A370206 - OEIS

%I #15 Feb 17 2024 12:44:33

%S 78,102,114,138,174,186,222,246,258,282,318,348,354,366,372,402,426,

%T 438,444,474,492,498,516,534,564,582,606,618,636,642,654,678,708,732,

%U 762,786,804,820,822,834,852,860,876,894,906,940,942,948,978,996,1002,1038,1060,1068,1074

%N Numbers j whose symmetric representation of sigma(j) consists of two copies of unimodal width pattern 121 separated by 0.

%C Each term has 4 odd divisors and has the form 2^k * p * q, k > 0, p and q prime,  2 < p < 2^(k+1) <  2^(k+1) * p < q. The inequalities ensure that the four 1's in row a(n) of triangle in A237048 are in positions 1, p, 2^(k+1), and 2^(k+1) * p <= floor( (sqrt(8*a(n)+1) - 1)/2 ) < q and establish width pattern 1210 in SRS(a(n)) up to the diagonal. Also since p < 2^(k+1), numbers of the form 2^k * p^3 force p^2 < 2^(k+1) * p which creates a width pattern of the form 1212121.

%C When a(n) satisfies q = 2^(k+1) * p + 1 it is the smallest number with prime factor p  whose two parts of SRS(a(n)) meet at the diagonal since in this case 2^(k+1) * p = floor( (sqrt(8*a(n)+1) - 1)/2 ). The first 4 numbers with p = 3 are  2* 3 * 13 = 78, 2^4 * 3 * 97 = 4656, 2^5 * 3 * 193 = 18528 and 2^7 * 3 * 769 = 295296. The smallest number with prime factor p = 47 has 355 digits.

%C Conjecture: The subsequence of numbers m whose two parts of SRS(m) meet at the diagonal is infinite.

%e a(1) = 78 = 2 * 3 * 13 = A262259(3) and SRS(78) consists of 2 unimodal parts of width pattern 121 that meet at diagonal position (54, 54).

%e a(38) = 4 * 5 * 41 = 820 = A262259(6)  is the smallest number in the sequence divisible by 5 and the two parts of SRS(a(38)) meet at diagonal position (570, 570).

%t (* function based on conditions for the odd divisors - fast computation *)

%t a370206Q[n_] := Module[{f=FactorInteger[n], d=Divisors[NestWhile[#/2&, n, EvenQ[#]&]]}, Length[f]==3&&f[[1, 1]]==2&&Length[d]==4&&f[[2, 1]]<2^(f[[1, 2]]+1)&&2^(f[[1, 2]]+1)*f[[2, 1]]<f[[3, 1]]]

%t a370206[m_, n_] := Select[Range[m, n], a370206Q]

%t a370206[1, 1074]

%t (* widthPattern[ ] and support functions are defined in A367377 - slow computation *)

%t a370206[m_, n_] :=  Select[Range[m, n], widthPattern[#]=={1, 2, 1, 0, 1, 2, 1}&]

%t a370206[1,1074]

%Y Cf. A082662, A235791, A237048, A237270, A237271, A237591, A237593, A249223, A262045, A262259, A341969, A342592, A342594, A342595, A342596, A367377, A370205.

%K nonn

%O 1,1

%A _Hartmut F. W. Hoft_, Feb 11 2024