A082662 -id:A082662 - cp's OEIS frontend

A116882 A number k is included if (highest odd divisor of k)^2 <= k.

1, 2, 4, 8, 12, 16, 24, 32, 40, 48, 56, 64, 80, 96, 112, 128, 144, 160, 176, 192, 208, 224, 240, 256, 288, 320, 352, 384, 416, 448, 480, 512, 544, 576, 608, 640, 672, 704, 736, 768, 800, 832, 864, 896, 928, 960, 992, 1024, 1088, 1152, 1216, 1280, 1344, 1408

Offset: 1

Leroy Quet, Feb 24 2006

Also k is included if (and only if) the greatest power of 2 dividing k is >= the highest odd divisor of k. All terms of the sequence are even besides the 1.
Equivalently, positive integers of the form k*2^m, where odd k <= 2^m. - Thomas Ordowski, Oct 19 2014
If we define a divisor d|n to be superior if d >= n/d, then superior divisors are counted by A038548 and listed by A161908. This sequence consists of 1 and all numbers without a superior odd divisor. - Gus Wiseman, Feb 18 2021
Numbers k such that A006519(k) >= A000265(k), with equality only when k = 1. - Amiram Eldar, Jan 24 2023

			40 = 8 * 5, where 8 is highest power of 2 dividing 40 and 5 is the highest odd dividing 40. 8 is >= 5 (so 5^2 <= 40), so 40 is in the sequence.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 38.
Milan Janjic and Boris Petkovic, A Counting Function, arXiv 1301.4550 [math.CO], 2013.
Amelia Carolina Sparavigna, Discussion of the groupoid of Proth numbers (OEIS A080075), Politecnico di Torino, Italy (2019).

Cf. A000265, A006519, A080075, A112714.
The complement is A116883.
Positions of zeros (and 1) in A341675.
A051283 = numbers without a superior prime-power divisor (zeros of A341593).
A059172 = numbers without a superior squarefree divisor (zeros of A341592).
A063539 = numbers without a superior prime divisor (zeros of A341591).
A333805 counts strictly inferior odd divisors.
A341594 counts strictly superior odd divisors.
- Inferior: A033676, A038548, A063962, A066839, A069288, A161906, A217581.
- Superior: A033677, A063538, A070038, A072500, A161908, A341676.
- Strictly Inferior: A056924, A060775, A070039, A333806, A341596, A341674.
- Strictly Superior: A056924, A064052/A048098, A140271, A238535, A341642, A341643, A341673.
Cf. A000005, A000203, A001248, A006530, A020639, A026804, A027193, A340101, A340854 (zeros of A340832), A001008, A002805.
Subsequence of A082662, {1} U A363122.

f[n_] := Select[Divisors[n], OddQ[ # ] &][[ -1]]; Insert[Select[Range[2, 1500], 2^FactorInteger[ # ][[1]][[2]] > f[ # ] &], 1, 1] (* Stefan Steinerberger, Apr 10 2006 *)
q[n_] := 2^(2*IntegerExponent[n, 2]) >= n; Select[Range[1500], q] (* Amiram Eldar, Jan 24 2023 *)

isok(n) = vecmax(select(x->((x % 2)==1), divisors(n)))^2 <= n; \\ Michel Marcus, Sep 06 2016

isok(n) = 2^(valuation(n,2)*2) >= n \\ Jeppe Stig Nielsen, Feb 19 2019

from itertools import count, islice
def A116882_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:(n&-n)**2>=n,count(max(startvalue,1)))
A116882_list = list(islice(A116882_gen(),20)) # Chai Wah Wu, May 17 2023

a(n) = A080075(n-1)-1. - Klaus Brockhaus, Georgi Guninski and M. F. Hasler, Aug 16 2010
a(n) ~ n^2/2. - Thomas Ordowski, Oct 19 2014
Sum_{n>=1} 1/a(n) = 1 + (3/4) * Sum_{k>=1} H(2^k-1)/2^k = 2.3388865091..., where H(k) = A001008(k)/A002805(k) is the k-th harmonic number. - Amiram Eldar, Jan 24 2023

