A281005 -id:A281005 - cp's OEIS frontend

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

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

A082662 Numbers k such that the odd part of k is less than sqrt(2k).

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 20, 24, 28, 32, 40, 48, 56, 64, 72, 80, 88, 96, 104, 112, 120, 128, 144, 160, 176, 192, 208, 224, 240, 256, 272, 288, 304, 320, 336, 352, 368, 384, 400, 416, 432, 448, 464, 480, 496, 512, 544, 576, 608, 640, 672, 704, 736, 768, 800

Offset: 1

Naohiro Nomoto, May 18 2003

Theorem: The following eight definitions are equivalent.
(P1) Numbers k such that the odd part of k (A000265(k)) is < sqrt(2k).
  (P1) is the new definition, repeated here for convenience. Note that this is not the same as saying A000265(k) < A172471(k), since A172471(k) = floor(sqrt(2*k)).
(P2) Numbers k such that the odd divisors of k are < sqrt(2k).
  (P2) and (P1) are obviously equivalent.
(P3) The numbers 1, S_0, S_1, S_2, ..., where
  S_m = { 2^(m+1)*(2^m+i) : i = 0 .. 3*2^m - 1 }.
So S_0 = {2,4,6}, S_1 = {8,12,16,20,24,28}, S_2 = {32,40,48,...,120}, S_3 = {128,144,...,496}, ...
  The proof that (P3) and (P1) are the same sequence is not difficult and will be added later. (P3) is equivalent to a formula stated without proof (it may have been only an empirical observation) in the original version of this entry.
(P4) Numbers k such that the odd part of k is <= A003056(k).
  That is, the odd part of k is <= floor((sqrt(1+8*n)-1)/2). It is more difficult to show this is equivalent to (P1), but it is true.
(P5) Numbers k such that the odd divisors of k are <= A003056(k).
  (P5) and (P4) are obviously equivalent.
(P6) Numbers k such that A001227(k) = A082647(k).
  (P6) was the original definition. In words, it says that the number of odd divisors of k is equal to the number of ways to write k as a sum of an odd number of consecutive positive integers, or equivalently as a sum of d consecutive positive integers for some d dividing k.  To show that (P6) is equivalent to (P1) one makes use of the Hirschhorn-Hirschhorn article.
(P7) Numbers k such that the odd part of k is <= the sum of divisors of the even part.
  (P7) was contributed by Jaycob Coleman, Jun 21 2014. To show (P7) is equivalent to (P1), write k as 2^m*s where s is odd.  Equality holds if and only if k is an even perfect number.
(P8) Numbers k such that A000265(k) <= A000203(A006519(k)) or also such that A000265(k) <= A038712(k).
  (P8) was contributed by Michel Marcus, Aug 14 2014. It is a restatement of (P7).
(End of theorem)
A further equivalent property, (P9), follows at once from (P4). This was conjectured by  Omar E. Pol, Apr 18 2017
(P9) These are the numbers k such that the sequence of successive widths in the symmetric representation of sigma(k) is unimodal.
Yet another equivalent property:
(P10) Numbers k >= 1 such if k = i + (i+1) + (i+2) + ... + (i+j-1) for some i >= 1 and j >= 1 then j is odd [Caballero, 2019]. - Michel Marcus, Jan 16 2020
This is a subsequence of A005153. - Jaycob Coleman, Jun 21 2014
The complement of this sequence is A281005. - Omar E. Pol, Apr 18 2017
Subsequence of A174973. - Omar E. Pol, Feb 01 2021

Amiram Eldar, Table of n, a(n) for n = 1..10000
José Manuel Rodríguez Caballero, Integers Which Cannot Be Partitioned Into an Even Number of Consecutive Parts, INTEGERS, Volume 19 (2019), #A20.
M. D. Hirschhorn and P. M. Hirschhorn, Partitions into Consecutive Parts, Mathematics Magazine: 2003, Volume 76, Number 4, pp. 306-308.
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.
Eric Weisstein's World of Mathematics, Even Part
Eric Weisstein's World of Mathematics, Odd Part

Cf. A281005 (complement).

