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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
105, 165, 273, 345, 357, 385, 777, 897, 1045, 1173, 1353, 1653, 1677, 1705, 2193, 2233, 2373, 2905, 3157, 3237, 3333, 3417, 3445, 3553, 3565, 3945, 4053, 4585, 4953, 5665, 5817, 6097, 6513, 6693, 7077, 7833, 8437, 8565, 8845, 10153, 11005, 11433 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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

LINKS

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

EXAMPLE

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.

From Hartmut F. W. Hoft, Jan 09 2021: (Start)

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.

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)

MATHEMATICA

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

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

a128284[11433] (* Hartmut F. W. Hoft, Jan 09 2021 *)

CROSSREFS

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

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

Subsequence of A046389.

Sequence in context: A046405 A128278 A234103 * A161962 A046887 A026066

Adjacent sequences:  A128281 A128282 A128283 * A128285 A128286 A128287

KEYWORD

nonn

AUTHOR

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

STATUS

approved

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Last modified March 3 23:16 EST 2021. Contains 341772 sequences. (Running on oeis4.)