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A128284 Numbers of the form m = p1 * p2 * p3 where for each d|m we have (d+m/d)/2 prime and p1 < p2 < p3 each prime. 5

%I

%S 105,165,273,345,357,385,777,897,1045,1173,1353,1653,1677,1705,2193,

%T 2233,2373,2905,3157,3237,3333,3417,3445,3553,3565,3945,4053,4585,

%U 4953,5665,5817,6097,6513,6693,7077,7833,8437,8565,8845,10153,11005,11433

%N Numbers of the form m = p1 * p2 * p3 where for each d|m we have (d+m/d)/2 prime and p1 < p2 < p3 each prime.

%C The symmetric representation of sigma (cf. A237593), SRS(a(n)), of any number in this sequence has between 4 and 8 regions with 3 regions impossible because p1 < p2 < p3 implies 2*p3 < p1*p2. When there are 8 regions they all have width 1 and their areas are the prime numbers (d+a(n)/d)/2 for the 4 respective pairs of divisors of a(n). In general, the areas of the regions in SRS(a(n)) need not be prime, except for the two symmetric outer regions (n+1)/2. - _Hartmut F. W. Hoft_, Jan 09 2021

%H David A. Corneth, <a href="/A128284/b128284.txt">Table of n, a(n) for n = 1..10000</a>

%e 165=3*5*11 and (3*5*11+1)/2=83, (3+5*7)/2=19, (5+3*7)/2=13, (7+3*5)/2=11 are all primes, so 165 is a term.

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

%e a(1) = 105 = 3*5*7 and SRS(a(1)) consists of four regions with areas ( 53, 43, 43, 53 ); the center areas have maximum width 2 and represent the sum of primes (3+35)/2 + (5+21)/2 + (7+15)/2 = 43.

%e a(17) = 2373 = 3*7*17 is the first number in the sequence whose symmetric representation of sigma consists of 8 regions, all of width 1 and the respective symmetric regions have areas: (2373 + 1)/2 = 1187, (791 + 3)/2 = 397, (339 + 7)/2 = 173, (21 + 113)/2 = 67. (End)

%t (* function goodL[] is defined in A128283 *)

%t a128284[n_] := goodL[{1, n}, 3]

%t a128284[11433] (* _Hartmut F. W. Hoft_, Jan 09 2021 *)

%Y Cf. A128281, A005383, A128283, A128285, A128286.

%Y Cf. A237048, A237591, A237593, A249223, A262045.

%Y Subsequence of A046389.

%K nonn

%O 1,1

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

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Last modified April 11 09:24 EDT 2021. Contains 342886 sequences. (Running on oeis4.)