More terms from Stefan Steinerberger, Apr 10 2006

A082647 Number of ways n can be expressed as the sum of d consecutive positive integers where d>0 is a divisor of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 1, 2, 2, 1, 3, 1, 1, 2, 1, 3, 2, 1, 1, 2, 2, 1, 3, 1, 1, 4, 1, 1, 2, 2, 2, 2, 1, 1, 3, 2, 2, 2, 1, 1, 3, 1, 1, 4, 1, 2, 3, 1, 1, 2, 3, 1, 3, 1, 1, 3, 1, 3, 2, 1, 2, 3, 1, 1, 3, 2, 1, 2, 2, 1, 4, 3, 1, 2, 1, 2, 2, 1, 2, 4, 2, 1, 2, 1, 2, 4

Offset: 1

Naohiro Nomoto, May 15 2003

Number of ways to write n as the sum of an odd number of consecutive integers. - Vladeta Jovovic, Aug 28 2007
Number of odd divisors of n less than sqrt(2*n). - Vladeta Jovovic, Sep 16 2007
Conjecture: a(n) is also the number of subparts in an octant of the symmetric representation of sigma(n). - Omar E. Pol, Feb 22 2017

			For n=6: 6 has two ways -- (d=3; 3|6), 1+2+3=6; and (d=1; 1|6), 6=6 -- so a(6)=2.

Peter Kagey, Table of n, a(n) for n = 1..10000
M. D. Hirschhorn and P. M. Hirschhorn, Partitions into Consecutive Parts, Mathematics Magazine: 2003, Volume 76, Number 4, pp. 306-308.
William Lowell Putnam Competition, Function A(k) in Problem B6, 2015.

Cf. A001227, A054843, A082637, A082662, A237593, A285901.

N:= 1000: # to get a(1) to a(N)
g:= add(x^(k*(2*k-1))/(1-x^(2*k-1)), k=1..floor(sqrt(N/2))):
S:= series(g,x,N+1):
seq(coeff(S,x,n),n=1..N); # Robert Israel, Dec 08 2015

a(n) = my(q = sqrt(2*n)); sumdiv(n, d, (d%2) && (d < q)); \\ Michel Marcus, Jul 04 2014

G.f.: Sum_{k>0} x^(k*(2*k-1))/(1-x^(2*k-1)). - Vladeta Jovovic, Aug 25 2004
Conjecture: a(n) = A067742(n) + A131576(n). - Omar E. Pol, Feb 22 2017
Conjecture: a(n) = A001227(n) - A131576(n). - Omar E. Pol, Apr 18 2017

A131576 Number of ways to represent n as a sum of an even number of consecutive integers.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 2, 1, 1, 0, 1, 1, 2, 0, 1, 1, 1, 0, 2, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 2, 1, 1, 0, 1, 1, 2, 1, 1, 1, 2, 0, 2, 1, 1, 1, 1, 1, 2, 0, 2, 1, 1, 1, 2, 1, 1, 0, 1, 1, 3, 1, 1, 2, 1, 0, 2, 1, 1, 1, 2, 1, 2, 0, 1, 2, 1, 1, 2, 1, 2, 0, 1, 1, 2, 1, 1, 2, 1, 0, 4

Offset: 1

Vladeta Jovovic, Aug 28 2007, Sep 16 2007

Equals number of odd divisors of n greater than sqrt(2*n). [Hirschhorn and Hirschhorn]
a(n) + A082647(n) = A001227. This follows immediately from the definitions. - N. J. A. Sloane, Dec 07 2020
Conjecture: a(n) is also the number of pairs of subparts in the symmetric representation of sigma(n) which are mirror images of each other in the main diagonal. (Cf. A279387). - Omar E. Pol, Feb 22 2017 [Conjecture clarified by N. J. A. Sloane, Dec 16 2020]
Indices of nonzero terms give A281005. - Omar E. Pol, Mar 04 2018
Indices of zero terms give A082662. - Omar E. Pol, Mar 20 2022

			a(11)=1 because we have 11=5+6; a(21)=2 because we have 21=10+11=1+2+3+4+5+6; a(75)=3 because we have 75=37+38=10+11+12+13+14+15=3+4+5+6+7+8+9+10+11+12.

