A308735 -id:A308735 - cp's OEIS frontend

A074173 Numbers n such that n and n+2 are of the form p^2*q where p and q are distinct primes.

18, 50, 242, 423, 475, 603, 637, 722, 845, 925, 1682, 1773, 2007, 2523, 2525, 2527, 3175, 3177, 4203, 4475, 4525, 4923, 5823, 6725, 6811, 6962, 7299, 7442, 7675, 8425, 8957, 8973, 9457, 9925, 10051, 10082, 10467, 11673, 11709, 12427, 12482, 12591

Offset: 1

Amarnath Murthy, Aug 30 2002

nonn

			18 is a member as 18 = 3^2*2 and 20 = 2^2*5.

Ray Chandler, Table of n, a(n) for n = 1..10000
Index to sequences related to prime signature

Cf. A048161, A054753, A074172, A074174, A308735, A308736.

lst={}; Do[f1=FactorInteger[n]; If[Sort[Transpose[f1][[2]]]=={1, 2}, f2=FactorInteger[n+2]; If[Sort[Transpose[f2][[2]]]=={1, 2}, AppendTo[lst, n]]], {n, 3, 10000}]; lst

Even terms in sequence are 2*A048161(n)^2. - Ray Chandler, Jun 24 2019

More terms from T. D. Noe, Oct 04 2004

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.

Original entry on oeis.org

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

Offset: 1

Ray Chandler, Jun 24 2019

nonn

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

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

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

Cf. A074173, A308735.

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

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.

cp's OEIS Frontend

A074173 Numbers n such that n and n+2 are of the form p^2*q where p and q are distinct primes.

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