Leroy Quet - cp's OEIS frontend

A175872 Write n in binary. Consider the 0's and 1 as a list. () If the new list consists entirely of 1's, then a(n) = the number of 1's, and end. Otherwise, construct a new list made up of the lengths, written in order, of the runs of similarly-valued terms from the previous list. Go to .

1, 2, 2, 2, 3, 2, 3, 2, 3, 4, 2, 1, 2, 2, 4, 2, 3, 2, 2, 2, 5, 2, 2, 2, 2, 2, 3, 2, 2, 2, 5, 2, 3, 2, 3, 4, 2, 3, 3, 2, 2, 6, 2, 1, 3, 2, 2, 2, 3, 1, 1, 3, 2, 4, 3, 1, 3, 2, 3, 2, 2, 2, 6, 2, 3, 2, 3, 4, 2, 4, 2, 4, 5, 2, 2, 2, 2, 4, 3, 2, 2, 3, 2, 2, 7, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 2, 3, 2, 3, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2

Offset: 1

Leroy Quet, Oct 03 2010

n appears for the first time at A000975(n). - Sean A. Irvine and N. J. A. Sloane, Dec 02 2010

A217921(n) = number of steps needed to calculate a(n). - Reinhard Zumkeller, Mar 26 2013

			100 (decimal) in binary is 1100100. The lengths of the runs are: 2,2,1,2. The lengths of the runs in the latest list are: 2,1,1. The lengths of the runs in the latest list are: 1,2. The lengths of the runs in the latest list are: 1,1. This last list consists entirely of 1's. There are two 1's, so a(100) = 2.

Sean A. Irvine, Table of n, a(n) for n = 1..100000

Cf. A101211, A000975.

Cf. A030308.

import Data.List (group, genericLength)
a175872 = f . a030308_row where
   f xs | all (== 1) xs = length xs
        | otherwise     = f $ map genericLength $ group xs
-- Reinhard Zumkeller, Mar 26 2013

f[n_Integer] := IntegerDigits[n, 2]; f[nn:{1..}] := nn; f[nn_List] := Length /@ Split[nn]; a[n_] := FixedPoint[f, n] // Length; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Nov 26 2013 *)

a(3) corrected by Leroy Quet, Oct 06 2010

More terms from Sean A. Irvine, Dec 02 2010

A180388 Number of permutations of 1..n with number of rises (p(i+1)>p(i)) different from number of rises in the inverse permutation.

Original entry on oeis.org

0, 0, 0, 0, 2, 24, 228, 2088, 19732, 197890, 2131060, 24729108, 309027560, 4148115048, 59611031304, 913915160628, 14897443573244, 257369478424890, 4698494631208500, 90393611677083892, 1828153614233855024, 38778554791649355232, 860921156414834534368

Offset: 0

Leroy Quet, D. S. McNeil and R. H. Hardin in the Sequence Fans Mailing List Sep 01 2010

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..200

a(n) + A180389(n) = n! = A000142(n).

More terms from Vaclav Kotesovec, Jun 10 2015

a(0)=0 prepended by Alois P. Heinz, Jun 10 2015

A180389 Number of permutations of 1..n with number of rises (p(i+1)>p(i)) the same as number of rises in the inverse permutation.

Original entry on oeis.org

1, 1, 2, 6, 22, 96, 492, 2952, 20588, 164990, 1497740, 15187692, 169974040, 2078905752, 27567259896, 393759207372, 6025346314756, 98317949671110, 1703879074519500, 31251488731748108, 604748393942784976, 12312387380060084768, 263079571362773145632

Offset: 0

Leroy Quet, D. S. McNeil and R. H. Hardin in the Sequence Fans Mailing List Sep 01 2010

nonn

Also equals sum of squares of the coefficients of the (numerators of) the G.F. for the count of monomials in the Schur polynomials of degree n (all partitions of weight n), in function of the number of variables v. - Wouter Meeussen, Dec 27 2010

Studied by Carlitz, Roselle, and Scoville in 'Permutations and Sequences with Repetitions by Number of Increases'. They refer to a rise/ascent as a 'jump', and consider the first entry of a permutation to always be a jump, so #jumps=#rises+1. Similarly, the number of rises in the inverse permutation/number of inverse descents corresponds to what they call the 'number of readings', and follow a convention so that #rises in inverse permutation+1=#readings. Formula can be attained by R_m(k,r), setting t=k, and summing k from 1 to m+1. - Kevin Dilks, Jun 09 2015

			For n=4, a(4)=22 are all permutations of length 4 except for 3142 (which has only one ascent, and two inverse ascents) and 2413 (which has two ascents, and only one inverse ascent). - _Kevin Dilks_, Jun 09 2015

