A395899 Sphenic numbers k = p*q*r, primes p < q < r, such that r < p^3.

30, 42, 70, 105, 165, 195, 231, 255, 273, 285, 345, 357, 385, 399, 429, 455, 483, 561, 595, 627, 663, 665, 715, 741, 759, 805, 897, 935, 969, 1001, 1015, 1045, 1085, 1105, 1173, 1235, 1265, 1295, 1309, 1311, 1435, 1463, 1495, 1505, 1547, 1595, 1615, 1645, 1705

Offset

1,1

Comments

Proper subset of A393644.

Superset of A383177.

Analogous to A138109, a sequence which instead pertains to squarefree semiprimes A006881.

Let lpf(n) = A020639(n) be the smallest prime factor of n.

Smallest term with lpf(a(n)) = prime(i) is prime(i)*prime(i+1)*prime(i+2) = A046301(n).

Largest term divisible by p = prime(i) is p*q*r, with r = prime(pi(p^3)) and q = r = prime(pi(p^3)-1).

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Michael De Vlieger, Plot prime(i) | a(n) at (x,y) = (n,i), for n = 1..2400, 8X vertical exaggeration.

Example

Let s = A383177 and lpf = A020639.
Table of n, a(n) for select n:
      n      a(n)                   lpf(a(n))^3
  ---------------------------------------------
      1       30 =  2 *   3 *   5      8 =  2^3
      2       42 =  2 *   3 *   7      8 =  2^3
      3       70 =  2 *   5 *   7      8 =  2^3
      4      105 =  3 *   5 *   7     27 =  3^3
     13      385 =  5 *   7 *  11    125 =  5^3
     29      969 =  3 *  17 *  19     27 =  3^3
     30     1001 =  7 *  11 *  13    343 =  7^3 = s(1)
     39     1309 =  7 *  11 *  17    343 =  7^3 = s(2)
     40     1311 =  3 *  19 *  23     27 =  3^3
     45     1547 =  7 *  13 *  17    343 =  7^3 = s(3)
     68     2431 = 11 *  13 *  17   1331 = 11^3
   1980    61585 =  5 * 109 * 113    125 =  5^3
  24929   780829 =  7 * 331 * 337    343 =  7^3
.
Let p(m) = prime(m), i the smallest index such that p(m) | a(i), j the smallest index such that lpf(a(j)) = p(m), and k the largest index such that p(m) | a(k). Let c(m) be the number of terms divisible by prime(m).
Table of m, p(m), i, j, k, c(m) for m = 1..5:
  m  p(m)    i    j        k     c(m)
  -----------------------------------
  1    2     1    1        3       3
  2    3     1    4       40      23
  3    5     1   13     1980     359
  4    7     2   30    24929    2050
  5   11     5   68   580381   22461

Maple

N:= 2000: # for terms <= N
P:= select(isprime, [2, seq(i, i=3..N/6, 2)]): nP:= nops(P):
sort(select(`<=`, [seq(seq(seq(P[i]*P[j]*P[k], i=1..j-1), j=2..k-1), k = 3..nP)], N)); # Robert Israel, May 11 2026

Mathematica

s = Select[Range[1800], PrimeNu[#] == PrimeOmega[#] == 3 &]; Select[s, Function[{p, q}, AllTrue[q, # < p^3 &]] @@ {First[#1], #2} & @@ TakeDrop[FactorInteger[#][[;; , 1]], 1] &]

Crossrefs

Cf. A005117, A007304, A020639, A046301, A120944, A138109, A383177, A393644.

Sequence in context: A219742 A348557 A382873 * A376800 A257832 A390326

Adjacent sequences: A395896 A395897 A395898 * A395900 A395901 A395902

Keyword

nonn,easy

Author

Michael De Vlieger, May 09 2026

Status

approved