A166133 -id:A166133 - cp's OEIS frontend

A256423 Consider numbers n such that A166133(n) sets a new record and also satisfies A166133(n)=A166133(n-1)^2-1; sequence gives values of A166133(n).

8, 39203, 72899, 324899, 359999, 675683, 777923, 1127843, 4536899, 6718463, 10036223, 30272003, 44916803, 54022499, 100761443, 108743183, 110249999, 212051843, 233722943, 289952783, 326163599, 388011203, 395612099, 431475983

Offset: 1

N. J. A. Sloane, Apr 08 2015

nonn

A subsequence of A256403.

The points (A256422(i), A256423(i)) form the upper envelope of A166133.

			A166133(5) = 8 = A166133(4)^2-1 = 9-1.

N. J. A. Sloane and Ray Chandler, Table of n, a(n) for n = 1..127 (first 100 terms from N. J. A. Sloane)
N. J. A. Sloane and others, "Blog" about A166133

Cf. A166133, A256422, A256403, A256404.

A256541 First differences of A166133.

Original entry on oeis.org

1, 2, -1, 5, -1, -1, -1, 7, -1, -1, -1, 7, -1, -1, -1, 8, -1, -1, -1, -1, 7, -1, -1, 47, -41, -1, -1, -1, 14, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 133, -121, -1, -1

Offset: 1

Reinhard Zumkeller, Apr 01 2015

sign

a(n) != 0; abs(a(A256543(n))) = 1.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A166133, A256543.

a256541 n = a256541_list !! (n-1)
a256541_list = zipWith (-) (tail a166133_list) a166133_list

A256557 a(n) = A166133(n)^2 - 1.

Original entry on oeis.org

0, 3, 15, 8, 63, 48, 35, 24, 143, 120, 99, 80, 255, 224, 195, 168, 440, 399, 360, 323, 288, 575, 528, 483, 4760, 783, 728, 675, 624, 1520, 1443, 1368, 1295, 1224, 1155, 1088, 1023, 960, 899, 840, 1599, 1680, 1763, 1848, 1935, 2024, 2115, 2208

Offset: 1

Bob Selcoe, Apr 01 2015

nonn

Let A166133 = B; A166133 is defined as: After b(1)=1, b(2)=2, and b(3)=4, b(n+1) is the smallest divisor of b(n)^2-1 that has not yet appeared in the sequence.

Since it is conjectured that A166133 is a permutation of the natural numbers, it is therefore conjectured that this sequence is a permutation of all numbers of the form n^2-1.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A166133, A256559.

s = {1, 2, 4}; Do[d = Divisors[Last[s]^2 - 1]; i = 1; While[i <= Length[d] && MemberQ[s, d[[i]]], i++]; s = Append[s, d[[i]]], {5000}]; t = Table[s[[k]], {k, 1, 5000}]; #^2 - 1 & /@ t; (* Michael De Vlieger, Apr 02 2015, after Hans Havermann at A166133 *)

a(n) = A166133(n+1)*A256559(n)

A256561 Indices of primes in A166133.

Original entry on oeis.org

2, 4, 8, 6, 10, 16, 21, 19, 23, 40, 38, 32, 42, 44, 48, 54, 60, 62, 68, 80, 78, 72, 91, 111, 114, 88, 86, 120, 118, 130, 137, 133, 150, 152, 168, 162, 97, 192, 188, 182, 176, 186, 160, 215, 280, 291, 122, 226, 222, 220, 240, 263, 275, 300, 245, 277, 329, 257

Offset: 1

Reinhard Zumkeller, Apr 02 2015

nonn

Reinhard Zumkeller and Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from Reinhard Zumkeller)

Cf. A166133, A256563, A255833, A000040.

a256561 = (+ 1) . fromJust . (`elemIndex` a166133_list) . a000040

a(n) = A255833(A000040(n)). - Ray Chandler, Apr 14 2015

A256563 Indices of semiprimes in A166133.

Original entry on oeis.org

3, 7, 12, 11, 15, 14, 17, 24, 29, 28, 36, 35, 34, 31, 30, 47, 50, 52, 56, 58, 59, 63, 66, 25, 77, 74, 92, 89, 102, 101, 109, 107, 106, 105, 83, 116, 128, 125, 124, 143, 142, 141, 135, 131, 147, 154, 155, 156, 103, 171, 99, 96, 95, 145, 189, 259, 178, 177

Offset: 1

Reinhard Zumkeller, Apr 02 2015

nonn

Reinhard Zumkeller and Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from Reinhard Zumkeller)

Cf. A166133, A256561, A255833, A001358.

a256563 = (+ 1) . fromJust . (`elemIndex` a166133_list) . a001358

a(n) = A255833(A001358(n)). - Ray Chandler, Apr 14 2015

A256564 Smallest prime factor of A166133(n).

Original entry on oeis.org

1, 2, 2, 3, 2, 7, 2, 5, 2, 11, 2, 3, 2, 3, 2, 13, 3, 2, 19, 2, 17, 2, 23, 2, 3, 2, 3, 2, 5, 3, 2, 37, 2, 5, 2, 3, 2, 31, 2, 29, 2, 41, 2, 43, 2, 3, 2, 47, 2, 7, 2, 3, 2, 53, 2, 5, 2, 3, 2, 59, 2, 61, 2, 3, 2, 5, 2, 67, 2, 3, 2, 79, 2, 7, 2, 3, 2, 73, 2, 71

Offset: 1

Reinhard Zumkeller, Apr 02 2015

a(n) = A020639(A166133(n)).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A166133, A020639, A244080.