M. D. Hirschhorn and P. M. Hirschhorn, Partitions into Consecutive Parts, Mathematics Magazine, 78:5 (2005), 396-398. [Please do not delete this reference. - N. J. A. Sloane, Dec 16 2020]

Antti Karttunen, Table of n, a(n) for n = 1..65537
M. D. Hirschhorn and P. M. Hirschhorn, Partitions into Consecutive Parts, Mathematics Magazine: 2005, Volume 78, Number 5, Pages: 396-397.

Cf. A082647, A082662, A001227, A237593, A279387, A281005.

G:=sum(x^(k*(2*k+1))/(1-x^(2*k)), k=1..10): Gser:=series(G,x=0,85): seq(coeff(Gser,x,n),n=1..80); # Emeric Deutsch, Sep 08 2007
A131576 := proc(n) local dvs,a,k,r; dvs := numtheory[divisors](n) ; a := 0 ; for k in dvs do r := n/k+1 ; if r mod 2 = 0 then if r/2-k >= 1 then a := a+1 ; fi ; fi ; od: RETURN(a) ; end: seq(A131576(n),n=1..120) ; # R. J. Mathar, Sep 13 2007

With[{m = 105}, Rest@ CoefficientList[Series[Sum[x^(k (2 k + 1))/(1 - x^(2 k)), {k, m}], {x, 0, m}], x]] (* Michael De Vlieger, Mar 04 2018 *)

a(n) = my(s=sqrt(2*n)); sumdiv(n, d, (d % 2) && (d > s)); \\ Michel Marcus, Jan 15 2020

G.f.: Sum_{k>=1} x^(k*(2*k+1))/(1-x^(2*k)). [Corrected by N. J. A. Sloane, Dec 18 2020]
a(A000040(i))=1 for i=1,2,3,... a(A000079(j))=0 for j=0,1,2,3,... - R. J. Mathar, Sep 13 2007
Conjectures: a(n) = (A001227(n) - A067742(n))/2 = A082647(n) - A067742(n). - Omar E. Pol, Feb 22 2017

A281005 Numbers n having at least one odd divisor greater than sqrt(2*n).

Original entry on oeis.org

3, 5, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 84, 85, 86, 87, 89, 90, 91, 92, 93, 94, 95, 97, 98, 99, 100, 101, 102, 103, 105

Offset: 1

Omar E. Pol, Feb 06 2017

nonn

Conjecture 1: also numbers n such that the symmetric representation of sigma(n) has at least one pair of equidistant subparts.
Conjecture 2: the number of pairs of equidistant subparts in the symmetric representation of sigma(k) equals the number of odd divisors of k greater than sqrt(2*k), with k >= 1.
For more information about the subparts see A279387.

