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A370206
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Numbers j whose symmetric representation of sigma(j) consists of two copies of unimodal width pattern 121 separated by 0.
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3
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78, 102, 114, 138, 174, 186, 222, 246, 258, 282, 318, 348, 354, 366, 372, 402, 426, 438, 444, 474, 492, 498, 516, 534, 564, 582, 606, 618, 636, 642, 654, 678, 708, 732, 762, 786, 804, 820, 822, 834, 852, 860, 876, 894, 906, 940, 942, 948, 978, 996, 1002, 1038, 1060, 1068, 1074
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OFFSET
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1,1
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COMMENTS
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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.
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.
Conjecture: The subsequence of numbers m whose two parts of SRS(m) meet at the diagonal is infinite.
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LINKS
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EXAMPLE
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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).
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).
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MATHEMATICA
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(* function based on conditions for the odd divisors - fast computation *)
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]]]
a370206[m_, n_] := Select[Range[m, n], a370206Q]
a370206[1, 1074]
(* widthPattern[ ] and support functions are defined in A367377 - slow computation *)
a370206[m_, n_] := Select[Range[m, n], widthPattern[#]=={1, 2, 1, 0, 1, 2, 1}&]
a370206[1, 1074]
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CROSSREFS
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Cf. A082662, A235791, A237048, A237270, A237271, A237591, A237593, A249223, A262045, A262259, A341969, A342592, A342594, A342595, A342596, A367377, A370205.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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