A286617 -id:A286617 - cp's OEIS frontend

A075157 Run lengths in the binary expansion of n gives the vector of exponents in prime factorization of a(n)+1, with the least significant run corresponding to the exponent of the least prime, 2; with one subtracted from each run length, except for the most significant run of 1's.

0, 1, 2, 3, 5, 4, 8, 7, 11, 14, 6, 9, 17, 24, 26, 15, 23, 44, 34, 29, 13, 10, 20, 19, 35, 74, 48, 49, 53, 124, 80, 31, 47, 134, 174, 89, 69, 76, 104, 59, 27, 32, 12, 21, 41, 54, 62, 39, 71, 224, 244, 149, 97, 120, 146, 99, 107, 374, 342, 249, 161, 624, 242, 63, 95, 404

Offset: 0

Antti Karttunen, Sep 13 2002

nonn

To make this a permutation of nonnegative integers, we subtract one from each run count except for the most significant run, e.g. a(11) = 9, as 11 = 1011 and 9+1 = 10 = 5^1 * 3^(1-1) * 2^(2-1).

Paul Tek (terms 0..10000) & Antti Karttunen, Table of n, a(n) for n = 0..16384
Index entries for sequences that are permutations of the natural numbers

Inverse of A075158.
Cf. A008578, A056539, A059900, A075162.
Cf. A000040, A000975, A006093, A030308, A007088, A075159, A278217, A286617.

import Data.List (group)
a075157 0 = 0
a075157 n = product (zipWith (^) a000040_list rs') - 1 where
   rs' = reverse $ r : map (subtract 1) rs
   (r:rs) = reverse $ map length $ group $ a030308_row n
-- Reinhard Zumkeller, Aug 04 2014

A005811(n) = hammingweight(bitxor(n, n>>1));  \\ This function from Gheorghe Coserea, Sep 03 2015
A286468(n) = { my(p=((n+1)%2), i=0, m=1); while(n>0, if(((n%2)==p), m *= prime(i), p = (n%2); i = i+1); n = n\2); m };
A075157(n) = if(!n,n,(prime(A005811(n))*A286468(n))-1);

(define (A075157 n) (if (zero? n) n (+ -1 (* (A000040 (A005811 n)) (fold-left (lambda (a r) (* (A003961 a) (A000079 (- r 1)))) 1 (binexp->runcount1list n))))))
(define (binexp->runcount1list n) (if (zero? n) (list) (let loop ((n n) (rc (list)) (count 0) (prev-bit (modulo n 2))) (if (zero? n) (cons count rc) (if (eq? (modulo n 2) prev-bit) (loop (floor->exact (/ n 2)) rc (1+ count) (modulo n 2)) (loop (floor->exact (/ n 2)) (cons count rc) 1 (modulo n 2)))))))
;; Or, using the code of A286468:
(define (A075157 n) (if (zero? n) n (- (* (A000040 (A005811 n)) (A286468 n)) 1)))

a(n) = A075159(n+1) - 1.
a(0) = 0; for n >= 1, a(n) = (A000040(A005811(n)) * A286468(n)) - 1.
Other identities. For all n >= 1:
a(A000975(n)) = A006093(n) = A000040(n)-1.

Entry revised, PARI-program added and the old incorrect Scheme-program replaced with a new one by Antti Karttunen, May 17 2017

A286602 Restricted growth sequence transform of A286601.

Original entry on oeis.org

1, 2, 3, 2, 4, 2, 3, 5, 6, 5, 3, 7, 4, 7, 4, 2, 4, 2, 4, 7, 4, 8, 9, 7, 6, 10, 9, 5, 11, 7, 3, 7, 11, 7, 3, 7, 11, 12, 13, 7, 6, 14, 15, 10, 9, 5, 9, 12, 4, 7, 4, 2, 9, 7, 4, 8, 13, 7, 9, 14, 4, 8, 13, 8, 16, 8, 13, 17, 4, 8, 13, 8, 13, 18, 13, 7, 19, 14, 9, 17, 4, 8, 9, 7, 4, 2, 4, 7, 13, 8, 4, 8, 9, 14, 13, 7, 11, 12, 13, 7, 11, 7, 3, 7

Offset: 0

Antti Karttunen, Jun 04 2017

The scatter plot looks complex.

Antti Karttunen, Table of n, a(n) for n = 0..65535

Cf. A193231, A234022.

Cf. A286581, A286589, A286597, A286599, A286600, A286601, A286617, A286619, A286622 for similarly formed sequences.

