

A128283


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


8



21, 33, 57, 85, 93, 133, 145, 177, 205, 213, 217, 253, 393, 445, 553, 565, 633, 697, 793, 817, 865, 913, 933, 973, 1137, 1285, 1345, 1417, 1437, 1465, 1477, 1513, 1537, 1717, 1765, 1837, 1857, 1893, 2101, 2173, 2245, 2305, 2517, 2577, 2581, 2605, 2641, 2653, 2733, 2761
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OFFSET

1,1


COMMENTS

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
Old name was: "a(n) is the nth 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


LINKS

David A. Corneth, Table of n, a(n) for n = 1..10000


EXAMPLE

85=5 * 17, (5 * 17+1)/2=43, (5+17)/2=11 are both primes and 85 is in the sequence.
From Hartmut F. W. Hoft, Jan 09 2021: (Start)
9=3*3 is not in the sequence even though (1+9)/2 and (3+3)/2 are primes, see also A340482.
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)


MATHEMATICA

ppQ[s_, k_] := Last[Transpose[FactorInteger[s]]==Table[1, k]
dQ[s_] := Module[{d=Divisors[s]}, AllTrue[Map[(d[[#]]+d[[#]])/2&, Range[Length[d]/2]], PrimeQ]]
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]
a128283[n_] := goodL[{1, n}, 2]
a128283[2653] (* Hartmut F. W. Hoft, Jan 09 2021 *)


CROSSREFS

Cf. A128281, A005383, A128284, A128285, A128286.
Cf. A100484, A233562, A237591, A237593, A249223, A262045, A340482.
Subsequence of A046388.
Sequence in context: A191683 A032603 A233562 * A280878 A033901 A033656
Adjacent sequences: A128280 A128281 A128282 * A128284 A128285 A128286


KEYWORD

nonn


AUTHOR

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


EXTENSIONS

Added "distinct" for clarification since 9 satisfies the divisor property. See also A340482.  Hartmut F. W. Hoft, Jan 09 2021
New name from David A. Corneth, Jan 09 2021


STATUS

approved



