OFFSET
1,1
COMMENTS
The sequence is the intersection of A239929 (sigma(j) has two parts) and of A241008 (sigma(j) has an even number of parts of width one).
The numbers in the sequence are precisely those defined by the formula for the triangle, see the link. The symmetric representation of sigma(j) has two parts, each part having width one, precisely when j = 2^(k - 1) * p where 2^k <= row(j) < p, p is prime and row(j) = floor((sqrt(8*j + 1) - 1)/2). Therefore, the sequence can be written naturally as a triangle as shown in the Example section.
The symmetric representation of sigma(j) = 2*j - 2 consists of two regions of width 1 that meet on the diagonal precisely when j = 2^(2^m - 1)*(2^(2^m) + 1) where 2^(2^m) + 1 is a Fermat prime (see A019434). This subsequence of numbers j is 3, 10, 136, 32896, 2147516416, ...[?]... (A191363).
The k-th column of the triangle starts in the row whose initial entry is the first prime larger than 2^(k+1) (that sequence of primes is A014210, except for 2).
Observation: at least the first 82 terms coincide with the numbers j with no middle divisors whose largest divisor <= sqrt(j) is a power of 2, or in other words, coincide with the intersection of A071561 and A365406. - Omar E. Pol, Oct 11 2023
LINKS
Hartmut F. W. Hoft, Equivalence proof of sequence and triangle
Hartmut F. W. Hoft, Image for sigma(n) values
FORMULA
Formula for the triangle of numbers associated with the sequence:
P(n, k) = 2^k * prime(n) where n >= 2, 0 <= k <= floor(log_2(prime(n)) - 1).
EXAMPLE
We show portions of the first eight columns, 0 <= k <= 7, of the triangle.
0 1 2 3 4 5 6 7
3
5 10
7 14
11 22 44
13 26 52
17 34 68 136
19 38 76 152
23 46 92 184
29 58 116 232
31 62 124 248
37 74 148 296 592
41 82 164 328 656
43 86 172 344 688
47 94 188 376 752
53 106 212 424 848
59 118 236 472 944
61 122 244 488 976
67 134 268 536 1072 2144
71 142 284 568 1136 2272
. . . . . .
. . . . . .
127 254 508 1016 2032 4064
131 262 524 1048 2096 4192 8384
137 274 548 1096 2192 4384 8768
. . . . . . .
. . . . . . .
251 502 1004 2008 4016 8032 16064
257 514 1028 2056 4112 8224 16448 32896
263 526 1052 2104 4208 8416 16832 33664
Since 2^(2^4) + 1 = 65537 is the 6543rd prime, column k = 15 starts with 2^15*(2^(2^16) + 1) = 2147516416 in row 6542 with 65537 in column k = 0.
For an image of the symmetric representations of sigma(m) for all values m <= 137 in the triangle see the link.
The first column is the sequence of odd primes, see A065091.
The second column is the sequence of twice the primes starting with 10, see A001747.
The third column is the sequence of four times the primes starting with 44, see A001749.
For related references also see A033676 (largest divisor of n less than or equal to sqrt(n)).
MATHEMATICA
(* functions path[] and a237270[ ] are defined in A237270 *)
atmostOneDiagonalsQ[n_]:=SubsetQ[{0, 1}, Union[Flatten[Drop[Drop[path[n], 1], - 1] - path[n - 1], 1]]]
(* data *)
Select[Range[200], Length[a237270[#]]==2 && atmostOneDiagonalsQ[#]&]
(* function for computing triangle in the Example section through row 55 *)
TableForm[Table[2^k Prime[n], {n, 2, 56}, {k, 0, Floor[Log[2, Prime[n]] - 1]}], TableDepth->2]
CROSSREFS
KEYWORD
nonn
AUTHOR
Hartmut F. W. Hoft, Sep 08 2014
STATUS
approved