A278217 Filter-sequence related to base-2 run-length encoding: a(n) = A046523(A075159(1+n)) = A046523(1+A075157(n)).

Original entry on oeis.org

1, 2, 2, 4, 6, 2, 4, 8, 12, 6, 2, 6, 12, 4, 8, 16, 24, 12, 6, 30, 6, 2, 6, 12, 36, 12, 4, 12, 24, 8, 16, 32, 48, 24, 12, 60, 30, 6, 30, 60, 12, 6, 2, 6, 30, 6, 12, 24, 72, 36, 12, 60, 12, 4, 12, 36, 72, 24, 8, 24, 48, 16, 32, 64, 96, 48, 24, 120, 60, 12, 60, 180, 60, 30, 6, 30, 210, 30, 60, 120, 24, 12, 6, 30, 6, 2, 6, 12, 60, 30, 6, 30, 60, 12, 24, 48, 144, 72

Offset: 0

Antti Karttunen, Nov 16 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16384

Cf. A046523, A075157, A075159, A286617 (rgs-version of this filter).
Other base-2 related filter sequences: A278219, A278222.
Sequences that partition N into same or coarser equivalence classes are at least these: A092339, A227185.

A005811(n) = hammingweight(bitxor(n, n>>1));  \\ This function from Gheorghe Coserea, Sep 03 2015
A286468(n) = { my(p=((n+1)%2), i=0, m=1); while(n>0, if(((n%2)==p), m *= prime(i), p = (n%2); i = i+1); n = n\2); m };
A075157(n) = if(!n,n,(prime(A005811(n))*A286468(n))-1);
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ This function from Charles R Greathouse IV, Aug 17 2011
A278217(n) = A046523(1+A075157(n)); \\ From Antti Karttunen, May 17 2017

(define (A278217 n) (A046523 (+ 1 (A075157 n))))
(define (A278217 n) (A046523 (A075159 (+ 1 n))))

a(n) = A046523(1+A075157(n)) = A046523(A075159(1+n)).

A286539 Restricted growth sequence of A286538.

Original entry on oeis.org

1, 1, 2, 1, 3, 4, 2, 1, 3, 5, 6, 3, 5, 4, 2, 1, 3, 5, 7, 8, 3, 9, 10, 5, 7, 6, 3, 5, 5, 4, 2, 1, 3, 5, 7, 11, 12, 3, 9, 13, 14, 5, 13, 14, 7, 11, 8, 3, 9, 10, 9, 13, 10, 5, 7, 7, 6, 3, 5, 5, 5, 4, 2, 1, 3, 5, 7, 11, 15, 16, 3, 9, 13, 17, 18, 5, 13, 19, 20, 7, 17, 18, 11, 15, 12, 3, 9, 13, 14, 9, 21, 19, 13, 19, 14, 5, 13, 14, 13, 17, 14, 7, 11, 11, 8, 3, 9, 10

Offset: 1

Antti Karttunen, May 17 2017

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A267111, A286537, A286538, A286617, A286619, A278222.

A286581 Restricted growth sequence transform of A286580.

Original entry on oeis.org

1, 2, 3, 2, 3, 4, 5, 2, 3, 6, 4, 5, 7, 6, 5, 2, 3, 6, 6, 4, 8, 9, 5, 7, 10, 6, 5, 11, 9, 6, 5, 2, 3, 6, 6, 6, 8, 12, 4, 8, 9, 9, 5, 13, 13, 10, 7, 10, 14, 6, 5, 12, 12, 11, 9, 10, 6, 5, 15, 14, 9, 6, 5, 2, 3, 6, 6, 6, 8, 12, 6, 8, 12, 12, 4, 13, 16, 9, 8, 9, 16, 9, 5, 13, 17, 13, 13, 10, 10, 7, 18, 19, 12, 14, 10, 14, 20, 6, 5, 18, 12, 12, 12, 10, 11, 9

Offset: 0

Antti Karttunen, Jun 03 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 0..65536

Cf. A233275, A286580, A286537, A286539, A286617, A286619, A286622.

cp's OEIS Frontend

A075157 Run lengths in the binary expansion of n gives the vector of exponents in prime factorization of a(n)+1, with the least significant run corresponding to the exponent of the least prime, 2; with one subtracted from each run length, except for the most significant run of 1's.

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A286602 Restricted growth sequence transform of A286601.

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A278217 Filter-sequence related to base-2 run-length encoding: a(n) = A046523(A075159(1+n)) = A046523(1+A075157(n)).

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Programs

PARI

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A286539 Restricted growth sequence of A286538.

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A286581 Restricted growth sequence transform of A286580.

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