A102371 Numbers missing from A102370.

1, 2, 7, 12, 29, 62, 123, 248, 505, 1018, 2047, 4084, 8181, 16374, 32755, 65520, 131057, 262130, 524279, 1048572, 2097133, 4194286, 8388587, 16777192, 33554409, 67108842, 134217711, 268435428, 536870885

Offset: 1

Author

Philippe Deléham, Feb 13 2005

Keywords

• nonn
• base

Comments

Indices of negative numbers in A103122.

Write numbers in binary under each other; start at 2^k, read in upward direction with the first bit omitted and convert to decimal:

. . . . . . . . . . 0

. . . . . . . . . . 1

.. . . . . . . . . 10 < -- Starting here, the upward diagonal (first bit omitted) reads 1 -> 1

.. . . . . . . . . 11

. . . . . . . . . 100 < -- Starting here, the upward diagonal (first bit omitted) reads 10 -> 2

. . . . . . . . . 101

. . . . . . . . . 110

. . . . . . . . . 111

.. . . . . . . . 1000 < -- Starting here, the upward diagonal (first bit omitted) reads 111 -> 7

. . . . . . . . .1001

Thus a(n) = A102370(2^n - n) - 2^n.

Do we have a(n) = 2^n-1-A105033(n-1)? - David A. Corneth, May 07 2020

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
• David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers [pdf, ps].
• David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers, J. Integer Seq. 8 (2005), no. 3, Article 05.3.6, 15 pp.

Crossrefs

Cf. A102370, A103530, A103581, A103582, A103583, A105033.

Programs

Haskell

a102371 n = a102371_list !! (n-1)
a102371_list = map (a105027 . toInteger) $ tail a000225_list
-- Reinhard Zumkeller, Jul 21 2012

Maple

A102371:= proc (n) local t1, l; t1 := -n; for l to n do if `mod`(n-l,2^l) = 0 then t1 := t1+2^l end if end do; t1 end proc;

Python

a=1
for n in range(2,66):
    print(a, end=",")
    a ^= a+n
# Alex Ratushnyak, Apr 21 2012

Formula

a(n) = -n + Sum_{ k >= 1, k == n mod 2^k } 2^k. - N. J. A. Sloane and David Applegate, Mar 22 2005. E.g. a(5) = -5 + 2^1 + 2^5 = 29.

a(2^k + k) -a(k) = 2^(2^k + k) - 2^k, with k>= 1.

a(1)=1, for n>1, a(n) = a(n-1) XOR (a(n-1) + n), where XOR is the bitwise exclusive-or operator. - Alex Ratushnyak, Apr 21 2012

a(n) = A105027(A000225(n)). - Reinhard Zumkeller, Jul 21 2012

Extensions

More terms from Benoit Cloitre, Mar 20 2005

a(16)-a(22) from Robert G. Wilson v, Mar 21 2005

a(15)-a(29) from David Applegate, Mar 22 2005