This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A237271 #272 Aug 19 2025 09:32:12
%S A237271 1,1,2,1,2,1,2,1,3,2,2,1,2,2,3,1,2,1,2,1,4,2,2,1,3,2,4,1,2,1,2,1,4,2,
%T A237271 3,1,2,2,4,1,2,1,2,2,3,2,2,1,3,3,4,2,2,1,4,1,4,2,2,1,2,2,5,1,4,1,2,2,
%U A237271 4,3,2,1,2,2,4,2,3,2,2,1,5,2,2,1,4,2,4,1,2,1
%N A237271 Number of parts in the symmetric representation of sigma(n).
%C A237271 The diagram of the symmetry of sigma has been via A196020 --> A236104 --> A235791 --> A237591 --> A237593.
%C A237271 For more information see A237270.
%C A237271 a(n) is also the number of terraces at n-th level (starting from the top) of the stepped pyramid described in A245092. - _Omar E. Pol_, Apr 20 2016
%C A237271 a(n) is also the number of subparts in the first layer of the symmetric representation of sigma(n). For the definion of "subpart" see A279387. - _Omar E. Pol_, Dec 08 2016
%C A237271 Note that the number of subparts in the symmetric representation of sigma(n) equals A001227(n), the number of odd divisors of n. (See the second example). - _Omar E. Pol_, Dec 20 2016
%C A237271 From _Hartmut F. W. Hoft_, Dec 26 2016: (Start)
%C A237271 Using odd prime number 3, observe that the 1's in the 3^k-th row of the irregular triangle of A237048 are at index positions
%C A237271 3^0 < 2*3^0 < 3^1 < 2*3^1 < ... < 2*3^((k-1)/2) < 3^(k/2) < ...
%C A237271 the last being 2*3^((k-1)/2) when k is odd and 3^(k/2) when k is even. Since odd and even index positions alternate, each pair (3^i, 2*3^i) specifies one part in the symmetric representation with a center part present when k is even. A straightforward count establishes that the symmetric representation of 3^k, k>=0, has k+1 parts. Since this argument is valid for any odd prime, every positive integer occurs infinitely many times in the sequence. (End)
%C A237271 a(n) = number of runs of consecutive nonzero terms in row n of A262045. - _N. J. A. Sloane_, Jan 18 2021
%C A237271 Indices of odd terms give A071562. Indices of even terms give A071561. - _Omar E. Pol_, Feb 01 2021
%C A237271 a(n) is also the number of prisms in the three-dimensional version of the symmetric representation of k*sigma(n) where k is the height of the prisms, with k >= 1. - _Omar E. Pol_, Jul 01 2021
%C A237271 With a(1) = 0; a(n) is also the number of parts in the symmetric representation of A001065(n), the sum of aliquot parts of n. - _Omar E. Pol_, Aug 04 2021
%C A237271 The parity of this sequence is also the characteristic function of numbers that have middle divisors. - _Omar E. Pol_, Sep 30 2021
%C A237271 a(n) is also the number of polycubes in the 3D-version of the ziggurat of order n described in A347186. - _Omar E. Pol_, Jun 11 2024
%C A237271 Conjecture 1: a(n) is the number of odd divisors of n except the "e" odd divisors described in A005279. Thus a(n) is the length of the n-th row of A379288. - _Omar E. Pol_, Dec 21 2024
%C A237271 The conjecture 1 was checked up n = 10000 by _Amiram Eldar_. - _Omar E. Pol_, Dec 22 2024
%C A237271 The conjecture 1 is true. For a proof see A379288. - _Hartmut F. W. Hoft_, Jan 21 2025
%C A237271 From _Omar E. Pol_, Jul 31 2025: (Start)
%C A237271 Conjecture 2: a(n) is the number of 2-dense sublists of divisors of n.
%C A237271 We call "2-dense sublists of divisors of n" to the maximal sublists of divisors of n whose terms increase by a factor of at most 2.
%C A237271 In a 2-dense sublist of divisors of n the terms are in increasing order and two adjacent terms are the same two adjacent terms in the list of divisors of n.
%C A237271 Example: for n = 10 the list of divisors of 10 is [1, 2, 5, 10]. There are two 2-dense sublists of divisors of 10, they are [1, 2], [5, 10], so a(10) = 2.
%C A237271 The conjecture 2 is essentially the same as the second conjecture in the Comments of A384149. See also _Peter Munn_'s formula in A237270.