Cf. A000196, A000203, A000265, A001227, A003056, A005153, A006519, A038712, A053644, A082647, A116882, A172471, A174973, A237593, A279387.

cnt[n_] := DivisorSum[n, Boole[OddQ[#] && #>Sqrt[2n]]&]; Select[Range[800], cnt[#]==0&] (* Jean-François Alcover, Feb 16 2017 *)

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

G.f. = 1 + (1/(1-x)^2) * Sum_{m >= 0} (2^(m+1)*x^(3*2^m-2) * ( x^(3*2^m)*(2^(m+2)*(x-1)-x) - 2^m*(x-1) + x ) ). (This follows from (P3).) :w
- N. J. A. Sloane, Feb 02 2021
a(n+1) = a(n) + A053644(A000196(2*a(n))). - Peter Munn, Oct 03 2023

Edited by N. J. A. Sloane, Jan 28 2021: Replaced original indirect definition by simple direct definition; rearranged comments; provided proofs (not yet included here) that the various definitions are equivalent

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 *)

A281009 Number of odd divisors of n minus the number of middle divisors of n.

Original entry on oeis.org

0, 0, 2, 0, 2, 0, 2, 0, 2, 2, 2, 0, 2, 2, 2, 0, 2, 2, 2, 0, 4, 2, 2, 0, 2, 2, 4, 0, 2, 2, 2, 0, 4, 2, 2, 2, 2, 2, 4, 0, 2, 2, 2, 2, 4, 2, 2, 0, 2, 2, 4, 2, 2, 2, 4, 0, 4, 2, 2, 2, 2, 2, 4, 0, 4, 2, 2, 2, 4, 2, 2, 0, 2, 2, 6, 2, 2, 4, 2, 0, 4, 2, 2, 2, 4, 2, 4, 0, 2, 4, 2, 2, 4, 2, 4, 0, 2, 2, 4, 2, 2, 4, 2, 0, 8

Offset: 1

Omar E. Pol, Feb 20 2017

nonn

Conjecture 1: a(n) is also twice the number of odd divisors of n greater than sqrt(2*n).
Conjecture 2: a(n) is also the number of odd divisors of n less than sqrt(2*n) that are not middle divisors of n, plus the number of odd divisors of n greater than sqrt(2*n).
Conjecture 3: a(n) is also the total number of equidistant subparts in the symmetric representation of sigma(n).
The "equidistant subparts" are the subparts that are not the "central subparts".
For more information of the "subparts" see A279387.

			For n = 45 the divisors of 45 are [1, 3, 5, 9, 15, 45]. There are 6 odd divisors, and two of them [5 and 9] are also the middle divisors of 45, so a(45) = 6 - 2 = 4.
Other examples (conjectured):
2) There are two odd divisors of 45 that are greater than the square root of 2*45 = 9.4..., so a(45) = 2*2 = 4.
3) The 45th row of A237593 is [23, 8, 5, 2, 2, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 5, 8, 23], and the 44th row of the same triangle is [23, 8, 4, 3, 2, 1, 1, 2, 2, 1, 1, 2, 3, 4, 8, 23], therefore between both symmetric Dyck paths (described in A237593 and A279387) there are two central subparts [27 and 1] and two pairs of equidistant subparts ([23, 23] and [2, 2]). The total number of equidistant subparts is equal to 4, so a(45) = 4. (the diagram of the symmetric representation of sigma(45) is too large to include).
4) The 45th row of A196020 is [89, 43, 27, 0, 13, 9, 0, 0, 1], hence the 45th row of A280850 is [23, 23, 27, 0, 2, 2, 0, 0, 1]. There are two central subparts [27 and 1] and two pairs of equidistant subparts ([23, 23] and [2, 2]). The total number of equidistant subparts is equal to 4, so a(45) = 4.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001227, A067742, A082647, A131576, A196020, A236104, A237048, A237593, A245092, A249351, A261699, A262626, A279667, A280849, A280850, A280940, A281005, A281007, A281008.

N:= 200: # to get a(1)..a(N)
A:= Vector(N):
for m from 1 to N by 2 do
  R:= [seq(k*m,k=1..N/m)];
  A[R]:= A[R] + Vector(nops(R),1);
od:
for m from 1 to N do
  R:= [seq(k*m, k= floor(m/2)+1..min(2*m,N/m))];
  A[R]:= A[R] - Vector(nops(R),1);
od:
convert(A,list); # Robert Israel, Feb 20 2017

Table[Count[#, d_ /; OddQ@ d] - Count[#, d_ /; Sqrt[n/2] <= d < Sqrt[2 n]] &@ Divisors@ n, {n, 120}] (* Michael De Vlieger, Feb 20 2017 *)

a(n) = A001227(n) - A067742(n).

Conjecture: a(n) = 2*A131576(n).

A281008 Least positive integer k with exactly n odd divisors greater than sqrt(2*k).

