cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

User: Paul Boddington

Paul Boddington's wiki page.

Paul Boddington has authored 67 sequences. Here are the ten most recent ones:

A255312 Positive integers m for which the m-th row of A088643 in reverse agrees with the first m terms of A132075.

Original entry on oeis.org

1, 2, 3, 4, 26, 27, 35, 36, 37, 47, 153, 206, 254, 255, 267, 326, 334, 543, 544, 550, 573, 590, 604, 719, 720, 965, 1327, 1340, 1353, 1354, 1414, 1423, 1453, 1474, 1579, 1589, 1598, 1762, 1856, 2105, 2840, 2921, 2922, 2928
Offset: 1

Views

Author

Paul Boddington, Feb 20 2015

Keywords

Comments

I do not know if this sequence is infinite. Given any term it is easy to compute all earlier terms. The largest number I know of in the sequence is 21977.

Links

Crossrefs

Cf. A088643, A132075.

A255309 Number of times log_2 can be applied to n until the result is either 1 or not a power of 2. Here log_2 means the base-2 logarithm.

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1
Offset: 0

Views

Author

Paul Boddington, Feb 20 2015

Keywords

Links

Crossrefs

Cf. A000079 (2^n), A007814 (2-adic valuation of n), A209229, A255308.

Programs

  • PARI
    nbi(n) = {my(nb = 0); if ((ispower(n, ,&m) && (m==2)) || (n==2), return(nbi(valuation(n, 2))+1);); nb;}
a(n) = { my(nb = 0); if ((ispower(n, ,&m) && (m==2)) || (n==2), return(nbi(valuation(n, 2))+1);); nb;} \\ Michel Marcus, Mar 11 2015; corrected Jun 13 2022
  • PARI
    A255309(n) = { my(k=0); while((n>1)&&!bitand(n,n-1),n = valuation(n,2); k++); (k); }; \\ Antti Karttunen, Sep 30 2018

Formula

a(n) = 1 + a(log_2(n)) if n is a power of 2 except 1, 0 otherwise.

Extensions

Extended up to a(128) by Antti Karttunen, Sep 30 2018

A255516 a(n) = A239839(n)/n! where A239839(n) is the number of ordered pairs of permutation functions on n elements where f(f(f(x))) = g(f(g(x))).

Original entry on oeis.org

1, 1, 2, 3, 7, 13, 29, 53, 126, 258, 602, 1250, 3031, 6703, 16367, 36698, 91731, 214315, 542225, 1284284, 3305371, 8063704, 21017382, 51961767, 137324923, 347282636, 928913378, 2379051372, 6439629060, 16796126329, 45974995358, 121341267340, 335665581851
Offset: 0

Views

Author

Paul Boddington, Feb 24 2015

Keywords

Comments

The fact that A239839(n) is a multiple of n! follows from a general result in group theory due to Solomon.

Links

Crossrefs

Cf. A239839.

Extensions

a(14)-a(32) from Hiroaki Yamanouchi, Mar 12 2015

A255310 Positive integers not the sum of iterated binary logs.

Original entry on oeis.org

2, 5, 6, 11, 20, 21, 22, 39, 72, 137, 266, 267, 524, 1037, 2062, 4111, 8208, 16401, 32786, 65555, 65556, 65557, 65558, 131095, 262168, 524313, 1048602, 2097179, 4194332, 8388637, 16777246, 33554463, 67108896
Offset: 1

Views

Author

Paul Boddington, Feb 20 2015

Keywords

Examples

			Clearly A232779 is increasing, and A232779(n) equals 1 + A232779(n - 1) unless n is a power of 2. Therefore this sequence consists of all numbers strictly between A232779(2^r - 1) and A232779(2^r) for some r. For example, A232779(15) = 15 + 3 + 1 = 19, whereas A232779(16) = 16 + 4 + 2 + 1 = 23, so this sequence includes the terms 20, 21, 22.
The sequence can also be obtained using the sequence b(n) = A255309(n).
Suppose t >= 2 is a power of 2. Let s be the sum of b(r) for r from 1 to t - 1.
Then the numbers t + s (inclusive) to t + s + b(t) (exclusive) are in this sequence, and all terms can be obtained in this way.
For example, if t = 16, then s = b(1) + b(2) + ... + b(15) = 4, and b(16) = 3, so the bounds are 16 + 4 = 20 and 16 + 4 + 3 = 23, producing the terms 20, 21, 22.

Crossrefs

Cf. A232779, A255309.

A255440 Number of ordered pairs of permutation functions on n elements satisfying f(g(g(g(x)))) = g(g(g(f(x)))).

Original entry on oeis.org

1, 1, 4, 24, 288, 3240, 74160, 1970640, 55762560, 2271628800, 118676275200, 5949399456000, 389282684313600, 31351069535385600, 2431871996666112000, 227068393053325824000, 25547764411931553792000, 2795442523571128983552000, 350437557048098924371968000
Offset: 0

Views

Author

Paul Boddington, Feb 24 2015

Keywords

Comments

See A239841 for more information and a proof that a(n) is a multiple of n!.
a(n) appears be n! * A232218(n).

Links

Crossrefs

Cf. A239841, A232218

Extensions

a(18) from Hiroaki Yamanouchi, Mar 13 2015

A255525 1/n! times the number of ordered pairs of permutation functions f,g on n elements where f(g(g(x))) = g(f(f(x))).