Haskell
```
a256564 = a020639 . a166133
```

A256578 Largest proper divisor of A166133(n).

Original entry on oeis.org

1, 1, 2, 1, 4, 1, 3, 1, 6, 1, 5, 3, 8, 5, 7, 1, 7, 10, 1, 9, 1, 12, 1, 11, 23, 14, 9, 13, 5, 13, 19, 1, 18, 7, 17, 11, 16, 1, 15, 1, 20, 1, 21, 1, 22, 15, 23, 1, 24, 7, 25, 17, 26, 1, 27, 11, 28, 19, 29, 1, 30, 1, 31, 21, 32, 13, 33, 1, 34, 67, 40, 1, 39, 11

Offset: 1

Reinhard Zumkeller, Apr 02 2015

nonn

a(n) = A032742(A166133(n)).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A166133, A032742, A244080.

Haskell
```
a256578 = a032742 . a166133
```

A256703 Numbers m such that A166133(m+1) = A166133(m)^2 - 1.

Original entry on oeis.org

4, 292, 330, 615, 625, 744, 982, 1075, 1671, 2176, 2389, 2820, 2937, 3228, 3598, 4187, 6590, 7803, 9960, 10173, 11628, 13140, 13396, 14035, 15588, 16396, 17766, 18813, 19858, 21111, 21115, 21246, 22808, 23241, 24784, 25050, 25149, 25167, 25384

Offset: 1

Reinhard Zumkeller, Apr 08 2015

nonn

Ray Chandler, Table of n, a(n) for n = 1..6575

Cf. A166133.

import Data.List (findIndices)
a256703 n = a256703_list !! (n-1)
a256703_list = map (+ 1) $ findIndices (\(u, v) -> v == u^2-1) $
                           zip a166133_list (tail a166133_list)

A256542 Number of divisors of A166133(n).

Original entry on oeis.org

1, 2, 3, 2, 4, 2, 4, 2, 6, 2, 4, 3, 5, 4, 4, 2, 4, 6, 2, 6, 2, 8, 2, 4, 4, 6, 4, 4, 3, 4, 4, 2, 9, 4, 4, 4, 6, 2, 8, 2, 8, 2, 8, 2, 6, 6, 4, 2, 10, 3, 6, 4, 6, 2, 8, 4, 8, 4, 4, 2, 12, 2, 4, 6, 7, 4, 8, 2, 6, 4, 10, 2, 8, 4, 6, 6, 4, 2, 12, 2, 8, 4, 4, 8, 8

Offset: 1

Reinhard Zumkeller, Apr 03 2015

nonn

a(n) = A000005(A166133(n)).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A166133, A032742, A244080.

Haskell
```
a256542 = a000005 . a166133
```

A256559 a(n) = A256557(n)/A166133(n+1), n>=3.

Original entry on oeis.org

5, 1, 9, 8, 7, 2, 13, 12, 11, 5, 17, 16, 15, 8, 22, 21, 20, 19, 12, 25, 24, 7, 170, 29, 28, 27, 16, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 21, 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, 66, 23, 505, 81, 80, 79, 78, 77, 76, 75, 74, 73

Offset: 3

Bob Selcoe, Apr 01 2015

nonn

Let A166133 = B; A166133 is defined as: After b(1)=1, b(2)=2, and b(3)=4, b(n+1) is the smallest divisor of b(n)^2-1 that has not yet appeared in the sequence.

A256557(n) = A166133(n)^2-1. Therefore, a(n) = (A166133(n)^2-1)/A166133(n+1), n>=3; that is, a(n) is A256557(n) divided by the smallest divisor of A166133(n+1)^2-1 which has not yet appeared in A166133. For example, a(12) = 5 means that 5 is A256557(12) = A166133(12)^2-1 = 80 divided its smallest divisor which has not yet appeared in A166133 (i.e., 16).

			a(13) = 17 because A256557(13)/A166133(14) = 255/15 = 17.

Cf. A166133, A256557.

lim = 200; s = {1, 2, 4}; Do[d = Divisors[Last[s]^2 - 1]; i = 1; While[i <= Length[d] && MemberQ[s, d[[i]]], i++]; s = Append[s, d[[i]]], {lim}]; a166133 = Table[s[[k]], {k, 1, lim}]; a256557 = #^2 - 1 & /@ a166133; t = PadLeft[Most@a256557, lim]; Drop[t/a166133, 3] (* Michael De Vlieger, Apr 02 2015, after Hans Havermann at A166133 *)

cp's OEIS Frontend

A256423 Consider numbers n such that A166133(n) sets a new record and also satisfies A166133(n)=A166133(n-1)^2-1; sequence gives values of A166133(n).

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A256541 First differences of A166133.

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A256557 a(n) = A166133(n)^2 - 1.

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Mathematica

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A256561 Indices of primes in A166133.

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Haskell

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A256563 Indices of semiprimes in A166133.

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A256564 Smallest prime factor of A166133(n).

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A256578 Largest proper divisor of A166133(n).

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A256703 Numbers m such that A166133(m+1) = A166133(m)^2 - 1.

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Haskell

A256542 Number of divisors of A166133(n).

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