A219665 -id:A219665 - cp's OEIS frontend

A230432 Simple self-inverse permutation of natural numbers: after zero, list each block of A219661(n) numbers in reverse order, from A226061(n+1) to A219665(n).

0, 1, 3, 2, 8, 7, 6, 5, 4, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 110, 109, 108, 107, 106, 105, 104, 103, 102, 101, 100, 99, 98, 97, 96, 95, 94, 93, 92, 91, 90, 89, 88, 87, 86, 85, 84, 83, 82, 81, 80, 79, 78, 77, 76, 75, 74, 73

Offset: 0

Antti Karttunen, Oct 22 2013

nonn

This permutation can be used to map between the sequences A219666 and A230416. E.g. A219666(n) = A230416(a(n)) and vice versa: A230416(n) = A219666(a(n)).

Antti Karttunen, Table of n, a(n) for n = 0..3149
Index entries for sequences that are permutations of the natural numbers

Cf. A219661, A219665, A226061, A230411, A219666, A230416, A230431.

Analogous sequence for binary system: A218602.

(define (A230432 n) (if (zero? n) n (- (A219665 (A230411 (+ 1 n))) (A230431 n) 1)))

a(n) = A219665(A230411(n+1)) - A230431(n) - 1.

A230411 a(n) = minimal k for which A219665(k) >= n; a(n) = one more than the factorial base width (A084558) of the (n-1)th term in the infinite trunk of factorial beanstalk (A219666).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Oct 22 2013

nonn

a(1)=1, after which each term n occurs A219661(n-1) times.

Auxiliary function for computing A219666, A230431 and A230432.

Antti Karttunen, Table of n, a(n) for n = 1..5000

Cf. A219661, A219665, A219666, A230431, A230432.

Analogous sequence for binary system: A213711.

a(n) = 1 + A084558(A219666(n-1)) = 1 + A084558(A230416(n-1)). [Each a(n) is one more than the number of digits needed in factorial base to write the (n-1)-th term in the infinite trunk of factorial beanstalk]

A255062 Number of steps to reach 0 when starting from (2^n)-1 and iterating the map x -> x - (number of runs in binary representation of x): a(n) = A255072(A000225(n)).

Original entry on oeis.org

0, 1, 2, 4, 7, 12, 21, 37, 66, 119, 216, 394, 722, 1330, 2464, 4590, 8591, 16143, 30435, 57550, 109115, 207389, 395046, 754028, 1441972, 2762765, 5303467, 10200386, 19656529, 37948282, 73384081, 142115377, 275551756, 534790473, 1038702981, 2018626773, 3924923938, 7634733313

Offset: 0

Antti Karttunen, Feb 14 2015

nonn

Also, for n >= 1, the number of steps to reach 0 when starting from 2^n and iterating the map x -> x minus A005811(x), the number of runs in binary representation of x.

One more than A255061.
First differences: A255071 (after the zero term).
Cf. A000079, A000225, A005811, A255071, A255072.
Analogous sequences: A213710 (A218600), A219665.

(define (A255062 n) (A255072 (A000225 n)))
(define (A255062 n) (if (<= n 1) n (+ (A255062 (- n 1)) (A255071 (- n 1))))) ;; Assuming that A255071 has been already computed, with e.g. the PARI-program given in that entry.

a(n) = A255072(A000225(n)).
a(0) = 0, a(1) = 1; for n > 1, a(n) = a(n-1) + A255071(n-1).
Other identities. For all n >= 1:
a(n) = A255072(A000079(n)). [See the Comments section.]
a(n) = 1 + A255061(n).

A213710 Number of steps to reach 0 when starting from 2^n and iterating the map x -> x - (number of 1's in binary representation of x): a(n) = A071542(2^n) = A218600(n)+1.

Original entry on oeis.org

1, 2, 3, 5, 8, 13, 22, 39, 69, 123, 221, 400, 730, 1344, 2494, 4656, 8728, 16406, 30902, 58320, 110299, 209099, 397408, 757297, 1446946, 2771952, 5323983, 10250572, 19780123, 38243221, 74058514, 143592685, 278661809, 541110612, 1051158028, 2042539461, 3969857206

Offset: 0

Antti Karttunen, Oct 26 2012

nonn

Conjecture: A179016(a(n))= 2^n for all n apart from n=2. This is true if all powers of 2 except 2 itself occur in A179016 as in that case they must occur at positions given by this sequence.

This is easy to prove: It suffices to note that after 3 no integer of form (2^k)+1 can occur in A005187, thus for all k >= 2, A213725((2^k)+1) = 1 or equally: A213714((2^k)+1) = 0. - Antti Karttunen, Jun 12 2013

One more than A218600, which is the partial sums of A213709, thus the latter also gives the first differences of this sequence.
Cf. A000079, A005187, A071542, A218616, A233271.
Analogous sequences: A219665, A255062.

a(n) = A071542(A000079(n)) = A071542(2^n).

a(n) = 1 + A218600(n).

a(29)-a(36) from Alois P. Heinz, Jul 03 2022

A219661 Number of steps to go from (n+1)!-1 to n!-1 with map x -> x - (sum of digits in factorial base representation of x).