			18 is in the sequence because one of its odd divisors is 9, and 9 is greater than 6, the square root of 2*18.
On the other hand the symmetric representation of sigma(18) has only one part of size 39, which is formed by a central subpart of size 35 and a pair of equidistant subparts [2, 2]. Since there is at least one pair of equidistant subparts, so 18 is in the sequence.
From _Omar E. Pol_, Dec 18 2020: (Start)
The 17th row of triangle A237593 is [9, 4, 2, 1, 1, 1, 1, 2, 4, 9] and the 18th row of the same triangle is [10, 3, 2, 2, 1, 1, 2, 2, 3, 10], so the diagram of the symmetric representation of sigma(18) = 39 is constructed as shown below in figure 1:
.                                     _                                      _
.                                    | |                                    | |
.                                    | |                                    | |
._                                   | |                                    | |
.                                    | |                                    | |
.                                    | |                                    | |
.                                    | |                                    | |
.                                    | |                                    | |
.                                    | |                                    | |
.                             _ _ _ _| |                             _ _ _ _| |
.                            |    _ _ _|                            |  _ _ _ _|
.                           _|   |                                 _| | |
.                         _|  _ _|                               _|  _|_|
.                     _ _|  _|                               _ _|  _|    2
.                    |     |  39                            |  _ _|
.                    |  _ _|                                | |_ _|
.                    | |                                    | |    2
.   _ _ _ _ _ _ _ _ _| |                   _ _ _ _ _ _ _ _ _| |
.  |_ _ _ _ _ _ _ _ _ _|                  |_ _ _ _ _ _ _ _ _ _|
.                                                              35
.
.   Figure 1. The symmetric               Figure 2. After the dissection
.   representation of sigma(18)           of the symmetric representation
.   has one part of size 39.              of sigma(18) into layers of
.                                         width 1 we can see three subparts.
.                                         The first layer has one subpart of
.                                         size 35. The second layer has
.                                         two equidistant subparts of size 2,
.                                         so 18 is in the sequence.
(End)

Indranil Ghosh, Table of n, a(n) for n = 1..1000

Complement of A082662.
Indices of positive terms in A131576.
Cf. A000203, A001227, A067742, A071561, A196020, A235791, A236104, A237048, A237270, A237271, A237571, A237593, A244050, A245092, A249351, A262626, A279387, A280850, A280851, A296508.

[k:k in [1..110] | not forall{d:d in Divisors(k)| IsEven(d) or d le Sqrt(2*k)}]; // Marius A. Burtea, Jan 15 2020

Select[Range@ 120, Count[Divisors@ #, d_ /; And[OddQ@ d, d > Sqrt[2 #]]] > 0 &] (* Michael De Vlieger, Feb 07 2017 *)

isok(n) = my(s=sqrt(2*n)); sumdiv(n, d, (d % 2) && (d > s)) > 0; \\ Michel Marcus, Jan 15 2020

A370206 Numbers j whose symmetric representation of sigma(j) consists of two copies of unimodal width pattern 121 separated by 0.

Original entry on oeis.org

78, 102, 114, 138, 174, 186, 222, 246, 258, 282, 318, 348, 354, 366, 372, 402, 426, 438, 444, 474, 492, 498, 516, 534, 564, 582, 606, 618, 636, 642, 654, 678, 708, 732, 762, 786, 804, 820, 822, 834, 852, 860, 876, 894, 906, 940, 942, 948, 978, 996, 1002, 1038, 1060, 1068, 1074

Offset: 1

Hartmut F. W. Hoft, Feb 11 2024

nonn

Each term has 4 odd divisors and has the form 2^k * p * q, k > 0, p and q prime,  2 < p < 2^(k+1) <  2^(k+1) * p < q. The inequalities ensure that the four 1's in row a(n) of triangle in A237048 are in positions 1, p, 2^(k+1), and 2^(k+1) * p <= floor( (sqrt(8*a(n)+1) - 1)/2 ) < q and establish width pattern 1210 in SRS(a(n)) up to the diagonal. Also since p < 2^(k+1), numbers of the form 2^k * p^3 force p^2 < 2^(k+1) * p which creates a width pattern of the form 1212121.
When a(n) satisfies q = 2^(k+1) * p + 1 it is the smallest number with prime factor p  whose two parts of SRS(a(n)) meet at the diagonal since in this case 2^(k+1) * p = floor( (sqrt(8*a(n)+1) - 1)/2 ). The first 4 numbers with p = 3 are  2* 3 * 13 = 78, 2^4 * 3 * 97 = 4656, 2^5 * 3 * 193 = 18528 and 2^7 * 3 * 769 = 295296. The smallest number with prime factor p = 47 has 355 digits.
Conjecture: The subsequence of numbers m whose two parts of SRS(m) meet at the diagonal is infinite.

			a(1) = 78 = 2 * 3 * 13 = A262259(3) and SRS(78) consists of 2 unimodal parts of width pattern 121 that meet at diagonal position (54, 54).
a(38) = 4 * 5 * 41 = 820 = A262259(6)  is the smallest number in the sequence divisible by 5 and the two parts of SRS(a(38)) meet at diagonal position (570, 570).