%C A237271 The indices where a(n) = 1 give A174973 (2-dense numbers). See the proof there. (End)
%C A237271 Conjecture 3: a(n) is the number of divisors p of n such that p is greater than twice the adjacent previous divisor of n. The divisors p give the n-th row of A379288. - _Omar E. Pol_, Aug 02 2025
%H A237271 Amiram Eldar, <a href="/A237271/b237271.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..5000 from Michel Marcus)
%F A237271 a(n) = A001227(n) - A239657(n). - _Omar E. Pol_, Mar 23 2014
%F A237271 a(p^k) = k + 1, where p is an odd prime and k >= 0. - _Hartmut F. W. Hoft_, Dec 26 2016
%F A237271 Theorem: a(n) <= number of odd divisors of n (cf. A001227). The differences are in A239657. - _N. J. A. Sloane_, Jan 19 2021
%F A237271 a(n) = A340846(n) - A340833(n) + 1 (Euler's formula). - _Omar E. Pol_, Feb 01 2021
%F A237271 a(n) = A000005(n) - A243982(n). - _Omar E. Pol_, Aug 02 2025
%e A237271 Illustration of initial terms (n = 1..12):
%e A237271 ---------------------------------------------------------
%e A237271 n A000203 A237270 a(n) Diagram
%e A237271 ---------------------------------------------------------
%e A237271 . _ _ _ _ _ _ _ _ _ _ _ _
%e A237271 1 1 1 1 |_| | | | | | | | | | | |
%e A237271 2 3 3 1 |_ _|_| | | | | | | | | |
%e A237271 3 4 2+2 2 |_ _| _|_| | | | | | | |
%e A237271 4 7 7 1 |_ _ _| _|_| | | | | |
%e A237271 5 6 3+3 2 |_ _ _| _| _ _|_| | | |
%e A237271 6 12 12 1 |_ _ _ _| _| | _ _|_| |
%e A237271 7 8 4+4 2 |_ _ _ _| |_ _|_| _ _|
%e A237271 8 15 15 1 |_ _ _ _ _| _| |
%e A237271 9 13 5+3+5 3 |_ _ _ _ _| | _|
%e A237271 10 18 9+9 2 |_ _ _ _ _ _| _ _|
%e A237271 11 12 6+6 2 |_ _ _ _ _ _| |
%e A237271 12 28 28 1 |_ _ _ _ _ _ _|
%e A237271 ...
%e A237271 For n = 9 the sum of divisors of 9 is 1+3+9 = A000203(9) = 13. On the other hand the 9th set of symmetric regions of the diagram is formed by three regions (or parts) with 5, 3 and 5 cells, so the total number of cells is 5+3+5 = 13, equaling the sum of divisors of 9. There are three parts: [5, 3, 5], so a(9) = 3.
%e A237271 From _Omar E. Pol_, Dec 21 2016: (Start)
%e A237271 Illustration of the diagram of subparts (n = 1..12):
%e A237271 ---------------------------------------------------------
%e A237271 n A000203 A279391 A001227 Diagram
%e A237271 ---------------------------------------------------------
%e A237271 . _ _ _ _ _ _ _ _ _ _ _ _
%e A237271 1 1 1 1 |_| | | | | | | | | | | |
%e A237271 2 3 3 1 |_ _|_| | | | | | | | | |
%e A237271 3 4 2+2 2 |_ _| _|_| | | | | | | |
%e A237271 4 7 7 1 |_ _ _| _ _|_| | | | | |
%e A237271 5 6 3+3 2 |_ _ _| |_| _ _|_| | | |
%e A237271 6 12 11+1 2 |_ _ _ _| _| | _ _|_| |
%e A237271 7 8 4+4 2 |_ _ _ _| |_ _|_| _ _ _|
%e A237271 8 15 15 1 |_ _ _ _ _| _| _| |
%e A237271 9 13 5+3+5 3 |_ _ _ _ _| | _| _|
%e A237271 10 18 9+9 2 |_ _ _ _ _ _| |_ _|
%e A237271 11 12 6+6 2 |_ _ _ _ _ _| |
%e A237271 12 28 23+5 2 |_ _ _ _ _ _ _|
%e A237271 ...
%e A237271 For n = 6 the symmetric representation of sigma(6) has two subparts: [11, 1], so A000203(6) = 12 and A001227(6) = 2.
