A319796 -id:A319796 - cp's OEIS frontend

A361903 Numbers k for which sqrt(k/2) divides k and the symmetric representation of sigma(k) has a single part.

2, 8, 18, 32, 72, 128, 162, 200, 288, 392, 450, 512, 648, 800, 882, 968, 1152, 1352, 1458, 1568, 1800, 2048, 2178, 2592, 3042, 3200, 3528, 3872, 4050, 4608, 5000, 5202, 5408, 5832, 6272, 6498, 7200, 7938, 8192, 8712, 9248, 9522, 9800, 10368, 11250, 11552, 12168, 12800, 13122, 14112

Offset: 1

Hartmut F. W. Hoft, Mar 28 2023

nonn

Every number in this sequence has the form 2^(2*i + 1) * k^(2*j), i,j>=0, k>=1.

The number of 1's in row a(n) of the triangle in A237048 as well as the length of that row are odd.

			a(4) = 32 has 4 as its single middle divisor, and its symmetric representation of sigma consists of one part of width 1.
a(9) = 288 = 2^5 * 3^2 has 3 middle divisors - 12 = 2^2 * 3 , 16 = 2^4, 18 = 2 * 3^2 - and its symmetric representation of sigma consists of one part, the section of maximum width 3 of the single part includes the diagonal (see also A250068).

Intersection of A001105 and A174973.
Subsequence of A071562 and of A319796.
Cf. A235791, A237048, A237270, A237271, A237593, A250068.

(* Function a237271[ ] is defined in A237271 *)
a361903[n_] := Select[Range[n], IntegerQ[#/Sqrt[#/2]]&&a237271[#]==1&]
a361903[15000]

a(n) = k when A001105(n) = k and A237271(k) = 1.

A361905 Numbers k for which sqrt(k/2) divides k and the width at the diagonal of the symmetric representation of sigma(k) equals 1.

Original entry on oeis.org

2, 8, 18, 32, 50, 98, 128, 162, 200, 242, 338, 392, 512, 578, 722, 882, 968, 1058, 1250, 1352, 1458, 1682, 1922, 2048, 2178, 2312, 2738, 2888, 3042, 3362, 3698, 3872, 4232, 4418, 4802, 5000, 5202, 5408, 5618, 6050, 6498, 6728, 6962, 7442, 7688, 8192, 8450, 8978, 9248, 9522

Offset: 1

Hartmut F. W. Hoft, Mar 28 2023

nonn

Every number in this sequence has the form 2^(2*i + 1) * k^(2*j), i, j >= 0, k >= 1.

The number of 1's in row a(n) of the triangle in A237048 as well as the length of that row are odd.

			a(4) = 32 has 4 as its single middle divisor, and its symmetric representation of sigma consists of one part of width 1.
a(5) = 50 has 5 as its single middle divisor, and its symmetric representation of sigma consists of three parts of width 1.
a(9) = 200 = 2^3 * 5^2 has 10 = 2 * 5 as its single middle divisor, and its symmetric representation of sigma consists of one part of maximum width 2 (A250068), but has width 1 at the diagonal.
a(39) = 6050 = 2^1 * 5^2 * 11^2 has 55 as its single middle divisor; it is the first number in the sequence whose symmetric representation of sigma consists of 3 parts and its central part has maximum width 2, but has width 1 at the diagonal.

Intersection of A001105 and A320137.
Subsequence of A071562 and of A319796.
Cf. A004171, A235791, A237048, A237270, A237271, A237593, A250068.

(* Function a249223[ ] is defined in A320137 *)
a361905[n_] := Select[Range[n], IntegerQ[#/Sqrt[#/2]]&&Last[a249223[#]]==1&]
a361905[10000]

a(n) = k when A001105(n) = k and A320137(k) = 1.

A365265 Numbers k for which sqrt(k/2) divides k and the symmetric representation of sigma(k) consists of a single part and its width at the diagonal equals 1.

Original entry on oeis.org

2, 8, 18, 32, 128, 162, 200, 392, 512, 882, 968, 1352, 1458, 2048, 2178, 3042, 3872, 5000, 5202, 5408, 6498, 8192, 9248, 9522, 11552, 13122, 15138, 16928, 17298, 19208, 26912, 30752, 32768, 36992, 43218, 43808, 46208, 53792, 58482, 59168, 67712, 70688

Offset: 1

Hartmut F. W. Hoft, Aug 29 2023

nonn

Every number a(n) has the form 2^(2*i + 1) * s^2, i>= 0 and s odd, the single middle divisor of a(n) is sqrt(a(n)/2), and sqrt(2*a(n)) - 1 = floor((sqrt(8*n + 1) - 1)/2) = A003056(a(n)).
The least number in the sequence with 3 odd prime divisors is a(126) = 1630818 = 2^1 * 3^2 * 7^2 * 43^2.
Conjecture: Let a(n) = 2^(2i+1) * s^2, i>=0 and s odd, be a number in the sequence.
(1) For any odd prime divisor p of s, number a(n) * p^2 is in the sequence.
(2) For any odd prime p not a divisor of s, number a(n) * p^2 is in the sequence if p satisfies sqrt(2*a(n)) < p < 2*a(n).

			a(5) = 128 = 2^7  has 2^3 as its single middle divisor, and its symmetric representation of sigma consists of one part of width 1.
a(10) = 882 = 2 * 3^2 * 7^2 has 3 * 7 as its single middle divisor, its symmetric representation of sigma is the smallest in this sequence of maximum width 3, consists of one part, and has width 1 at the diagonal.
A table of ranges for the single odd prime factor p for numbers k in the sequence having the form 2^(2i+1) * p^(2j), i>=0 and j>0, indexed by exponent 2i+1 of 2 in number k. The lower bound is A014210(i+1) and the upper bound is A014234(2(i+1)) = A104089(i+1):
---------------------
  2i+1  /---- p ----/
---------------------
  1       3  ..    3
  3       5  ..   13
  5      11  ..   61
  7      17  ..  251
  9      37  .. 1021
...

Intersection of A361903 and A361905.
Also subsequence of the following sequences: A001105, A071562, A238443 = A174973, A319796, A320137.
The powers of 2 with an odd index (A004171) form a subsequence.
Cf. A003056, A014210, A014234, A104089, A235791, A237048, A237270, A237271, A237593, A249223, A250068.

(* a2[ ] and its support functions are defined in A249223 *)
a365265Q[n_] := Module[{list=If[Divisible[n, Sqrt[n/2]], a2[n], {0}]}, Last[list]==1&&AllTrue[list, #>0&]]
a365265[{m_, n_}] := Select[Range[m, n], a365265Q]
a365265[{1,75000}]

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A361903 Numbers k for which sqrt(k/2) divides k and the symmetric representation of sigma(k) has a single part.

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A361905 Numbers k for which sqrt(k/2) divides k and the width at the diagonal of the symmetric representation of sigma(k) equals 1.

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A365265 Numbers k for which sqrt(k/2) divides k and the symmetric representation of sigma(k) consists of a single part and its width at the diagonal equals 1.

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