|
| |
|
|
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
|
|
|
|
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.
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]
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
Kok Seng Chua (chuakokseng(AT)hotmail.com), Mar 05 2007
|
|
|
STATUS
|
approved
|
| |
|
|
