A257512 -id:A257512 - cp's OEIS frontend

A256490 First differences of A257512: a(n) = A257512(n+1) - A257512(n).

8, 7, 9, 7, 13, 2, 10, 7, 13, 2, 14, 8, 7, 2, 11, 7, 13, 2, 14, 8, 7, 2, 15, 8, 7, 9, 7, 14, 1, 2, 12, 7, 13, 2, 14, 8, 7, 2, 15, 8, 7, 9, 7, 14, 1, 2, 16, 8, 7, 9, 7, 14, 1, 10, 7, 14, 1, 15, 8, 7, 1, 2, 13, 7, 13, 2, 14, 8, 7, 2, 15, 8, 7, 9, 7, 14, 1, 2, 16, 8, 7, 9, 7, 14, 1, 10, 7, 14, 1, 15, 8, 7, 1

Offset: 1

Antti Karttunen, May 03 2015

nonn

It seems that for all n >= 1, Sum_{k=1 .. 2^n} a(k) = 2^(n+3) - 1 = A000225(n+3).

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000225, A257512, A256489.

(define (A256490 n) (- (A257512 (+ n 1)) (A257512 n)))

a(n) = A257512(n+1) - A257512(n).

A257508 Next-to-leaf vertices in binary beanstalk; Numbers n for which A257265(n) = 1.

Original entry on oeis.org

1, 3, 4, 7, 10, 11, 15, 18, 22, 23, 25, 26, 31, 34, 38, 39, 41, 46, 47, 49, 50, 54, 56, 57, 63, 66, 70, 71, 73, 78, 79, 81, 82, 86, 88, 94, 95, 97, 98, 102, 104, 105, 110, 113, 116, 117, 119, 120, 127, 130, 134, 135, 137, 142, 143, 145, 146, 150, 152, 158, 159, 161, 162, 166, 168, 169, 174, 177, 180, 181

Offset: 1

Antti Karttunen, May 03 2015

nonn

Numbers n for which A257265(n) = 1, in other words, numbers n for which a descendant leaf nearest to n in binary beanstalk is one edge away.
Numbers n such that either A079559(A213723(n)) or A079559(A213724(n)) (or both) are zero.
Equal to A257507 with duplicate terms removed.

			3 is present because it has an immediate leaf-child 5, as A011371(5) = 3.
4 is present because it has an immediate leaf-child 6, as A011371(6) = 4.
10 is present because it has two immediate leaf-children, 12 and 13, as A011371(12) = A011371(13) = 10.
See also Paul Tek's illustration.

Antti Karttunen, Table of n, a(n) for n = 1..12289
Paul Tek, Illustration of how natural numbers in range 0 .. 133 are organized as a binary tree in the binary beanstalk

Positions of 1's in A257265.
Subsequence of A005187.
Cf. A011371, A079559, A213723, A213724, A257507, A257509, A257512 (a subsequence).

a257508 n = a257508_list !! (n-1)
a257508_list = filter ((== 1) . a257265) [0..]
-- Reinhard Zumkeller, May 06 2015

A257507 Row 2 of A257264: a(n) = A011371(A055938(n)).

Original entry on oeis.org

1, 3, 4, 7, 10, 10, 11, 15, 18, 18, 22, 23, 25, 25, 26, 31, 34, 34, 38, 39, 41, 41, 46, 47, 49, 50, 54, 54, 56, 56, 57, 63, 66, 66, 70, 71, 73, 73, 78, 79, 81, 82, 86, 86, 88, 88, 94, 95, 97, 98, 102, 102, 104, 105, 110, 110, 113, 116, 117, 117, 119, 119, 120, 127, 130, 130, 134, 135, 137, 137, 142, 143, 145, 146

Offset: 1

Antti Karttunen, May 03 2015

nonn

The sequence gives the parent node of each leaf-vertex (A055938) in binary beanstalk.

			Terms of A055938 are the leaf-nodes in Paul Tek's illustration. This sequence gives the corresponding parent-node (in that illustration a node immediately below where the arrow points), for each term of A055938[1..]: 2, 5, 6, 9, 12, 13, 14, ...
As A055938(4) = 9, and 9's parent node is 7 (because A011371(9) = 7), a(4) = 7.
As A055938(5) = 12, and 12's parent node is 10, a(5) = 10.
As A055938(6) = 13, and 13's parent node is 10, a(6) = 10.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Paul Tek, Illustration of how natural numbers in range 0 .. 133 are organized as a binary tree in the binary beanstalk

Row 2 of A257264.
Cf. A011371, A055938.
Cf. A257508 (same sequence with duplicates removed), A257512 (the terms which occur twice).

(define (A257507 n) (A011371 (A055938 n)))

a(n) = A011371(A055938(n)).

A262903 Numbers that are not leaves but all of whose children are leaves in the tree generated by edge-relation A049820(child) = parent.

Original entry on oeis.org

4, 5, 14, 16, 32, 41, 44, 77, 80, 92, 101, 110, 119, 128, 139, 148, 158, 161, 169, 176, 191, 192, 199, 215, 224, 227, 234, 238, 249, 262, 264, 277, 296, 311, 317, 327, 350, 351, 352, 360, 363, 382, 385, 389, 392, 395, 396, 411, 427, 430, 437, 448, 449, 461, 464, 483, 488, 518, 523, 531, 532, 542, 552, 561, 568, 570, 577, 579, 600, 601, 613, 619, 632, 634, 636, 645, 648, 659, 665, 666, 671, 682, 683, 696, 705, 723

Offset: 1

Antti Karttunen, Oct 06 2015

nonn

Numbers n for which A060990(n) > 0 and A060990(n) = A262900(n).

Numbers n for which A262695(n) = 2.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000005, A049820, A060990, A262695, A262900.
Subsequence of A262901 and A236562.
No common terms with A259934.
Cf. also A257512.

cp's OEIS Frontend

A256490 First differences of A257512: a(n) = A257512(n+1) - A257512(n).

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A257508 Next-to-leaf vertices in binary beanstalk; Numbers n for which A257265(n) = 1.

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A257507 Row 2 of A257264: a(n) = A011371(A055938(n)).

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A262903 Numbers that are not leaves but all of whose children are leaves in the tree generated by edge-relation A049820(child) = parent.

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