A296096 Numbers k with the property that k is the product of two distinct primes whose difference is also the product of two distinct primes.

34, 39, 46, 51, 55, 74, 82, 87, 91, 95, 106, 111, 118, 119, 123, 134, 142, 155, 158, 178, 183, 187, 194, 203, 215, 226, 247, 262, 267, 287, 291, 299, 314, 326, 327, 335, 358, 371, 391, 395, 407, 411, 422, 446, 447, 478, 502, 527, 538, 543, 551, 586, 591, 611, 614

Offset

1,1

Comments

The number of times this process can be repeated would be called the order of the number. For example, 82 is the smallest number of this type of order 2.

Links

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

Example

51 is a number of this type since 51=3*17 and 17-3=14 are also the product of two distinct primes.

Maple

N:= 10^4: # to get all terms <= N
P:= select(isprime, [2, seq(i, i=3..N, 2)]):
P2:= {}:
for i from 1 to nops(P) while P[i]^2 <= N do
  for j from i+1 while P[i]*P[j] <= N do od:
  P2:= P2 union {seq(P[i]*P[k], k=i+1..j-1)};
od:
P3:= {}:
for i from 1 to nops(P) while P[i]^2 <= N do
  for j from i+1 while P[i]*P[j] <= N do od:
  Q:= map(`-`, convert(P[i+1..j-1], set), P[i]) intersect P2;
  P3:= P3 union map(t -> (t+P[i])*P[i], Q);
od:
sort(convert(P3, list)); # Robert Israel, Jan 05 2018

Mathematica

Select[Range[6, 614], And[AllTrue[#, PrimeQ], Length@ # == 2, FactorInteger[First@ Differences@ #][[All, -1]] == {1, 1}] &@ Most@ Rest@ Divisors@ # &] (* Michael De Vlieger, Dec 13 2017 *)

Prog

(Sage)
for n in range(1, 100):
    L=list(factor(n))
    itsemiprime=false
    degree=-1
    if len(L)==2 and L[0][1]==1 and L[1][1]==1:
        itsemiprime=true
    while len(L)==2:
        if L[0][1]==1 and L[1][1]==1:
            L=list(factor(L[1][0]-L[0][0]))
            temp=len(L)
            degree=degree+1
        else:
            break
    if itsemiprime:
        n, degree
(PARI) isok1(n) = (bigomega(n)==2) && issquarefree(n);
isok(n) = isok1(n) && (f=factor(n)) && isok1(f[2, 1]-f[1, 1]); \\ Michel Marcus, Dec 21 2017

Crossrefs

Cf. A006881.

Sequence in context: A281239 A295750 A141699 * A337695 A045044 A108303

Adjacent sequences: A296093 A296094 A296095 * A296097 A296098 A296099

Keyword

nonn

Author

Carl Lienert and Pamela K. M. Smith, Dec 04 2017

Status

approved