A375168 Numbers k with omega(k) = A001221(k) > 2 such that the set of distinct primes dividing k arranged in ascending order is an arithmetic progression.

105, 231, 315, 525, 627, 693, 735, 897, 935, 945, 1575, 1581, 1617, 1729, 1881, 2079, 2205, 2465, 2541, 2625, 2691, 2835, 2967, 3675, 4123, 4301, 4675, 4715, 4725, 4743, 4851, 5145, 5487, 5643, 6237, 6615, 6897, 7623, 7685, 7875, 7881, 8073, 8505, 8901, 9717

Offset

1,1

Comments

The corresponding common differences are 2, 4, 2, 2, 8, 4, 2, 10, 6, 2, 2, 14, 4, 6, 8,...

Links

Table of n, a(n) for n=1..45.

Example

105 is in the sequence because the prime divisors are {3, 5, 7} with the common difference = 2.
21505 is in the sequence because the set of the prime divisors is {5, 11, 17, 23} with the common difference = 6.
623645 is in the sequence because the set of the prime divisors is {5, 11, 17, 23, 29} with the common difference = 6.

Maple

with(numtheory):
for n from 1 to 10000 do:
 r:={}:d:=factorset(n):n0:=nops(d):
    for k from 1 to n0-1 do:
     r:=r union
{d[k+1]-d[k]}:
    od:
    if n0>2 and nops(r)= 1
     then printf(`%d, `, n):
     else
     fi:
    od:

Prog

(PARI) isok(k) = my(f=factor(k)); if ((nb=omega(f)) > 2, my(v = vector(nb-1, i, f[i+1, 1]-f[i, 1]), w = vector(nb-2, i, v[i+1]-v[i])); w == vector(nb-2); ); \\ Michel Marcus, Aug 20 2024

Crossrefs

Cf. A001221.

Sequence in context: A179143 A176878 A088595 * A308643 A229094 A307108

Adjacent sequences: A375165 A375166 A375167 * A375169 A375170 A375171

Keyword

nonn

Author

Michel Lagneau, Aug 05 2024

Status

approved