Vaclav Kotesovec, Table of n, a(n) for n = 0..200
L. Carlitz et al., Permutations and sequences with repetitions by number of increases, J. Combin. Theory, 1 (1966), 350-374.
Wouter Meeussen, Schur Polynomials
Wouter Meeussen, Some notes on the link between A180389 and Schur polynomials

A180388(n) + a(n) = n! = A000142(n).

seq(add(add(add((-1)^(i+j)*binomial(n+1,i)*binomial(n+1,j)*binomial((m-i)*(m-j)+n-1,n),i=0..m),j=0..m),m=0..n+1), n=0..30); # Kevin Dilks, Jun 09 2015

Table[Sum[Sum[Sum[(-1)^(i+j)*Binomial[n+1,i]*Binomial[n+1,j]*Binomial[(m-i)*(m-j)+n-1,n],{i,0,m}],{j,0,m}],{m,0,n+1}],{n,0,10}] (* Kevin Dilks, Jun 09 2015 *)

a(n) = Sum_{m=0..n+1} Sum_{i=0..m} Sum_{j=0..m} (-1)^{i+j} binomial(n+1,i) binomial(n+1,j) binomial((m-i)*(m-j)+n-1,n). - Kevin Dilks, Jun 09 2015

a(n) ~ sqrt(3) * n! / sqrt(Pi*n). - Vaclav Kotesovec, Jun 10 2015

a(15)-a(20) from Wouter Meeussen, Dec 27 2010

a(0)=1 prepended by Alois P. Heinz, Jun 10 2015

A175494 a(n) = floor(n^(1/d(n))), where d(n) = number of divisors of n.

Original entry on oeis.org

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

Offset: 1

Leroy Quet, May 30 2010

nonn

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A034884, A175495.

Table[Floor[n^(1/DivisorSigma[0, n])], {n, 100}] (* T. D. Noe, May 14 2013 *)

More terms from R. J. Mathar, May 31 2010

A175500 a(1) = 1. a(n) = the smallest integer not yet occurring such that if d(a(n)) = d(a(k)), then d(a(n-1)) doesn't equal d(a(k-1)) for any k where 2<= k <= n-1, where d(m) = the number of divisors of m.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 12, 8, 9, 10, 14, 16, 11, 24, 13, 36, 15, 18, 17, 48, 19, 60, 20, 25, 28, 30, 21, 40, 32, 44, 64, 22, 72, 23, 81, 26, 80, 27, 100, 29, 120, 31, 144, 33, 168, 34, 180, 35, 192, 37, 240, 38, 324, 41, 252, 42, 49, 54, 56, 84, 39, 336, 43

Offset: 1

Leroy Quet, May 31 2010

This sequence is a permutation of the positive integers.

The derived sequence 2^d(a(n))*3^d(a(n+1)), where d(m) = the number of divisors of m, contains only distinct terms. - Paul Tek, Mar 05 2014

Paul Tek, Table of n, a(n) for n = 1..2473
Paul Tek, C++ program for this sequence

Cf. A175501, A175502

ok(j, va, vs, n) = {if (vecsearch(vs, j), return (0)); for (k=1, n-1, if ((numdiv(j) == numdiv(va[k])) && (numdiv(va[k-1]) == numdiv(va[n-1])), return (0));); 1;}
findnew(va, vs, n) = {my(j = 1); my(vs = vecsort(va)); until (ok(j, va, vs, n), j++); j;}
lista(nn) = {my(va = [1]); for (n=2, nn, vs = vecsort(va); newa = findnew(va, vs, n); va = concat(va, newa);); va;} \\ Michel Marcus, May 04 2016

a(26)-a(64) from Paul Tek, Mar 05 2014

A175502 a(1) = 1. a(n) = the smallest integer not yet occurring such that each unordered pair {d(a(k)),d(a(k-1))} occurs only once at most, for 2<= k <= n, where d(m) = the number of divisors of m.

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 12, 8, 10, 16, 7, 24, 9, 18, 20, 30, 14, 36, 11, 48, 15, 60, 13, 64, 21, 120, 17, 144, 22, 180, 19, 192, 25, 49, 72, 28, 80, 40, 42, 81, 32, 100, 54, 84, 90, 112, 121, 168, 44, 240, 23, 360, 26, 320, 45, 252, 56, 210, 96, 196

Offset: 1

Leroy Quet, May 31 2010

nonn

This sequence seems likely to be a permutation of the positive integers. Is it?

Ivan Neretin, Table of n, a(n) for n = 1..1000

Cf. A175500, A175503

More terms from Sean A. Irvine, Jun 14 2011

A175503 a(n) = the number of divisors of A175502(n).

