A103318 -id:A103318 - cp's OEIS frontend

A102370 "Sloping binary numbers": write numbers in binary under each other (right-justified), read diagonals in upward direction, convert to decimal.

0, 3, 6, 5, 4, 15, 10, 9, 8, 11, 14, 13, 28, 23, 18, 17, 16, 19, 22, 21, 20, 31, 26, 25, 24, 27, 30, 61, 44, 39, 34, 33, 32, 35, 38, 37, 36, 47, 42, 41, 40, 43, 46, 45, 60, 55, 50, 49, 48, 51, 54, 53, 52, 63, 58, 57, 56, 59, 126, 93, 76, 71, 66, 65, 64, 67, 70, 69

Offset: 0

Philippe Deléham, Feb 13 2005

All terms are distinct, but certain terms (see A102371) are missing. But see A103122.

Trajectory of 1 is 1, 3, 5, 15, 17, 19, 21, 31, 33, ..., see A103192.

			........0
........1
.......10
.......11
......100
......101
......110
......111
.....1000
.........
The upward-sloping diagonals are:
0
11
110
101
100
1111
1010
.......
giving 0, 3, 6, 5, 4, 15, 10, ...
The sequence has a natural decomposition into blocks (see the paper): 0; 3; 6, 5, 4; 15, 10, 9, 8, 11, 14, 13; 28, 23, 18, 17, 16, 19, 22, 21, 20, 31, 26, 25, 24, 27, 30; 61, ...
Reading the array of binary numbers along diagonals with slope 1 gives this sequence, slope 2 gives A105085, slope 0 gives A001477 and slope -1 gives A105033.

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe)
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. Preprint versions: [pdf, ps].
Index entries for sequences related to binary expansion of n

Related sequences (1): A103542 (binary version), A102371 (complement), A103185, A103528, A103529, A103530, A103318, A034797, A103543, A103581, A103582, A103583.
Related sequences (2): A103584, A103585, A103586, A103587, A103127, A103192 (trajectory of 1), A103122, A103588, A103589, A103202 (sorted), A103205 (base 10 version).
Related sequences (3): A103747 (trajectory of 2), A103621, A103745, A103615, A103842, A103863, A104234, A104235, A103813, A105023, A105024, A105025, A105026, A105027, A105028.
Related sequences (4): A105029, A105030, A105031, A105032, A105033, A105034, A105035, A105108.
Related sequences (5): A105229, A105271, A104378, A104401, A104403, A104489, A104490, A104853, A104893, A104894, A105085.

a102370 n = a102370_list !! n
a102370_list = 0 : map (a105027 . toInteger) a062289_list
-- Reinhard Zumkeller, Jul 21 2012

A102370:=proc(n) local t1,l; t1:=n; for l from 1 to n do if n+l mod 2^l = 0 then t1:=t1+2^l; fi; od: t1; end;

f[n_] := Block[{k = 1, s = 0, l = Max[2, Floor[Log[2, n + 1] + 2]]}, While[k < l, If[ Mod[n + k, 2^k] == 0, s = s + 2^k]; k++ ]; s]; Table[ f[n] + n, {n, 0, 71}] (* Robert G. Wilson v, Mar 21 2005 *)

A102370(n)=n-1+sum(k=0,ceil(log(n+1)/log(2)),if((n+k)%2^k,0,2^k)) \\ Benoit Cloitre, Mar 20 2005

{a(n) = if( n<1, 0, sum( k=0, length( binary( n)), bitand( n + k, 2^k)))} /* Michael Somos, Mar 26 2012 */

def a(n): return 0 if n<1 else sum([(n + k)&(2**k) for k in range(len(bin(n)[2:]) + 1)]) # Indranil Ghosh, May 03 2017

a(n) = n + Sum_{ k >= 1 such that n + k == 0 mod 2^k } 2^k. (Cf. A103185.) In particular, a(n) >= n. - N. J. A. Sloane, Mar 18 2005

a(n) = A105027(A062289(n)) for n > 0. - Reinhard Zumkeller, Jul 21 2012

More terms from Benoit Cloitre, Mar 20 2005

A034797 a(0) = 0; a(n+1) = a(n) + 2^a(n).

