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A134344
Composite numbers such that the arithmetic mean of their prime factors (counted with multiplicity) is prime.
28
4, 8, 9, 16, 20, 21, 25, 27, 32, 33, 44, 49, 57, 60, 64, 68, 69, 81, 85, 93, 105, 112, 116, 121, 125, 128, 129, 133, 145, 156, 169, 177, 180, 188, 195, 205, 212, 213, 217, 220, 231, 237, 243, 249, 253, 256, 265, 272, 275, 289, 297, 309, 332, 336, 343, 356, 361
OFFSET
1,1
COMMENTS
Originally, the definition started with "Nonprime numbers ...". This may be misleading, since 1 is also nonprime, but has no prime factors. - Hieronymus Fischer, May 05 2013
LINKS
Harvey P. Dale and Hieronymus Fischer, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harvey P. Dale)
EXAMPLE
a(1) = 4, since 4 = 2*2 and the arithmetic mean (2+2)/2 = 2 is prime.
a(5) = 20, since 20 = 2*2*5 and the arithmetic mean (2+2+5)/3 = 3 is prime.
MATHEMATICA
ampfQ[n_]:=PrimeQ[Mean[Flatten[Table[#[[1]], {#[[2]]}]&/@FactorInteger[ n]]]]; nn=400; Select[Complement[Range[nn], Prime[Range[ PrimePi[nn]]]], ampfQ] (* Harvey P. Dale, Nov 06 2012 *)
PROG
(PARI) is(n)=if(n<4, return(0)); my(f=factor(n), s=sum(i=1, #f~, f[i, 1]*f[i, 2])/sum(i=1, #f~, f[i, 2])); (#f~>1 || f[1, 2]>1) && denominator(s)==1 && isprime(s) \\ Charles R Greathouse IV, Sep 14 2015
KEYWORD
nonn
AUTHOR
Hieronymus Fischer, Oct 23 2007
EXTENSIONS
Definition clarified by Hieronymus Fischer, May 05 2013
STATUS
approved