The On-Line Encyclopedia of Integer Sequences (OEIS)

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A286612

Numbers n for which A019565(n) <= A087207(n) < n.
(history; published version)

#10 by N. J. A. Sloane at Tue Jun 20 23:11:35 EDT 2017

STATUS

proposed

approved

#9 by Antti Karttunen at Tue Jun 20 17:47:32 EDT 2017

STATUS

editing

proposed

#8 by Antti Karttunen at Tue Jun 20 17:34:00 EDT 2017

LINKS

Antti Karttunen, <a href="/A286612/b286612.txt">Table of n, a(n) for n = 1..97</a>

#7 by Antti Karttunen at Tue Jun 20 17:33:22 EDT 2017

COMMENTS

Any 2-cycle of A087207 and also any 2-cycle of A019565 (in which case A019565(x) = A087207(x) for both members of the cycle), if such cycles exist at all, must have the other larger one of its members included in this sequence.

#6 by Antti Karttunen at Tue Jun 20 16:26:25 EDT 2017

DATA

65, 513, 1026, 4107, 8200, 8256, 16400, 16450, 16512, 16544, 16641, 32800, 32900, 33024, 33282, 33920, 49184, 65600, 65800, 66048, 66080, 131200, 131334, 132096, 132160, 163968, 262400, 262668, 264192, 264320, 274432, 327936, 524342, 524610, 524800, 524832, 525826, 528384, 532500, 540736, 548864, 655872, 786467, 1048617

Discussion

Tue Jun 20

17:07

Antti Karttunen: (Above question is red herring. If A019565(n) = A048675(n) then we already have found 2-cycle of A019565(n)).

17:10

Antti Karttunen: So, let's ponder instead generic 2-cycles of A087207: Starting from other member a, one first applies A019565 (to obtain a squarefree number c), then applies zero or more times A065642 to obtain number b which has same prime divisors as c, then applies A019565 to b (to obtain a squarefree number d), and then applies zero or more times A065642 to obtain number a again.

17:25

Antti Karttunen: Because iterating A065642 from any starting value n > 1 is a monotonic process, it means that b >= c and a >= d. AND A087207(b) = a, A087207(a) = b. Thus A019565(a) <= b that is, A019565(a) <= A087207(a).

17:28

Antti Karttunen: But also A019565(b) <= a that is A019565(b) <= A087207(b). So we see that both members of such hypothetical 2-cycles should be present in A286611, and because in any 2-cycle the other member is larger and the other member is smaller, the larger one should be present in this sequence.

#5 by Antti Karttunen at Tue Jun 20 16:04:14 EDT 2017

CROSSREFS

Subsequence of A285315.

Discussion

Tue Jun 20

16:12

Antti Karttunen: For 2-cycles of  A019565, A087207(n) cannot be less than A019565(n), because it must be equal to A019565(n).

16:15

Antti Karttunen: For 2-cycles of A019565, A087207(n) cannot be >= n because A087207(n) is never equal to n (different parity), and cannot be more than n, if A019565(n) <= n, because we require that A019565(n) = A087207(n). (And in these cases actually A087207(n) = A048675(n)).

16:16

Antti Karttunen: So the question is: How sparse is the intersection of this sequence with A285319 ?

#4 by Antti Karttunen at Tue Jun 20 16:02:31 EDT 2017

CROSSREFS

Cf. A019565, A087207, A285319, A286609.

#3 by Antti Karttunen at Tue Jun 20 15:59:27 EDT 2017

COMMENTS

Any 2-cycle of A087207 and also any 2-cycle of A019565 (in which case A019565(x) = A087207(x) for both members of the cycle), if such cycles exist at all, must have the other one of its members included in this sequence.

#2 by Antti Karttunen at Tue Jun 20 15:51:37 EDT 2017

NAME

allocated Numbers n for Antti Karttunenwhich A019565(n) <= A087207(n) < n.

DATA

65, 513, 1026, 4107, 8200, 8256, 16400, 16450, 16512, 16544, 16641, 32800, 32900, 33024, 33282, 33920, 49184, 65600, 65800, 66048, 66080

OFFSET

1,1

LINKS

<a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>

<a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>

PROG

(Scheme, with Antti Karttunen's IntSeq-library)

(define A286612 (MATCHING-POS 1 1 (lambda (n) (and (< (A087207 n) n) (<= (A019565 n) (A087207 n))))))

CROSSREFS

Intersection of A286608 and A286611.

Cf. A019565, A087207, A286609.

KEYWORD

allocated

nonn,base

AUTHOR

Antti Karttunen, Jun 20 2017

STATUS

approved

editing

#1 by Antti Karttunen at Thu May 11 12:51:34 EDT 2017

NAME

allocated for Antti Karttunen

KEYWORD

allocated

STATUS

approved