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 A372180 #18 Apr 28 2024 22:27:53
%S A372180 6,12,78,20,102,1014,24,114,1734,12246,28,138,2166,12714,171366,40,
%T A372180 174,3174,13026,501126,1922622,48,186,5046,13182,781926,2057406,
%U A372180 28960854,56,222,5766,13494,1679046,2067546,144825414,300014754,80,246,8214,13962,4243686,2072382,282275286,300137214,4174476774
%N A372180 Square array read by antidiagonals upwards in which T(n,m) is the n-th number whose symmetric representation of sigma consists of m copies of unimodal pattern 121 (separated by 0's if m > 1).
%C A372180 Every number in this sequence is even since the symmetric representation of sigma for an odd number q starts 101. Each number in column m of T(n,m) has 2*m odd divisors.
%C A372180 Since u(m) = 2 * 3 * 13^(m-1), m>=1, has 2m odd divisors and 1 < 3 < 4 < 4*3 < 13 < 3*13 < 4*13 < 3*4*13 < 13^2 < ..., the symmetric representation of sigma for u(m) consists of m copies of unimodal pattern 121. Therefore, every column in the table T(n,m), m>=1, contains infinitely many entries. Number u(m) is the smallest entry in the m-th column when m is prime.
%C A372180 In general: If m>1 then T(n,m) = 2^k * q, k>=1, q odd, has at least 4 odd divisors which satisfy
%C A372180 d_(2i+2) < 2^(k+1) * d_(2i+1) < 2^(k+1) * d_(2i+2) < d_(2i+3), i>=0,
%C A372180 with the odd divisors d_j of n in increasing order.
%F A372180 T(n,1) = 2^k * p with odd prime p satisfying p < 2^(k+1), see A370205.
%F A372180 T(n,2) = 2^k * p * q, k > 0, p and q prime, 2 < p < 2^(k+1) < 2^(k+1) * p < q, see A370206.
%e A372180 a(1) = T(1,1) = 6, its symmetric representation of sigma, SRS(6), has unimodal pattern 121 and a single unit of width 2 at the diagonal.
%e A372180 a(3) = T(1,2) = 78, SRS(78) has unimodal pattern 1210121;
%e A372180 a(10) = T(1,4) = 12246, SRS(12246) has unimodal pattern 121012101210121;
%e A372180 both symmetric representations of sigma have width 0 at the diagonal where two parts meets.
%e A372180 Each number in the m-th column has 2m odd divisors. T(1,9) = 4174476774.
%e A372180 -------------------------------------------------------------------------
%e A372180 n\m 1 2 3 4 5 6 7 8
%e A372180 -------------------------------------------------------------------------
%e A372180 1| 6 78 1014 12246 171366 1922622 28960854 300014754 ...
%e A372180 2| 12 102 1734 12714 501126 2057406 144825414 300137214 ...
%e A372180 3| 20 114 2166 13026 781926 2067546 282275286 300235182 ...
%e A372180 4| 24 138 3174 13182 1679046 2072382 888215334 300357642 ...
%e A372180 5| 28 174 5046 13494 4243686 2081742 3568939926 300431118 ...
%e A372180 6| 40 186 5766 13962 5541126 2091882 ... 300602562 ...
%e A372180 7| 48 222 8214 14118 8487372 2097966 300651546 ...
%e A372180 8| 56 246 10086 14898 11082252 2110134 300896466 ...
%e A372180 9| 80 258 10092 15054 11244966 2112162 301165878 ...
%e A372180 10| 88 282 11094 15366 16954566 2116218 301386306 ...
%e A372180 ...
%t A372180 divQ[k_, {d1_, d2_, d3_}] := d2<2^(k+1)d1&&2^(k+1)d2<d3
%t A372180 seqQ[t_, m_] := Module[{oP, k, dL}, oP=NestWhile[#/2&, t, EvenQ[#]&]; k=Log[2, t/oP]; dL=Divisors[oP]; Length[dL]==2m&&If[Length[dL]==2, Last[dL]<2^(k+1), AllTrue[Partition[dL, 3, 2], divQ[k, #]&]]]
%t A372180 a372180[{r_, s_}, m_] := Select[Range[r, s], seqQ[#, m]&] (* range r...s of numbers in column m *)
%t A372180 a372180[{1, 15366}, 4] (* computes column 4 in the table *)
%t A372180 a372180[{1, 2116218}, 6] (* computes column 6 in the table *)
%Y A372180 Row 1 gives A372181.
%Y A372180 Column 1 gives A370205.
%Y A372180 Column 2 gives A370206.
%Y A372180 Cf. A235791, A237048, A237270, A237271, A237591, A237593, A249223, A262045, A341969, A341971, A342592, A342594, A342595, A342596, A367377, A370209.
%K A372180 nonn,tabl
%O A372180 1,1
%A A372180 _Hartmut F. W. Hoft_, Apr 21 2024