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Squarefree products of k primes that are symmetrically distributed around their average. Case k = 5.
4

%I #22 Nov 10 2017 21:47:10

%S 53295,119301,229245,399993,608235,623645,1462731,2324495,3696189,

%T 3973145,4482879,5356445,5920971,6249633,7588977,8270385,10160943,

%U 10450121,10505373,13185969,13630011,13760929,14935029,19095395,20280795,22566271,23131549,23408259,24778401

%N Squarefree products of k primes that are symmetrically distributed around their average. Case k = 5.

%H Robert Israel, <a href="/A294752/b294752.txt">Table of n, a(n) for n = 1..3560</a>

%e 53295 = 3*5*11*17*19. Prime factors average is (3 + 5 + 11 + 17 + 19)/5 = 11 and 3 + 8 = 11 = 19 - 8, 5 + 6 = 11 = 17 - 6.

%p with(numtheory): P:=proc(q,h) local a,b,k,n,ok;

%p for n from 2*3*5*7*11 to q do if not isprime(n) and issqrfree(n) then a:=ifactors(n)[2];

%p if nops(a)=h then b:=2*add(a[k][1],k=1..nops(a))/nops(a); ok:=1;

%p for k from 1 to trunc(nops(a)/2) do if a[k][1]+a[nops(a)-k+1][1]<>b then ok:=0; break; fi; od; if ok=1 then print(n); fi; fi; fi; od; end: P(10^9,5);

%p # Alternative:

%p N:= 10^8: # to get all terms <= N

%p M:= floor((8*N/15)^(1/3)):

%p P:= select(isprime, [seq(i,i=3..M,2)]): nP:= nops(P):

%p Res:= NULL:

%p for i3 from 3 to nP-2 do

%p p3:= P[i3];

%p for i1 from 1 to i3-2 do

%p if isprime(2*p3 - P[i1]) then

%p for i2 from i1+1 to i3-1 do

%p if isprime(2*p3 - P[i2]) then

%p v:=P[i1]*P[i2]*p3*(2*p3-P[i2])*(2*p3-P[i1]);

%p if v <= N then Res:= Res, v fi

%p fi

%p od

%p fi

%p od

%p od:

%p sort([Res]): # _Robert Israel_, Nov 10 2017

%o (PARI) isok(n, nb=5) = {if (issquarefree(n) && (omega(n)==nb), f = factor(n)[, 1]~; avg = vecsum(f)/#f; for (k=1, #f\2, if (f[k] + f[#f-k+1] != 2*avg, return(0));); return (1););} \\ _Michel Marcus_, Nov 10 2017

%Y Subsequence of A046387, A203614.

%Y Cf. A006881 (k=2), A262723 (k=3), A294751 (k=4), A294776 (k=6).

%K nonn

%O 1,1

%A _Paolo P. Lava_, Nov 08 2017

%E More terms from _Giovanni Resta_, Nov 09 2017

%E Missing term 23131549 inserted by _Robert Israel_, Nov 10 2017