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 A246955 #40 Oct 22 2023 16:57:13
%S A246955 3,5,7,10,11,13,14,17,19,22,23,26,29,31,34,37,38,41,43,44,46,47,52,53,
%T A246955 58,59,61,62,67,68,71,73,74,76,79,82,83,86,89,92,94,97,101,103,106,
%U A246955 107,109,113,116,118,122,124,127,131,134,136,137,139,142,146,148,149,151,152,157,158,163,164,166,167,172,173,178,179,181,184,188,191,193,194,197,199
%N A246955 Numbers j for which the symmetric representation of sigma(j) has two parts, each of width one.
%C A246955 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).
%C A246955 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.
%C A246955 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).
%C A246955 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).
%C A246955 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
%H A246955 Hartmut F. W. Hoft, <a href="/A246955/a246955_1.pdf">Equivalence proof of sequence and triangle</a>
%H A246955 Hartmut F. W. Hoft, <a href="/A246955/a246955_2.pdf">Image for sigma(n) values</a>
%F A246955 Formula for the triangle of numbers associated with the sequence:
%F A246955 P(n, k) = 2^k * prime(n) where n >= 2, 0 <= k <= floor(log_2(prime(n)) - 1).
%e A246955 We show portions of the first eight columns, 0 <= k <= 7, of the triangle.
%e A246955 0 1 2 3 4 5 6 7
%e A246955 3
%e A246955 5 10
%e A246955 7 14
%e A246955 11 22 44
%e A246955 13 26 52
%e A246955 17 34 68 136
%e A246955 19 38 76 152
%e A246955 23 46 92 184
%e A246955 29 58 116 232
%e A246955 31 62 124 248
%e A246955 37 74 148 296 592
%e A246955 41 82 164 328 656
%e A246955 43 86 172 344 688
%e A246955 47 94 188 376 752
%e A246955 53 106 212 424 848
%e A246955 59 118 236 472 944
%e A246955 61 122 244 488 976
%e A246955 67 134 268 536 1072 2144
%e A246955 71 142 284 568 1136 2272
%e A246955 . . . . . .
%e A246955 . . . . . .
%e A246955 127 254 508 1016 2032 4064
%e A246955 131 262 524 1048 2096 4192 8384
%e A246955 137 274 548 1096 2192 4384 8768
%e A246955 . . . . . . .
%e A246955 . . . . . . .
%e A246955 251 502 1004 2008 4016 8032 16064
%e A246955 257 514 1028 2056 4112 8224 16448 32896
%e A246955 263 526 1052 2104 4208 8416 16832 33664
%e A246955 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.
%e A246955 For an image of the symmetric representations of sigma(m) for all values m <= 137 in the triangle see the link.
%e A246955 The first column is the sequence of odd primes, see A065091.
%e A246955 The second column is the sequence of twice the primes starting with 10, see A001747.
%e A246955 The third column is the sequence of four times the primes starting with 44, see A001749.
%e A246955 For related references also see A033676 (largest divisor of n less than or equal to sqrt(n)).
%t A246955 (* functions path[] and a237270[ ] are defined in A237270 *)
%t A246955 atmostOneDiagonalsQ[n_]:=SubsetQ[{0, 1}, Union[Flatten[Drop[Drop[path[n], 1], - 1] - path[n - 1], 1]]]
%t A246955 (* data *)
%t A246955 Select[Range[200], Length[a237270[#]]==2 && atmostOneDiagonalsQ[#]&]
%t A246955 (* function for computing triangle in the Example section through row 55 *)
%t A246955 TableForm[Table[2^k Prime[n], {n, 2, 56}, {k, 0, Floor[Log[2, Prime[n]] - 1]}], TableDepth->2]
%Y A246955 Cf. A000203, A033676, A071561, A163280, A237270, A237271, A237593, A241008, A241010, A247687, A250068, A250070, A250071, A365406.
%K A246955 nonn
%O A246955 1,1
%A A246955 _Hartmut F. W. Hoft_, Sep 08 2014