Original entry on oeis.org

1, 3, 21, 75, 105, 315, 495, 945, 1575, 2835, 3465, 4095, 11025, 17955, 10395, 23205, 17325, 24255, 31185, 36855, 51975, 61425, 45045, 108675, 143325, 121275, 184275, 155925, 135135, 176715, 239085, 315315, 294525, 225225, 606375, 626535, 405405, 700245, 1531530, 1351350, 2072070, 1289925, 855855

Offset: 0

Omar E. Pol, Feb 16 2017

nonn

Conjecture: a(n) is also the smallest number k having n pairs of equidistant subparts in the symmetric representation of sigma(k).
For more information about the "subparts" see A279387.
Observations about the known terms:
Observation 1: terms a(1)-a(51) are divisible by 3.
Observation 2: terms a(3)-a(51) are divisible by 5.

			a(3) = 75 because the divisors of 75 are [1, 3, 5, 15, 25, 75], and 75 has three odd divisors greater than the square root of 2*75 = 12.2..., and it is the smallest number with that property.
Other examples (conjectured):
2) The 75th row of A237593 is [38, 13, 7, 4, 3, 3, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 3, 3, 4, 7, 13, 38], and the 74th row of the same triangle is [38, 13, 6, 5, 3, 2, 2, 1, 2, 1, 1, 1, 1, 2, 1, 2, 2, 3, 5, 6, 13, 38], therefore between both symmetric Dyck paths (described in A237593 and A279387) there are three pairs of equidistant subparts: [38, 38], [21, 21] and [3, 3]. That is the first row with that property, so a(3) = 75. (The diagram of the symmetric representation of sigma(75) is too large to include).
3) The 75th row of A196020 is [149, 73, 47, 0, 25, 19, 0, 0, 0, 5, 0], hence the 75th row of A280850 is [38, 38, 21, 0, 3, 3, 0, 0, 0, 21, 0]. There are three pairs of equidistant subparts [38, 38], [21, 21] and [3, 3]. That is the first row with that property, so a(3) = 75.
4) The 75th row of A237048 is [1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0]. The sum of the even-indexed terms is equal to 3. That is the first row with that property, so a(3) = 75.
5) The 75th row of A261699 is [1, 75, 3, 0, 5, 25, 0, 0, 0, 15, 0]. There are three even-indexed terms that are positive integers: [75, 25, 15]. That is the first row with that property, so a(3) = 75.

Charles R Greathouse IV, Table of n, a(n) for n = 0..200

Row 1 of A280849.

Cf. A000203, A001227, A008585, A067742, A082262, A131576, A196020, A235791, A236104, A237048, A237270, A237271, A237591, A237593, A244050, A245092, A261699, A262626, A279387, A280850, A280940, A281005.

cnt[k_] := cnt[k] = DivisorSum[k, Boole[OddQ[#] && #>Sqrt[2k]]&]; a[n_] := a[n] = For[k = 1, True, k++, If[cnt[k]==n, Return[k]]]; Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 16 2017 *)

a(n,{s=0},{q=1},{k=2},{w=1})={if(n<1,return(1));my(z,ii,F,d,L:list,V,p,ans:list);ans=List();if(q<1,q=1);if(k<2,k=2);while(k++,p=sqrt(2*k);F=factor(k);ii=vecsum(F[1,]);F=F[,1]~;L=List([1]);for(i=1,ii,forvec(y=vector(i,t,[1,#F]),d=prod(u=1,#y,F[y[u]]);if((d<=k)&&!(k%d),listput(L,d)),1));V=Set(Vec(L));if(n==sum(u=1,#V,(V[u]>p)&&(V[u]%2==!!w)),if(s,print1(V","));listput(ans,k);if(z++==q,if(#ans==1,return(k),return(Vec(ans))),n++)))} \\ with n>=1, "s" set to 1 also prints the divisors (of "w" version: 1 odd, 0 even) for the first "q" terms from the n-th, resuming their search with k>=2. - R. J. Cano, Feb 20 2017

a(n)=my(k,s); while(k++, s=sqrtint(2*k); if(sumdiv(k>>valuation(k,2), d, d>s)==n, return(k))) \\ Charles R Greathouse IV, Feb 20 2017

a(10)-a(30) from Jean-François Alcover, Feb 16 2017

a(31)-a(43) from Michael De Vlieger, Feb 18 2017

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

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

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A082662 Numbers k such that the odd part of k is less than sqrt(2k).

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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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A281009 Number of odd divisors of n minus the number of middle divisors of n.

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A281008 Least positive integer k with exactly n odd divisors greater than sqrt(2*k).

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

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