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%I #72 Sep 28 2018 23:24:45
%S 1,2,4,8,9,16,25,32,49,50,64,81,98,121,128,169,242,256,289,338,361,
%T 484,512,529,578,625,676,722,729,841,961,1024,1058,1156,1250,1369,
%U 1444,1681,1682,1849,1922,2048,2116,2209,2312,2401,2738,2809,2888,3025,3249,3362,3364,3481,3698,3721,3844
%N Numbers n with the property that the number of parts in the symmetric representation of sigma(n) is odd, and that all parts have width 1.
%C The first eight entries in A071562 but not in this sequence are 6, 12, 15, 18, 20, 24, 28 & 30.
%C The first eight entries in A238443 but not in this sequence are 6, 12, 18, 20, 24, 28, 30 & 36.
%C The union of A241008 and of this sequence equals A174905 (for a proof see link in A174905).
%C Let n = 2^m * product(p_i^e_i, i=1,...,k) = 2^m * q with m >= 0, k >= 0, 2 < p_1, ...< p_k primes and e_i >= 1, for all 1 <= i <= k. For each number n in this sequence all e_i are even, and for any two odd divisors f < g of n, 2^(m+1) * f < g. The sum of the areas of the regions r(n, z) equals sigma(n). For a proof of the characterization and the formula see the theorem in the link below.
%C Numbers 3025 = 5^2 * 11^2 and 510050 = 2^1 * 5^2 * 101^2 are the smallest odd and even numbers, respectively, in the sequence with two distinct odd prime factors.
%C Among the 706 numbers in the sequence less than 1000000 (see link to the table) there are 143 that have two different odd prime factors, but none with three. All numbers with three different odd prime factors are larger than 500000000. Number 4450891225 = 5^2 * 11^2 * 1213^2 is in the sequence, but may not be the smallest one with three different odd prime factors. Note that 1213 is the first prime that extends the list of divisors of 3025 while preserving the property for numbers in this sequence.
%C The subsequence of numbers n = 2^(k-1) * p^2 satisfying the constraints above is A247687.
%C n = 3^(2*h) = 9^h = A001019(h), h>=0, is the smallest number for which the symmetric representation of sigma(n) has 2*h+1 regions of width one, for example for h = 1, 2, 3 and 5, but not for h = 4 in which case 3025 = 5^2 * 11^2 < 3^8 = 6561 is the smallest (see A318843). [Comment corrected by _Hartmut F. W. Hoft_, Sep 04 2018]
%C Computations using this characterization are more efficient than those of Dyck paths for the symmetric representations of sigma(n), e.g., the Mathematica code below.
%H Hartmut F. W. Hoft, <a href="/A241010/a241010.txt">Table of n, a(n) for n = 1..563</a> (values less than 1000000)
%H Hartmut F. W. Hoft, <a href="/A241010/a241010.pdf">Proof of characterization theorem</a>
%H Hartmut F. W. Hoft, <a href="/A241010/a241010_1.txt">Table of n, a(n) for n = 1...706(values less than 1000000)</a>
%H Hartmut F. W. Hoft, <a href="/A241010/a241010_1.pdf">Illustration of the symmetric representation of sigma(a(n)), for n=1..15</a>
%H Hartmut F. W. Hoft, <a href="/A241010/a241010_2.pdf">Proof of property for n and formula for regions of sigma(n)</a>
%F Formula for the z-th region in the symmetric representation of n = 2^m * q in this sequence, 1 <= z <= sigma_0(q) and q odd: r(n, z) = 1/2 * (2^(m+1) - 1) * (d_z + d_(2*x+2-z)) where 1 = d_1 < ... < d_(2*x+1) = q are the odd divisors of n.
%e This irregular triangle presents in each column those elements of the sequence that have the same factor of a power of 2.
%e row/col 2^0 2^1 2^2 2^3 2^4 2^5 ...
%e 2^k: 1 2 4 8 16 32 ...
