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%I #10 May 16 2026 00:08:03
%S 30,42,70,105,165,195,231,255,273,285,345,357,385,399,429,455,483,561,
%T 595,627,663,665,715,741,759,805,897,935,969,1001,1015,1045,1085,1105,
%U 1173,1235,1265,1295,1309,1311,1435,1463,1495,1505,1547,1595,1615,1645,1705
%N Sphenic numbers k = p*q*r, primes p < q < r, such that r < p^3.
%C Proper subset of A393644.
%C Superset of A383177.
%C Analogous to A138109, a sequence which instead pertains to squarefree semiprimes A006881.
%C Let lpf(n) = A020639(n) be the smallest prime factor of n.
%C Smallest term with lpf(a(n)) = prime(i) is prime(i)*prime(i+1)*prime(i+2) = A046301(n).
%C 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).
%H Michael De Vlieger, <a href="/A395899/b395899.txt">Table of n, a(n) for n = 1..10000</a>
%H Michael De Vlieger, <a href="/A395899/a395899.png">Plot prime(i) | a(n) at (x,y) = (n,i)</a>, for n = 1..2400, 8X vertical exaggeration.
%e Let s = A383177 and lpf = A020639.
%e Table of n, a(n) for select n:
%e n a(n) lpf(a(n))^3
%e ---------------------------------------------
%e 1 30 = 2 * 3 * 5 8 = 2^3
%e 2 42 = 2 * 3 * 7 8 = 2^3
%e 3 70 = 2 * 5 * 7 8 = 2^3
%e 4 105 = 3 * 5 * 7 27 = 3^3
%e 13 385 = 5 * 7 * 11 125 = 5^3
%e 29 969 = 3 * 17 * 19 27 = 3^3
%e 30 1001 = 7 * 11 * 13 343 = 7^3 = s(1)
%e 39 1309 = 7 * 11 * 17 343 = 7^3 = s(2)
%e 40 1311 = 3 * 19 * 23 27 = 3^3
%e 45 1547 = 7 * 13 * 17 343 = 7^3 = s(3)
%e 68 2431 = 11 * 13 * 17 1331 = 11^3
%e 1980 61585 = 5 * 109 * 113 125 = 5^3
%e 24929 780829 = 7 * 331 * 337 343 = 7^3
%e .
%e 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).
%e Table of m, p(m), i, j, k, c(m) for m = 1..5:
%e m p(m) i j k c(m)
%e -----------------------------------
%e 1 2 1 1 3 3
%e 2 3 1 4 40 23
%e 3 5 1 13 1980 359
%e 4 7 2 30 24929 2050
%e 5 11 5 68 580381 22461
%p N:= 2000: # for terms <= N
%p P:= select(isprime, [2,seq(i,i=3..N/6,2)]): nP:= nops(P):
%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
%t s = Select[Range[1800], PrimeNu[#] == PrimeOmega[#] == 3 &]; Select[s, Function[{p, q}, AllTrue[q, # < p^3 &]] @@ {First[#1], #2} & @@ TakeDrop[FactorInteger[#][[;; , 1]], 1] &]
%Y Cf. A005117, A007304, A020639, A046301, A120944, A138109, A383177, A393644.
%K nonn,easy
%O 1,1
%A _Michael De Vlieger_, May 09 2026