Original entry on oeis.org

1, 2, 5, 19, 83, 428, 2611, 18473, 150726, 1377548, 13851248, 152610108, 1835293041, 23925573979, 335859122743, 5049372125352, 80942722123544, 1378487515335424, 24858383452927384, 473228664468684846

Offset: 1

Antti Karttunen, Dec 03 2012

nonn

			(1!)-1 (0) is reached from (2!)-1 (1) with one step by subtracting A034968(1) from 1.
(2!)-1 (1) is reached from (3!)-1 (5) with two steps by first subtracting A034968(5) from 5 -> 2, and then subtracting A034968(2) from 2 -> 1.
(3!)-1 (5) is reached from (4!)-1 (23) with five steps by repeatedly subtracting the sum of digits in factorial expansion as: 23 - 6 = 17, 17 - 5 = 12, 12 - 2 = 10, 10 - 3 = 7, 7 - 2 = 5.
Thus a(1)=1, a(2)=2 and a(3)=5.

Hiroaki Yamanouchi, Fast Python-program for computing terms of this sequence

Row sums of A230420 and A230421.
Cf. A000142, A007623, A219651, A219652, A219662, A219663, A219665, A226061.
Cf. also A213709 (analogous sequence for base-2), A261234 (for base-3).

Table[Length@ NestWhileList[# - Total@ IntegerDigits[#, MixedRadix[Reverse@ Range[2, 120]]] &, (n + 1)! - 1, # > n! - 1 &] - 1, {n, 0, 8}] (* Michael De Vlieger, Jun 27 2016, Version 10.2 *)

(define (A219661 n) (if (zero? n) n (let loop ((i (-1+ (A000142 (1+ n)))) (steps 1)) (cond ((isA000142? (1+ (A219651 i))) steps) (else (loop (A219651 i) (1+ steps)))))))
(define (isA000142? n) (and (> n 0) (let loop ((n n) (i 2)) (cond ((= 1 n) #t) ((not (zero? (modulo n i))) #f) (else (loop (/ n i) (1+ i)))))))
;; Alternative definition:
(define (A219661 n) (- (A219652 (-1+ (A000142 (1+ n)))) (A219652 (-1+ (A000142 n)))))

a(n) = A219652((n+1)!-1) - A219652(n!-1).

a(n) = A219662(n) + A219663(n).

Terms a(16) - a(20) computed with Hiroaki Yamanouchi's Python-program by Antti Karttunen, Jun 27 2016

A226061 Partial sums of A219661.

Original entry on oeis.org

0, 1, 3, 8, 27, 110, 538, 3149, 21622, 172348, 1549896, 15401144, 168011252, 2003304293, 25928878272, 361788001015, 5411160126367, 86353882249911, 1464841397585335, 26323224850512719, 499551889319197565

Offset: 1

Antti Karttunen, May 28 2013

nonn

a(n) tells the position of (n!)-1 in A219666.

One less than A219665.
Cf. A219661, A219666.
Analogous sequence for binary system: A218600.
Cf. also A230410, A231719.

Accumulate@ Table[Length@ NestWhileList[# - Total@ IntegerDigits[#,
MixedRadix[Reverse@ Range[2, 120]]] &, (n + 1)! - 1, # > n! - 1 &] - 1, {n, 0, 8}] (* Michael De Vlieger, Jun 27 2016, Version 10.2 *)

a(n) = a(n-1) + A219661(n-1) with a(1) = 0.
a(n) = A219652(n!-1).
a(n) = A219665(n) - 1.

Terms a(16) - a(21) computed from the new terms of A219661 by Antti Karttunen, Jun 27 2016

A230431 After the first zero, integers from 0 to A219661(n)-1 followed by integers from 0 to A219661(n+1)-1, etc.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 2, 3, 4, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44

Offset: 0

Antti Karttunen, Oct 22 2013

nonn

Cf. A219661, A219665, A230411. Used to compute A230432.

Analogous sequence for binary system: A218601.

(define (A230431 n) (if (< n 2) 0 (- n (A219665 (-1+ (A230411 (+ n 1)))))))

a(0) = a(1) = 0, and for n > 1, a(n) = n - A219665(A230411(n+1)-1).

cp's OEIS Frontend

A230432 Simple self-inverse permutation of natural numbers: after zero, list each block of A219661(n) numbers in reverse order, from A226061(n+1) to A219665(n).

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A230411 a(n) = minimal k for which A219665(k) >= n; a(n) = one more than the factorial base width (A084558) of the (n-1)th term in the infinite trunk of factorial beanstalk (A219666).

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A255062 Number of steps to reach 0 when starting from (2^n)-1 and iterating the map x -> x - (number of runs in binary representation of x): a(n) = A255072(A000225(n)).

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A213710 Number of steps to reach 0 when starting from 2^n and iterating the map x -> x - (number of 1's in binary representation of x): a(n) = A071542(2^n) = A218600(n)+1.

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A219661 Number of steps to go from (n+1)!-1 to n!-1 with map x -> x - (sum of digits in factorial base representation of x).

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A226061 Partial sums of A219661.

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A230431 After the first zero, integers from 0 to A219661(n)-1 followed by integers from 0 to A219661(n+1)-1, etc.

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