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 A378471 #11 Dec 12 2024 23:28:44
%S A378471 3,5,7,9,10,11,13,14,15,17,19,21,22,23,25,26,27,29,31,33,34,35,37,38,
%T A378471 39,41,43,44,45,46,47,49,50,51,52,53,55,57,58,59,61,62,63,65,67,68,69,
%U A378471 70,71,73,74,75,76,77,79,81,82,83,85,86,87,89,91,92,93,94,95,97,98,99,101,103,105
%N A378471 Numbers m whose symmetric representation of sigma(m), SRS(m), has at least 2 parts the first of which has width 1.
%C A378471 Numbers m = 2^k * q, k >= 0 and q > 1 odd, without odd prime factors p < 2^(k+1).
%C A378471 This sequence is a proper subsequence of A238524. Numbers 78 = A370206(1) = A238524(55) and 102 = A237287(72) are not in this sequence since their width pattern (A341969) is 1210121.
%C A378471 A000079 is not a subsequence since SRS(2^k), k>=0, consists of a single part of width 1.
%C A378471 Let m = 2^k * q, k >= 0 and q > 1 odd, be a number in this sequence and s the size of the first part of SRS(m) which has width 1 and consists of 2^(k+1) - 1 legs of width 1. Therefore, s = Sum_{i=1..2^(k+1)-1} a237591(m, i) = a235791(m, 1) - a235791(m, 2^(k+1)) = ceiling((m+1)/1 - (1+1)/2) - ceiling((m+1)/2^(k+1) - (2^(k+1) + 1)/2) = (2^(k+1) - 1)(q+1)/2. In other words, point (m, s) is on the line s(m) = (2^(k+1) - 1)/2^(k+1) * m + (2^(k+1) - 1)/2.
%C A378471 For every odd number m in this sequence, the first part of SRS(m) has size (m+1)/2.
%C A378471 Let u = 2^k * Product_{i=1..PrimePi(2^(k+1)} p_i, where p_i is the i-th prime, and let v be the number of elements in this sequence that are in the set V = {m = 2^k * q | 1 < m <= u } then T(j + t*v, k) = T(j, k) + t*u, 1 <= j and 1 <= t, holds for the elements in column k.
%e A378471 a(5) = 10 is in the sequence since SRS(10) = {9, 9} consists of 2 parts of width 1 and of sizes 9 = (2^2 - 1)(5+1)/2.
%e A378471 a(15) = 25 is in the sequence since the first part of SRS(25) = {13, 5, 13} has width 1 and has size 13 = (2^1 - 1)(25+1)/2.
%e A378471 a(28) = 44 is in the sequence since SRS(44) = {42, 42} has width 1 and has size 42 = (2^3 - 1)(11+1)/2.
%e A378471 The upper left hand 11 X 11 section of array T(j, k) shows the j-th number m in this sequence of the form m = 2^k * q with q odd. The first part of SRS(m) of every number in column k consists of 2^(k+1) - 1 legs of width 1.
%e A378471 j\k| 0 1 2 3 4 5 6 7 8 9 10 ...
%e A378471 ------------------------------------------------------------------------
%e A378471 1 | 3 10 44 136 592 2144 8384 32896 133376 527872 2102272
%e A378471 2 | 5 14 52 152 656 2272 8768 33664 133888 528896 2112512
%e A378471 3 | 7 22 68 184 688 2336 8896 34432 138496 531968 2118656
%e A378471 4 | 9 26 76 232 752 2528 9536 34688 140032 537088 2130944
%e A378471 5 | 11 34 92 248 848 2656 9664 35456 142592 538112 2132992
%e A378471 6 | 13 38 116 296 944 2848 10048 35968 144128 543232 2137088
%e A378471 7 | 15 46 124 328 976 3104 10432 36224 145664 544256 2139136
%e A378471 8 | 17 50 148 344 1072 3232 10688 37504 146176 547328 2149376
%e A378471 9 | 19 58 164 376 1136 3296 11072 39296 147712 556544 2161664
%e A378471 10 | 21 62 172 424 1168 3424 11456 39808 150272 558592 2163712
%e A378471 11 | 23 70 188 472 1264 3488 11584 40064 151808 559616 2180096
%e A378471 ...
%e A378471 Row 1 is A246956(n), n>=1.
%e A378471 Column 0 is A005408(n) with T(j + 1, 0) = T(j, 0) + 2, n>=1.
%e A378471 Column 1 is A091999(n) with T(j + 2, 1) = T(j, 1) + 12, n>=2.
%e A378471 Column 2 is A270298(n) with T(j + 48, 2) = T(j, 2) + 840, n>=1.
%e A378471 Column 3 is A270301(n) with T(j + 5760, 3) = T(j, 3) + 240240, n>=1.
%t A378471 (* partsSRS[] and widthPattern[ ] are defined in A377654 *)
%t A378471 a378471[m_, n_] := Select[Range[m, n], Length[partsSRS[#]]>1&&widthPattern[#][[1;;2]]=={1, 0}&]
%t A378471 a378471[1, 105]
%Y A378471 Cf. A000079, A005408, A091999, A235791, A237287, A237591, A237593, A238524, A246956, A270298, A270301, A341969, A370206, A377654.
%K A378471 nonn
%O A378471 1,1
%A A378471 _Hartmut F. W. Hoft_, Nov 27 2024