%e 3^2: 9
%e 5^2: 25 50
%e 7^2: 49 98
%e 3^4: 81
%e 11^2: 121 242 484
%e 13^2: 169 338 676
%e 17^2: 289 578 1156 2312
%e 19^2: 361 722 1444 2888
%e 23^2: 529 1058 2116 4232
%e 5^4: 625 1250
%e 3^6: 729
%e 29^2: 841 1682 3364 6728
%e 31^2: 961 1922 3844 7688
%e 37^2: 1369 2738 5476 10952 21904
%e 41^2: 1681 3362 6724 13448 26896
%e 43^2: 1849 3698 7396 14792 29584
%e 47^2: 2209 4418 8836 17672 35344
%e 7^4: 2401 4802
%e 53^2: 2809 5618 11236 22472 44944
%e 5^2*11^2: 3025
%e 3^2*19^2: 3249
%e 59^2: 3481 6962 13924 27848 55696
%e 61^2: 3721 7442 14884 29768 59536
%e 67^2: 4489 8978 17956 35912 71824 143648
%e 3^2*23^2: 4761
%e 71^2: 5041
%e ...
%e 5^2*101^2:225025 510050
%e ...
%e Number 3025 = 5^2 * 11^2 is in the sequence since its divisors are 1, 5, 11, 25, 55, 121, 275, 605 and 3025. Number 6050 = 2^1 * 5^2 * 11^2 is not in the sequence since 2^2 * 5 > 11 while 5 < 11.
%e Number 510050 = 2^1 * 5^2 * 101^2 is in the sequence since its 9 odd divisors 1, 5, 25, 101, 505, 2525, 10201, 51005 and 225025 are separated by factors larger than 2^2. The areas of its 9 regions are 382539, 76515, 15339, 3939, 1515, 3939, 15339, 76515 and 382539. However, 2^2 * 5^2 * 101^2 is not in the sequence.
%e The first row is A000079.
%e The rows, except the first, are indexed by products of even powers of the odd primes satisfying the property, sorted in increasing order.
%e The first column is a subsequence of A244579.
%e A row labeled p^(2*h), h>=1 and p>=3 with p = A000040(n), has A098388(n) entries.
%e Starting with the second column, dividing the entries of a column by 2 creates a proper subsequence of the prior column.
%e See A259417 for references to other sequences of even powers of odd primes that are subsequences of column 1.
%e The first entry greater than 16 in column labeled 2^4 is 21904 since 37 is the first prime larger than 2^5. The rightmost entry in the row labeled 19^2 is 2888 in the column labeled 2^3 since 2^4 < 19 < 2^5.
%t (* path[n] and a237270[n] are defined in A237270 *)
%t atmostOneDiagonalsQ[n_] := SubsetQ[{0, 1}, Union[Flatten[Drop[Drop[path[n], 1], -1] - path[n-1], 1]]]
%t Select[Range[1000], atmostOneDiagonalsQ[#] && OddQ[Length[a237270[#]]]&] (* data *)
%t (* more efficient code based on numeric characterization *)
%t divisorPairsQ[m_, q_] := Module[{d = Divisors[q]}, Select[2^(m + 1)*Most[d] - Rest[d], # >= 0 &] == {}]
%t a241010AltQ[n_] := Module[{m, q, p, e}, m=IntegerExponent[n, 2]; q=n/2^m; {p, e} = Transpose[FactorInteger[q]]; q==1||(Select[e, EvenQ]==e && divisorPairsQ[m, q])]
%t a241010Alt[m_,n_] := Select[Range[m, n], a241010AltQ]
%t a241010Alt[1,4000] (* data *)
%Y Cf. A000203, A174905, A236104, A237270 (symmetric representation of sigma(n)), A237271, A237593, A238443, A241008, A071562, A246955, A247687, A250068, A250070, A250071.
%Y Cf. A318843
%K nonn
%O 1,2
%A _Hartmut F. W. Hoft_, Aug 07 2014
%E More terms and further edited by _Hartmut F. W. Hoft_, Jun 26 2015 and Jul 02 2015 and corrected Oct 11 2015