Original entry on oeis.org

0, 1, 3, 11, 2059

Offset: 0

Joseph Shipman (shipman(AT)savera.com)

First impartial game with value n, using natural enumeration of impartial games.
The natural 1-1 correspondence between nonnegative numbers and hereditarily finite sets is given by f(A)=sum over members m of A of 2^f(m). A set can be considered an impartial game where the legal moves are the members. The value of an impartial game is always an ordinal (for finite games, an integer).
The next term, a(5) = 2^2059 + 2059, has 620 decimal digits and is too large to include. - Olivier Gérard, Jun 26 2001
Positions of records in A103318. - N. J. A. Sloane and David Applegate, Mar 21 2005
The first n terms in this sequence form the lexicographically earliest n-vertex clique in the Ackermann-Rado encoding of the Rado graph (an infinite graph in which vertex i is adjacent to vertex j, with iDavid Eppstein, Aug 22 2014
This sequence was used by Spiro to bound the density of refactorable numbers (A033950). - David Eppstein, Aug 22 2014
For any positive integer m, a(1), a(2), ..., a(3^m) modulo 3^m form a complete residue set. - Yifan Xie, Aug 19 2025

J. H. Conway, On Numbers and Games, Academic Press.

N. J. A. Sloane, Table of n, a(n) for n = 0..5
Wilhelm Ackermann, Die Widerspruchsfreiheit der allgemeinen Mengenlehre, Math. Ann. 114 (1939), no. 1, 305-315.
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. Preprint versions: [pdf, ps].
O. Kurganskyy and I. Potapov, Reachability problems for PAMs, arXiv preprint arXiv:1510.04121 [cs.NA], 2015.
Richard Rado, Universal graphs and universal functions, Acta Arith. 9 (1964), 331-340.
Claudia Spiro, How often is the number of divisors of n a divisor of n?, J. Number Theory 21 (1985), no. 1, 81-100.
Wikipedia, Rado graph.

Cf. A034798, A103318.

a=0;lst={a};Do[AppendTo[lst,a+=2^a],{n,0,4}];lst (* Vladimir Joseph Stephan Orlovsky, May 06 2010 *)
NestList[#+2^#&,0,5] (* Harvey P. Dale, Mar 22 2020 *)

A103319 Primes of the form p! + 1 where p is prime.

Original entry on oeis.org

3, 7, 39916801, 13763753091226345046315979581580902400000001, 33452526613163807108170062053440751665152000000001, 4470115461512684340891257138125051110076800700282905015819080092370422104067183317016903680000000000000001

Offset: 1

Jonathan Sondow, Jan 31 2005

The values of p are 2, 3, 11, 37, 41, 73 which is A093804 (with a different definition). Subsequence of A088332 (primes of the form n! + 1).

			2 and 2! + 1 = 3 are prime, so 3 is a member.

R. K. Guy, Unsolved Problems in Number Theory, Section A2.

R. Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670, 2012 - From N. J. A. Sloane, Jun 13 2012
Eric Weisstein's World of Mathematics, Factorial Prime
Index entries for sequences related to factorial numbers.

Cf. A093804, A002981, A088332, A103318.

Select[Table[p!+1,{p,Prime[Range[30]]}],PrimeQ] (* Harvey P. Dale, Nov 28 2019 *)

A105032 Binary equivalents of A103745.

Original entry on oeis.org

1, 10, 101, 1000, 10001, 100010, 1000001, 10000000, 100000001, 1000000010, 10000000101, 100000000000, 1000000000001, 10000000000010, 100000000000001, 1000000000000000, 10000000000000001, 100000000000000010, 1000000000000000101, 10000000000000001000, 100000000000000000001, 1000000000000000000010, 10000000000000000000001

Offset: 1

N. J. A. Sloane, David Applegate, Benoit Cloitre and Philippe Deléham, Apr 03 2005

nonn

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.

Cf. A102371, A103318, A105031.

a(n) = A007088(A103745(n)). - R. J. Mathar, May 02 2007

More terms from R. J. Mathar, May 02 2007

cp's OEIS Frontend

A102370 "Sloping binary numbers": write numbers in binary under each other (right-justified), read diagonals in upward direction, convert to decimal.

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A034797 a(0) = 0; a(n+1) = a(n) + 2^a(n).

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A103319 Primes of the form p! + 1 where p is prime.

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A105032 Binary equivalents of A103745.

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A105553 a(n) is the number of 1's in A103528(n) written in base 2.

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