Cf. A082662, A235791, A237048, A237270, A237271, A237591, A237593, A249223, A262045, A262259, A341969, A342592, A342594, A342595, A342596, A367377, A370205.

(* function based on conditions for the odd divisors - fast computation *)
a370206Q[n_] := Module[{f=FactorInteger[n], d=Divisors[NestWhile[#/2&, n, EvenQ[#]&]]}, Length[f]==3&&f[[1, 1]]==2&&Length[d]==4&&f[[2, 1]]<2^(f[[1, 2]]+1)&&2^(f[[1, 2]]+1)*f[[2, 1]]A367377 - slow computation *)
a370206[m_, n_] :=  Select[Range[m, n], widthPattern[#]=={1, 2, 1, 0, 1, 2, 1}&]
a370206[1,1074]

A280849 Square array T(j,k) read by antidiagonals upwards, in which column k lists the numbers n having k odd divisors greater than sqrt(2*n), with j >= 1, k >= 0.

Original entry on oeis.org

1, 2, 3, 4, 5, 21, 6, 7, 27, 75, 8, 9, 33, 135, 105, 12, 10, 39, 147, 189, 315, 16, 11, 45, 165, 225, 525, 495, 20, 13, 51, 171, 297, 675, 585, 945, 24, 14, 55, 175, 351, 693, 765, 1155, 1575, 28, 15, 57, 195, 385, 735, 855, 1365, 2475, 2835, 32, 17, 63, 207, 405, 819, 1071, 1485, 2625

Offset: 1

Omar E. Pol, Feb 15 2017

Conjecture: column k lists also the numbers n having k pairs of equidistant subparts in the symmetric representation of sigma(n).
For more information about the "subparts" see A279387.
This sequence is a permutation of the natural numbers.

			The upper-left corner of the square array begins:
   1,  3, 21,  75, 105, 315, 495,  945, 1575, 2835, ...
   2,  5, 27, 135, 189, 525, 585, 1155, 2475, ...
   4,  7, 33, 147, 225, 675, 765, 1365, ...
   6,  9, 39, 165, 297, 693, 855, ...
   8  10, 45, 171, 351, 735, ...
  12, 11, 51, 175, 385, ...
  16, 13, 55, 195, ...
  20, 14, 57, ...
  24, 15, ...
  28, ...
  ...

Index entries for sequences that are permutations of the natural numbers

Row 1 gives A281008.
Column 0 gives A082662. The rest of the terms are in A281005 in increasing order.
Cf. A000203, A001227, A067742, A131576, A196020, A235791, A236104, A237048, A237270, A237271, A237591, A237593, A244050, A245092, A261699, A262626, A279387, A280940.

jMax = 11; nMax = 5000; cnt[n_] := cnt[n] = DivisorSum[n, Boole[OddQ[#] && # > Sqrt[2n]]&]; col[k_] := Select[Range[nMax], cnt[#] == k&]; T[j_, k_] := col[k][[j]]; Table[T[j-k, k], {j, 1, jMax}, {k, 0, j-1}] // Flatten (* Jean-François Alcover, Feb 16 2017 *)

A370205 Numbers j whose symmetric representation of sigma(j) consists of the single unimodal width pattern 121.

