which n divides their difference.

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A205383 a(n) = (1/n)*A205382(n).

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Mathematica

A205850 [s(k)-s(j)]/4, where the pairs (k,j) are given by A205847 and A205848, and s(k) denotes the (k+1)-st Fibonacci number.

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A204897 a(n) = (p(n)-q(n))/n, where (p(n), q(n)) is the least pair of primes for which n divides p(n)-q(n).

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Mathematica

PARI

Extensions

A204914 Ordered differences of squared primes.

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Python

A204925 a(n) is the index j < k such that n divides s(k) - s(j), where k is the least index (A204924) for which such j exists, and s=(1,2,3,5,8,13,...), the Fibonacci numbers.

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Mathematica

A204928 s(k(n)) - s(j(n)), where (s(k(n)), s(j(n))) is the least pair of distinct Fibonacci numbers for which n divides s(k(n)) - s(j(n)).

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Mathematica

A204929 (s(k(n)) - s(j(n)))/n, where (s(k(n)), s(j(n))) is the least pair of distinct Fibonacci numbers for which n divides s(k(n)) - s(j(n)).

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Mathematica