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 4, 4, 6, 9, 13, 17, 27, 31, 42, 57, 83, 104
Offset: 0

Views

Author

Paul Boddington, Feb 24 2015

Keywords

Comments

The fact that A239836(n) is a multiple of n! follows from a general result in group theory due to Solomon.

Links

Crossrefs

Cf. A239836.

Formula

a(n) = A239836(n)/n!.

A255515 1/n! times the number of ordered pairs of permutation functions f,g on n elements where f(f(x)) = g(f(g(x))).

Original entry on oeis.org

1, 1, 1, 2, 4, 5, 9, 17, 26, 46, 87, 137, 246, 474, 791, 1420, 2783, 4839, 8806, 17361, 31252, 57613, 114497, 211496, 395637, 791597, 1495016, 2834242, 5710510, 10986325, 21101677, 42796060, 83674582, 162695172, 332141014, 658589710, 1295551257, 2661881254
Offset: 0

Views

Author

Paul Boddington, Feb 24 2015

Keywords

Comments

The fact that A239837(n) is a multiple of n! follows from a general result in group theory due to Solomon.

Links

Crossrefs

Cf. A239837.

Formula

a(n) = A239837(n)/n!.

Extensions

a(14)-a(37) from Hiroaki Yamanouchi, Mar 12 2015

A255308 Number of times log_2 can be applied to n until the result is not a power of 2. Here log_2 means the base-2 logarithm.

Original entry on oeis.org

0, 1, 2, 0, 3, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1
Offset: 0

Views

Author

Paul Boddington, Feb 20 2015

Keywords

Comments

Multiplicative because only powers of two have a nonzero value. - Andrew Howroyd, Jul 27 2018

Links

Crossrefs

Cf. A209229, A255309.
One less than A293668.

Programs

  • PARI
    a(n) = if(n < 1, 0, my(e=valuation(n, 2)); if(n == 2^e, 1 + a(e), 0)) \\ Andrew Howroyd, Jul 27 2018

Formula

a(n) = 1 + a(log_2(n)) if n is a power of 2, 0 otherwise.

Extensions

Extended up to a(128) by Antti Karttunen, Sep 30 2018

A132163 Triangle read by rows. For row n, start with 1 but from the second term onwards always choose the largest positive integer between 1 and n inclusive that i) has not already appeared in the row ii) gives a prime when added to the previous term. Stop if no such integer can be found.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 4, 3, 2, 1, 4, 3, 2, 5, 1, 6, 5, 2, 3, 4, 1, 6, 7, 4, 3, 2, 5, 1, 6, 7, 4, 3, 8, 5, 2, 1, 6, 7, 4, 9, 8, 5, 2, 3, 1, 10, 9, 8, 5, 6, 7, 4, 3, 2, 1, 10, 9, 8, 11, 6, 7, 4, 3, 2, 5, 1, 12, 11, 8, 9, 10, 7, 6, 5, 2, 3, 4
Offset: 1

Views

Author

Paul Boddington, Nov 04 2007

Keywords

Comments

The following statements are conjectural: 1) The n-th row is always a permutation of 1,...,n. 2) For the even rows, the last term is one less than a prime (so the row gives a solution to the prime circle problem - see A051252). 3) There exists a (unique) sequence b(2), b(3),... with the property that for every n > 1 there is a positive integer N such that every even row of the triangle from the 2N-th onwards ends b(n), ..., b(3), b(2) and every odd row from the (2N - 1)-th onwards ends b(n)+(-1)^n, ..., b(3)-1, b(2)+1. (If the sequence b(n) exists it is probably A132075 without the initial term 1.)

Links

Crossrefs

This sequence is a variation on A088643.

Programs

  • Haskell
    import Data.List (delete)
a132163_tabl = map a132163_row [1..]
a132163 n k = a132163_row n !! (k-1)
a132163_row n = 1 : f 1 [n, n-1 .. 2] where
   f u vs = g vs where
     g []                            = []
     g (x:xs) | a010051 (x + u) == 1 = x : f x (delete x vs)
              | otherwise            = g xs
-- Reinhard Zumkeller, Jan 05 2013
  • Mathematica
    t[, 1] = 1; t[n, k_] := t[n, k] = For[ j = n, j > 1, j--, If[ PrimeQ[ t[n, k-1] + j] && FreeQ[ Table[ t[n, m], {m, 1, k-1}], j], Return[j] ] ]; Table[ t[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Apr 02 2013 *)

A132126 Number of nonassociative subloops of order 8n of the Cayley octonions (up to isomorphism).

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 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, 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
Offset: 1

Views

Author

Paul Boddington, Oct 31 2007

Keywords

Comments

Every nonassociative subloop of the octonions has order a multiple of 8.

Links

Crossrefs

Cf. A090750.

Programs

  • Mathematica
    ReplacePart[PadRight[{0},120,1],{6->2,12->2,30->3}] (* Harvey P. Dale, Dec 18 2018 *)
  • PARI
    A132126(n) = if(1==n,0,if((6==n)||(12==n),2,if(30==n,3,1))); \\ Antti Karttunen, Sep 27 2018

Formula

a(1) = 0, a(6) = 2, a(12) = 2, a(30) = 3, otherwise a(n) = 1.

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