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A128283 Numbers of the form m = p1 * p2 where for each d|m we have (d+m/d)/2 prime and p1 < p2 both prime. 8

%I

%S 21,33,57,85,93,133,145,177,205,213,217,253,393,445,553,565,633,697,

%T 793,817,865,913,933,973,1137,1285,1345,1417,1437,1465,1477,1513,1537,

%U 1717,1765,1837,1857,1893,2101,2173,2245,2305,2517,2577,2581,2605,2641,2653,2733,2761

%N Numbers of the form m = p1 * p2 where for each d|m we have (d+m/d)/2 prime and p1 < p2 both prime.

%C The symmetric representation of sigma (A237593) for p1*p2, SRS(p1*p2), consists of either 4 or 3 regions. Let p1 < p2. Then 2*p1 < p2 implies that SRS(p1*p2), consists of 2 pairs of regions of widths 1 having respective sizes (p1*p2 + 1)/2 and (p1 + p2)/2; and p2 < 2*p1 implies that SRS(p1*p2) consists of 2 outer regions of width 1 and size (p1*p2 + 1)/2 and a central region of maximum width 2 of size p1 + p2 . Therefore, if SRS(p1*p2) has four regions, the area of each is a prime number (see A233562) and if it has three regions, the central area is an even semiprime (A100484). - _Hartmut F. W. Hoft_, Jan 09 2021

%C Old name was: "a(n) is the n-th smallest product of two distinct odd primes m=p1*p2 with the property that (d+m/d)/2 are all primes for each d dividing m.". - _David A. Corneth_, Jan 09 2021

%H David A. Corneth, <a href="/A128283/b128283.txt">Table of n, a(n) for n = 1..10000</a>

%e 85=5 * 17, (5 * 17+1)/2=43, (5+17)/2=11 are both primes and 85 is in the sequence.

%e From _Hartmut F. W. Hoft_, Jan 09 2021: (Start)

%e 9=3*3 is not in the sequence even though (1+9)/2 and (3+3)/2 are primes, see also A340482.

%e a(33) = 1537 = 29*53 is the first number for which the symmetric representation of sigma consists of three regions ( 769, 82, 769 ) with 5 units of width 2 straddling the diagonal in the central region; (1537+1)/2 = 769 and (29+53)/2 = 41 are primes. (End)

%t ppQ[s_, k_] := Last[Transpose[FactorInteger[s]]==Table[1, k]

%t dQ[s_] := Module[{d=Divisors[s]}, AllTrue[Map[(d[[#]]+d[[-#]])/2&, Range[Length[d]/2]], PrimeQ]]

%t goodL[{m_, n_}, k_] := Module[{i=m, list={}}, While[i<=n, If[ppQ[i, k] && dQ[i], AppendTo[list, i]]; i+=2]; list]/;OddQ[m]

%t a128283[n_] := goodL[{1, n}, 2]

%t a128283[2653] (* _Hartmut F. W. Hoft_, Jan 09 2021 *)

%Y Cf. A128281, A005383, A128284, A128285, A128286.

%Y Cf. A100484, A233562, A237591, A237593, A249223, A262045, A340482.

%Y Subsequence of A046388.

%K nonn

%O 1,1

%A Kok Seng Chua (chuakokseng(AT)hotmail.com), Mar 05 2007

%E Added "distinct" for clarification since 9 satisfies the divisor property. See also A340482. - _Hartmut F. W. Hoft_, Jan 09 2021

%E New name from _David A. Corneth_, Jan 09 2021

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Last modified April 13 02:47 EDT 2021. Contains 342934 sequences. (Running on oeis4.)