Original entry on oeis.org

1, 2, 2, 3, 4, 2, 6, 4, 4, 5, 2, 8, 3, 6, 6, 8, 4, 9, 2, 10, 4, 12, 2, 7, 4, 16, 2, 15, 4, 18, 2, 14, 3, 3, 12, 6, 10, 8, 8, 5, 6, 9, 8, 12, 12, 10, 3, 16, 6, 20, 2, 24, 4, 14, 6, 18, 8, 16, 12, 9, 10, 10, 16, 9, 3, 18, 12, 15, 6, 24, 8, 20, 4, 21, 2, 30, 4, 32, 2, 27

Offset: 1

Leroy Quet, May 31 2010

nonn

Each unordered pair {a(k),a(k-1)} occurs at most once in the sequence.

Ivan Neretin, Table of n, a(n) for n = 1..1000

Cf. A175502

More terms from Ivan Neretin, Jun 05 2016

A175501 a(n) = the number of divisors of A175500(n).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 6, 4, 3, 4, 4, 5, 2, 8, 2, 9, 4, 6, 2, 10, 2, 12, 6, 3, 6, 8, 4, 8, 6, 6, 7, 4, 12, 2, 5, 4, 10, 4, 9, 2, 16, 2, 15, 4, 16, 4, 18, 4, 14, 2, 20, 4, 15, 2, 18, 8, 3, 8, 8, 12, 4, 20, 2, 14, 6, 12, 8, 10, 6, 10, 8, 9, 6, 16, 6, 9, 8, 16, 8

Offset: 1

Leroy Quet, May 31 2010

nonn

Each ordered pair (a(k),a(k-1)) occurs at most once in the sequence.

Paul Tek, Table of n, a(n) for n = 1..2473

Cf. A175500.

Terms a(26) and beyond from b-file by Andrew Howroyd, Feb 05 2018

A175479 a(1)=1. For n >= 1, if a(n) = 0, then append to the sequence the digits of binary n+1 (most significant digits first and least significant digits last). If a(n) = 1, then append to the sequence the digits of binary n+1 in reverse order (least significant digits first and most significant digits last).

Original entry on oeis.org

1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0

Offset: 1

Leroy Quet, May 25 2010

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A175480.

a=[1]; for (n=1, 84, print1 (a[n] ", "); if (#a<84, a=concat(a, if (a[n], Vecrev(binary(n+1)), binary(n+1))))) \\ Rémy Sigrist, Nov 08 2018

Definition fixed by Leroy Quet, May 28 2010

A175480 a(1)=1. For n >= 1, if a(n) = 1, then append to the sequence the digits of binary n+1 (most significant digits first and least significant digits last). If a(n) = 0, then append to the sequence the digits of binary n+1 in reverse order (least significant digits first and most significant digits last).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0

Offset: 1

Leroy Quet, May 25 2010

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A175479.

a=[1]; for (n=1, 84, print1 (a[n] ", "); if (#a<84, a=concat(a, if (a[n], binary(n+1), Vecrev(binary(n+1)))))) \\ Rémy Sigrist, Nov 08 2018

Definition fixed by Leroy Quet, May 28 2010

cp's OEIS Frontend

User: Leroy Quet

A175872 Write n in binary. Consider the 0's and 1 as a list. (*) If the new list consists entirely of 1's, then a(n) = the number of 1's, and end. Otherwise, construct a new list made up of the lengths, written in order, of the runs of similarly-valued terms from the previous list. Go to *.

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A180388 Number of permutations of 1..n with number of rises (p(i+1)>p(i)) different from number of rises in the inverse permutation.

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A180389 Number of permutations of 1..n with number of rises (p(i+1)>p(i)) the same as number of rises in the inverse permutation.

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A175494 a(n) = floor(n^(1/d(n))), where d(n) = number of divisors of n.

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A175500 a(1) = 1. a(n) = the smallest integer not yet occurring such that if d(a(n)) = d(a(k)), then d(a(n-1)) doesn't equal d(a(k-1)) for any k where 2<= k <= n-1, where d(m) = the number of divisors of m.

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A175502 a(1) = 1. a(n) = the smallest integer not yet occurring such that each unordered pair {d(a(k)),d(a(k-1))} occurs only once at most, for 2<= k <= n, where d(m) = the number of divisors of m.

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A175503 a(n) = the number of divisors of A175502(n).

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A175501 a(n) = the number of divisors of A175500(n).

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A175872 Write n in binary. Consider the 0's and 1 as a list. () If the new list consists entirely of 1's, then a(n) = the number of 1's, and end. Otherwise, construct a new list made up of the lengths, written in order, of the runs of similarly-valued terms from the previous list. Go to .