Original entry on oeis.org

6, 12, 20, 24, 28, 40, 48, 56, 80, 88, 96, 104, 112, 160, 176, 192, 208, 224, 272, 304, 320, 352, 368, 384, 416, 448, 464, 496, 544, 608, 640, 704, 736, 768, 832, 896, 928, 992, 1088, 1184, 1216, 1280, 1312, 1376, 1408, 1472, 1504, 1536, 1664, 1696, 1792, 1856, 1888, 1952, 1984

Offset: 1

Hartmut F. W. Hoft, Feb 11 2024

nonn

Every term has 2 odd divisors and has the form 2^k * p, k > 0, p prime and 2 < p < 2^(k+1), and therefore is a subsequence of A082662. The two 1's in row a(n) of the triangle of A237048 occur in positions 1 and p up to the diagonal since p <= floor( (sqrt(8*a(n) + 1) - 1)/2 ) < 2^(k+1) which represents the unimodal width pattern 121 in SRS(a(n)).

Numbers in this sequence divisible by 5 have the form 2^(k+2) * 5, k >= 0, the least being a(3) = 20.

Cf. A082662, A235791, A237048, A237270, A237271, A237591, A237593, A249223, A262045, A341969, A342592, A342594, A342595, A342596.

(* function based on conditions for the odd divisors - fast computation *)
a370205Q[n_] := Module[{p=NestWhile[#/2&, n, EvenQ[#]&]}, PrimeQ[p]&&p^2<2n]
a370205[m_, n_] := Select[Range[m, n], a370205Q]
a370205[1, 1984]
(* widthPattern[ ] and support functions are defined in A341969 - slow computation *)
a370205[m_, n_] := Select[Range[m, n], widthPattern[#]=={1, 2, 1}&]
a370205[1, 1984]

A370209 a(n) is the smallest number of the form 2^k * p * (2^(k+1) * p + 1) where 2 < p < 2^(k+1) is the n-th prime and 2^(k+1) * p + 1 is prime, or -1 if no such number exists.

Original entry on oeis.org

78, 820, 6328, 62128, 5539456, 155155972096, 739936, 69342976, 431056, 31494016, 44864128, 3525354496, 3788128

Offset: 2

Hartmut F. W. Hoft, Feb 11 2024

nonn

a(n) is the smallest number of the form described above whose symmetric representation of sigma, SRS(a(n)), consists of 2 parts that have a unimodal width pattern of type 121 and that meet at the diagonal. Since floor( (sqrt(8*a(n) + 1) - 1)/2 ) = 2^(k+1) * p, the central 0 width extent of SRS(a(n)) equals 0.

Conjecture: The sequence is infinite.

			a(2) = 78 = 2 * 3 * 13 = A262259(3) and SRS(78) consists of 2 unimodal parts 121 that meet at diagonal position (54, 54).
a(4) = 6328 = 8 * 7 * 113 = A262259(11) which demonstrates that  2^k < p < 2^(k+1) need not be true.
a(15) with k = 582 and p = 47, its second prime factor 2^(k+1) * p + 1 has 178 digits so that a(15) has 355 digits.
a(16) = 24129129742336 = 2^16 * 53 * 6946817.
Table of records of number of digits a(2) through a(500):
sequence index    2  3  4  5  6   7   15    76   419    438
number of digits  2  3  4  5  7  12  355  3854  5856  20049

Chai Wah Wu, Table of n, a(n) for n = 2..75 [Notice that a(15) has 355 digits, so the graph cuts off after 14 terms. - _N. J. A. Sloane_, Feb 18 2024]

Cf. A082662, A235791, A237048, A237270, A237271, A237591, A237593, A249223, A262045, A262259, A341969, A342592, A342594, A342595, A342596, A367377, A370205, A370206.

minExp[p_] := Module[{k=Floor[Log[2, p]]}, NestWhile[#+1&, k+1, !PrimeQ[2^# p+1]&]-1]/;PrimeQ[p]
a370209[p_] := Module[{k=minExp[p]}, 2^k p(2^(k+1)p+1)]/;PrimeQ[p]
Map[a370209[Prime[#]]&, Range[2, 14]] (* a(15) is too large to list *)

from itertools import count
from sympy import prime, isprime
def A370209(n):
    p = prime(n)
    return next((p<Chai Wah Wu, Feb 17 2024

a(n) = min( 2^k * p * (2^(k+1) * p + 1) : p = prime(n), 2 < p < 2^(k+1), 2^(k+1) * p + 1 is prime ), n>=2.

