A294751 Squarefree products of k primes that are symmetrically distributed around their average. Case k = 4.

2145, 4641, 4845, 5005, 9177, 11305, 13485, 13585, 17017, 21489, 21505, 23529, 26445, 31465, 31857, 33649, 35409, 35581, 36685, 42441, 43401, 46189, 46345, 49569, 50065, 53985, 60697, 61705, 63085, 63597, 65569, 67821, 69745, 77745, 80845, 83049, 87505, 88881

Offset: 1

Author

Paolo P. Lava, Nov 08 2017

Keywords

• nonn

Examples

			2145 = 3*5*11*13. Prime factors average is (3 + 5 + 11 + 13)/4 = 8 and 3 + 5 = 8 = 13 - 5, 5 + 3 = 8 = 11 - 3.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Subsequence of A046386.

Cf. A006881 (k=2), A262723 (k=3), A294752 (k=5), A294776 (k=6).

Programs

Maple

with(numtheory): P:=proc(q,h) local a,b,k,n,ok;
for n from 2*3*5*7 to q do if not isprime(n) and issqrfree(n) then a:=ifactors(n)[2];
if nops(a)=h then b:=2*add(a[k][1],k=1..nops(a))/nops(a); ok:=1;
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(10^9,4);
# Alternative:
N:= 10^5: # to get terms <= N
M:= floor(max(fsolve(3*5*(M-5)*(M-3) = N))):
P:= select(isprime, [seq(i,i=3..M/2,2)]): nP:= nops(P):
Res:= NULL:
for m from 10 by 2 to M do
  for ix from 1 to nP-2 do
    x:= P[ix];
    if x >= m/2 or (x*(m-x))^2 >= N then break fi;
    if not isprime(m-x) then next fi;
    for iy from ix+1 to nP-1 do
      y:= P[iy];
      if y >= m/2 or x*(m-x)*y*(m-y) >= N then break fi;
      if not isprime(m-y) then next fi;
      Res:= Res, x*(m-x)*y*(m-y);
od od od:
sort([Res]); # Robert Israel, May 19 2019

PARI

isok(n, nb=4) = {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

Extensions

More terms from Giovanni Resta, Nov 09 2017