A241010 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.
1, 2, 4, 8, 9, 16, 25, 32, 49, 50, 64, 81, 98, 121, 128, 169, 242, 256, 289, 338, 361, 484, 512, 529, 578, 625, 676, 722, 729, 841, 961, 1024, 1058, 1156, 1250, 1369, 1444, 1681, 1682, 1849, 1922, 2048, 2116, 2209, 2312, 2401, 2738, 2809, 2888, 3025, 3249, 3362, 3364, 3481, 3698, 3721, 3844
Offset: 1
Keywords
Examples
This irregular triangle presents in each column those elements of the sequence that have the same factor of a power of 2. row/col 2^0 2^1 2^2 2^3 2^4 2^5 ... 2^k: 1 2 4 8 16 32 ... 3^2: 9 5^2: 25 50 7^2: 49 98 3^4: 81 11^2: 121 242 484 13^2: 169 338 676 17^2: 289 578 1156 2312 19^2: 361 722 1444 2888 23^2: 529 1058 2116 4232 5^4: 625 1250 3^6: 729 29^2: 841 1682 3364 6728 31^2: 961 1922 3844 7688 37^2: 1369 2738 5476 10952 21904 41^2: 1681 3362 6724 13448 26896 43^2: 1849 3698 7396 14792 29584 47^2: 2209 4418 8836 17672 35344 7^4: 2401 4802 53^2: 2809 5618 11236 22472 44944 5^2*11^2: 3025 3^2*19^2: 3249 59^2: 3481 6962 13924 27848 55696 61^2: 3721 7442 14884 29768 59536 67^2: 4489 8978 17956 35912 71824 143648 3^2*23^2: 4761 71^2: 5041 ... 5^2*101^2:225025 510050 ... 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. 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. The first row is A000079. The rows, except the first, are indexed by products of even powers of the odd primes satisfying the property, sorted in increasing order. The first column is a subsequence of A244579. A row labeled p^(2*h), h>=1 and p>=3 with p = A000040(n), has A098388(n) entries. Starting with the second column, dividing the entries of a column by 2 creates a proper subsequence of the prior column. See A259417 for references to other sequences of even powers of odd primes that are subsequences of column 1. 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.
Links
- Hartmut F. W. Hoft, Table of n, a(n) for n = 1..563 (values less than 1000000)
- Hartmut F. W. Hoft, Proof of characterization theorem
- Hartmut F. W. Hoft, Table of n, a(n) for n = 1...706(values less than 1000000)
- Hartmut F. W. Hoft, Illustration of the symmetric representation of sigma(a(n)), for n=1..15
- Hartmut F. W. Hoft, Proof of property for n and formula for regions of sigma(n)
Crossrefs
Programs
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Mathematica
(* path[n] and a237270[n] are defined in A237270 *) atmostOneDiagonalsQ[n_] := SubsetQ[{0, 1}, Union[Flatten[Drop[Drop[path[n], 1], -1] - path[n-1], 1]]] Select[Range[1000], atmostOneDiagonalsQ[#] && OddQ[Length[a237270[#]]]&] (* data *) (* more efficient code based on numeric characterization *) divisorPairsQ[m_, q_] := Module[{d = Divisors[q]}, Select[2^(m + 1)*Most[d] - Rest[d], # >= 0 &] == {}] 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])] a241010Alt[m_,n_] := Select[Range[m, n], a241010AltQ] a241010Alt[1,4000] (* data *)
Formula
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.
Extensions
More terms and further edited by Hartmut F. W. Hoft, Jun 26 2015 and Jul 02 2015 and corrected Oct 11 2015
Comments