A308736 Numbers n such that n, n+2, n+4, n+6 are of the form p^2*q where p and q are distinct primes.

2523, 3112819, 5656019, 10132171, 12167825, 16639567, 25302173, 31995475, 35158921, 37334419, 43890719, 44816821, 47715269, 53548223, 55534523, 90526075, 90533525, 127558319, 142929025, 143167073, 144989575, 147182225

Offset

1,1

Comments

All terms are odd. Proof: if n is even then out of the 4 numbers n, n+2, n+4, n+6, 2 of them must be either both of the form 2*p^2, 2*q^2, or both of the form 4*p, 4*q. In either case, for p != q and p, q prime, the difference between these 2 numbers are more than 6, reaching a contradiction. - Chai Wah Wu, Jun 24 2019

Links

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 123 terms from Ray Chandler)

Example

2523 = 3*29*29, 2525 = 5*5*101, 2527 = 7*19*19, 2529 = 3*3*281.

Mathematica

psx = Table[{0}, {7}]; nmax = 150000000; n = 1; lst = {};
While[n < nmax, n++;
  psx = RotateRight[psx];
  psx[[1]] = Sort[Last /@ FactorInteger[n]];
  If[Union[{psx[[1]], psx[[3]], psx[[5]], psx[[7]]}] == {{1, 2}}, AppendTo[lst, n - 6]]; ];
lst

Prog

(Python)
from sympy import factorint
A308736_list, n, mlist = [], 3, [False]*4
while len(A308736_list) < 100:
    if mlist[0] and mlist[1] and mlist[2] and mlist[3]:
        A308736_list.append(n)
    n += 2
    f = factorint(n+6)
    mlist = mlist[1:] + [(len(f), sum(f.values())) == (2, 3)] # Chai Wah Wu, Jun 24 2019, Jan 03 2022.

Crossrefs

Cf. A074173, A308735.

Sequence in context: A308735 A096026 A278003 * A031984 A045213 A251918

Adjacent sequences: A308733 A308734 A308735 * A308737 A308738 A308739

Keyword

nonn

Author

Ray Chandler, Jun 24 2019

Status

approved