A327188 For any n >= 0: consider the different ways to split the binary representation of n into two (possibly empty) parts, say with value x and y; a(n) is the greatest possible value of x AND y (where AND denotes the bitwise AND operator).

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 2, 2, 0, 1, 2, 3, 0, 1, 2, 2, 0, 1, 2, 2, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 2, 4, 4, 4, 4, 0, 1, 2, 2, 4, 5, 4, 5, 0, 1, 2, 3, 4, 4, 6, 6, 0, 1, 2, 3, 4, 5, 6, 7, 0, 1, 2, 2, 4, 4, 4, 4, 0, 1, 2, 2, 4, 4, 4, 4, 0, 1, 2, 2, 4, 5, 4

Offset: 0

Rémy Sigrist, Aug 25 2019

The first 10000 positive integers where the sequence equals zero match the first 10000 terms of A082662; is that true forever?

			For n=42:
- the binary representation of 42 is "101010",
- there are 7 ways to split it:
   - "" and "101010": x=0 and y=42: 0 AND 42 = 0,
   - "1" and "01010": x=1 and y=10: 1 AND 10 = 0,
   - "10" and "1010": x=2 and y=10: 2 AND 10 = 2,
   - "101" and "010": x=5 and y=2: 5 AND 2 = 0,
   - "1010" and "10": x=10 and y=2: 10 AND 2 = 2,
   - "10101" and "0": x=21 and y=0: 21 AND 0 = 0,
   - "101010" and "": x=42 and y=0: 42 AND 0 = 0,
- hence a(42) = 2.

Rémy Sigrist, Table of n, a(n) for n = 0..8192

See A327186 for other variants.

Cf. A082662.

a(n) = my (v=-oo, b=binary(n)); for (w=0, #b, v=max(v, bitand(fromdigits(b[1..w],2), fromdigits(b[w+1..#b],2)))); v

A352505 Sum of all parts of all partitions of n into an even number of consecutive parts.

Original entry on oeis.org

0, 0, 3, 0, 5, 0, 7, 0, 9, 10, 11, 0, 13, 14, 15, 0, 17, 18, 19, 0, 42, 22, 23, 0, 25, 26, 54, 0, 29, 30, 31, 0, 66, 34, 35, 36, 37, 38, 78, 0, 41, 42, 43, 44, 90, 46, 47, 0, 49, 50, 102, 52, 53, 54, 110, 0, 114, 58, 59, 60, 61, 62, 126, 0, 130, 66, 67, 68, 138, 70, 71, 0

Offset: 1

Omar E. Pol, Mar 19 2022

nonn

			For n = 21 the partitions of 21 into an even number of consecutive parts are [11, 10] and [6, 5, 4, 3, 2, 1], so a(21) = 11 + 10 + 6 + 5 + 4 + 3 + 2 + 1 = 21*2 = 42.

Indices of zero terms give A082662.
Indices of nonzero terms give A281005.
Cf. A066186, A131576, A245579, A285891, A286000, A286001, A299765, A352257.

a(n) = n*A131576(n).

a(n) = A245579(n) - A352257(n).

cp's OEIS Frontend

A116882 A number k is included if (highest odd divisor of k)^2 <= k.

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A082647 Number of ways n can be expressed as the sum of d consecutive positive integers where d>0 is a divisor of n.

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A131576 Number of ways to represent n as a sum of an even number of consecutive integers.

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A281005 Numbers n having at least one odd divisor greater than sqrt(2*n).

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A370206 Numbers j whose symmetric representation of sigma(j) consists of two copies of unimodal width pattern 121 separated by 0.

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A280849 Square array T(j,k) read by antidiagonals upwards, in which column k lists the numbers n having k odd divisors greater than sqrt(2*n), with j >= 1, k >= 0.

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A370205 Numbers j whose symmetric representation of sigma(j) consists of the single unimodal width pattern 121.

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