%e A237271 For n = 12 the symmetric representation of sigma(12) has two subparts: [23, 5], so A000203(12) = 28 and A001227(12) = 2. (End)
%e A237271 From _Hartmut F. W. Hoft_, Dec 26 2016: (Start)
%e A237271 Two examples of the general argument in the Comments section:
%e A237271 Rows 27 in A237048 and A249223 (4 parts)
%e A237271 i: 1 2 3 4 5 6 7 8 9 . . 12
%e A237271 27: 1 1 1 0 0 1 1's in A237048 for odd divisors
%e A237271 1 27 3 9 odd divisors represented
%e A237271 27: 1 0 1 1 1 0 0 1 1 1 0 1 blocks forming parts in A249223
%e A237271 Rows 81 in A237048 and A249223 (5 parts)
%e A237271 i: 1 2 3 4 5 6 7 8 9 . . 12. . . 16. . . 20. . . 24
%e A237271 81: 1 1 1 0 0 1 0 0 1 0 0 0 1's in A237048 f.o.d
%e A237271 1 81 3 27 9 odd div. represented
%e A237271 81: 1 0 1 1 1 0 0 0 1 1 1 1 1 1 1 1 0 0 0 1 1 1 0 1 blocks fp in A249223
%e A237271 (End)
%t A237271 a237271[n_] := Length[a237270[n]] (* code defined in A237270 *)
%t A237271 Map[a237271, Range[90]] (* data *)
%t A237271 (* _Hartmut F. W. Hoft_, Jun 23 2014 *)
%t A237271 a[n_] := Module[{d = Partition[Divisors[n], 2, 1]}, 1 + Count[d, _?(OddQ[#[[2]]] && #[[2]] >= 2*#[[1]] &)]]; Array[a, 100] (* _Amiram Eldar_, Dec 22 2024 *)
%o A237271 (PARI) fill(vcells, hga, hgb) = {ic = 1; for (i=1, #hgb, if (hga[i] < hgb[i], for (j=hga[i], hgb[i]-1, cell = vector(4); cell[1] = i - 1; cell[2] = j; vcells[ic] = cell; ic ++;););); vcells;}
%o A237271 findfree(vcells) = {for (i=1, #vcells, vcelli = vcells[i]; if ((vcelli[3] == 0) && (vcelli[4] == 0), return (i));); return (0);}
%o A237271 findxy(vcells, x, y) = {for (i=1, #vcells, vcelli = vcells[i]; if ((vcelli[1]==x) && (vcelli[2]==y) && (vcelli[3] == 0) && (vcelli[4] == 0), return (i));); return (0);}
%o A237271 findtodo(vcells, iz) = {for (i=1, #vcells, vcelli = vcells[i]; if ((vcelli[3] == iz) && (vcelli[4] == 0), return (i)); ); return (0);}
%o A237271 zcount(vcells) = {nbz = 0; for (i=1, #vcells, nbz = max(nbz, vcells[i][3]);); nbz;}
%o A237271 docell(vcells, ic, iz) = {x = vcells[ic][1]; y = vcells[ic][2]; if (icdo = findxy(vcells, x-1, y), vcells[icdo][3] = iz); if (icdo = findxy(vcells, x+1, y), vcells[icdo][3] = iz); if (icdo = findxy(vcells, x, y-1), vcells[icdo][3] = iz); if (icdo = findxy(vcells, x, y+1), vcells[icdo][3] = iz); vcells[ic][4] = 1; vcells;}
%o A237271 docells(vcells, ic, iz) = {vcells[ic][3] = iz; while (ic, vcells = docell(vcells, ic, iz); ic = findtodo(vcells, iz);); vcells;}
%o A237271 nbzb(n, hga, hgb) = {vcells = vector(sigma(n)); vcells = fill(vcells, hga, hgb); iz = 1; while (ic = findfree(vcells), vcells = docells(vcells, ic, iz); iz++;); zcount(vcells);}
%o A237271 lista(nn) = {hga = concat(heights(row237593(0), 0), 0); for (n=1, nn, hgb = heights(row237593(n), n); nbz = nbzb(n, hga, hgb); print1(nbz, ", "); hga = concat(hgb, 0););} \\ with heights() also defined in A237593; \\ _Michel Marcus_, Mar 28 2014
%o A237271 (Python)
%o A237271 from sympy import divisors
%o A237271 def a(n: int) -> int:
%o A237271 divs = list(divisors(n))
%o A237271 d = [divs[i:i+2] for i in range(len(divs) - 1)]
%o A237271 s = sum(1 for pair in d if len(pair) == 2 and pair[1] % 2 == 1 and pair[1] >= 2 * pair[0])
%o A237271 return s + 1
%o A237271 print([a(n) for n in range(1, 80)]) # _Peter Luschny_, Aug 05 2025
%Y A237271 Row lengths of A237270 and of A379288.
%Y A237271 Column 1 of A279387.
%Y A237271 Partial sums give A237590.
%Y A237271 Parity gives A347950.
%Y A237271 Cf. A000203, A000265, A001065, A001227, A005279, A024916, A060831, A061345, A067742, A071561, A071562, A175254, A196020, A221529, A235791, A236104, A237048, A237591, A237593, A239657, A244050, A244971, A245092, A249223, A250068, A261699, A262045, A262612, A262626, A274824, A279387, A279693, A319073, A340583, A340846, A342344, A347186, A379288.
%Y A237271 Cf. A027750, A174973 (2-dense numbers), A239663, A240062, A243982, A379379, A380580, A384149, A384222, A384225, A384226, A384230, A384930.
%K A237271 nonn
%O A237271 1,3
%A A237271 _Omar E. Pol_, Feb 25 2014