A338556 -id:A338556 - cp's OEIS frontend

A007304 Sphenic numbers: products of 3 distinct primes.

30, 42, 66, 70, 78, 102, 105, 110, 114, 130, 138, 154, 165, 170, 174, 182, 186, 190, 195, 222, 230, 231, 238, 246, 255, 258, 266, 273, 282, 285, 286, 290, 310, 318, 322, 345, 354, 357, 366, 370, 374, 385, 399, 402, 406, 410, 418, 426, 429, 430, 434, 435, 438

Offset: 1

Simon Plouffe

Note the distinctions between this and "n has exactly three prime factors" (A014612) or "n has exactly three distinct prime factors." (A033992). The word "sphenic" also means "shaped like a wedge" [American Heritage Dictionary] as in dentation with "sphenic molars." - Jonathan Vos Post, Sep 11 2005
Also the volume of a sphenic brick. A sphenic brick is a rectangular parallelepiped whose sides are components of a sphenic number, namely whose sides are three distinct primes. Example: The distinct prime triple (3,5,7) produces a 3x5x7 unit brick which has volume 105 cubic units. 3-D analog of 2-D A037074 Product of twin primes, per Cino Hilliard's comment. Compare with 3-D A107768 Golden 3-almost primes = Volumes of bricks (rectangular parallelepipeds) each of whose faces has golden semiprime area. - Jonathan Vos Post, Jan 08 2007
Sum(n>=1,  1/a(n)^s) = (1/6)*(P(s)^3 - P(3*s) - 3*(P(s)*P(2*s)-P(3*s))), where P is prime zeta function. - Enrique Pérez Herrero, Jun 28 2012
Also numbers n with A001222(n)=3 and A001221(n)=3. - Enrique Pérez Herrero, Jun 28 2012
n = 265550 is the smallest n with a(n) (=1279789) < A006881(n) (=1279793). - Peter Dolland, Apr 11 2020

			From _Gus Wiseman_, Nov 05 2020: (Start)
Also Heinz numbers of strict integer partitions into three parts, where the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). These partitions are counted by A001399(n-6) = A069905(n-3), with ordered version A001399(n-6)*6. The sequence of terms together with their prime indices begins:
     30: {1,2,3}     182: {1,4,6}     286: {1,5,6}
     42: {1,2,4}     186: {1,2,11}    290: {1,3,10}
     66: {1,2,5}     190: {1,3,8}     310: {1,3,11}
     70: {1,3,4}     195: {2,3,6}     318: {1,2,16}
     78: {1,2,6}     222: {1,2,12}    322: {1,4,9}
    102: {1,2,7}     230: {1,3,9}     345: {2,3,9}
    105: {2,3,4}     231: {2,4,5}     354: {1,2,17}
    110: {1,3,5}     238: {1,4,7}     357: {2,4,7}
    114: {1,2,8}     246: {1,2,13}    366: {1,2,18}
    130: {1,3,6}     255: {2,3,7}     370: {1,3,12}
    138: {1,2,9}     258: {1,2,14}    374: {1,5,7}
    154: {1,4,5}     266: {1,4,8}     385: {3,4,5}
    165: {2,3,5}     273: {2,4,6}     399: {2,4,8}
    170: {1,3,7}     282: {1,2,15}    402: {1,2,19}
    174: {1,2,10}    285: {2,3,8}     406: {1,4,10}
(End)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
"Sphenic", The American Heritage Dictionary of the English Language, Fourth Edition, Houghton Mifflin Company, 2000.

Products of exactly k distinct primes, for k = 1 to 6: A000040, A006881. A007304, A046386, A046387, A067885.
Cf. A002033, A010051, A020639, A037074, A046393, A061299, A067467, A071140, A096917, A096918, A096919, A100765, A103653, A107464, A107768, A179643, A179695.
Cf. A162143 (a(n)^2).
For the following, NNS means "not necessarily strict".
A014612 is the NNS version.
A046389 is the restriction to odds (NNS: A046316).
A075819 is the restriction to evens (NNS: A075818).
A239656 gives first differences.
A285508 lists terms of A014612 that are not squarefree.
A307534 is the case where all prime indices are odd (NNS: A338471).
A337453 is a different ranking of ordered triples (NNS: A014311).
A338557 is the case where all prime indices are even (NNS: A338556).
A001399(n-6) counts strict 3-part partitions (NNS: A001399(n-3)).
A005117 lists squarefree numbers.
A008289 counts strict partitions by sum and length.
A220377 counts 3-part pairwise coprime strict partitions (NNS: A307719).
Cf. A000212, A000217, A001840, A101271, A284825, A321773, A337599, A337605.

a007304 n = a007304_list !! (n-1)
a007304_list = filter f [1..] where
f u = p < q && q < w && a010051 w == 1 where
p = a020639 u; v = div u p; q = a020639 v; w = div v q
-- Reinhard Zumkeller, Mar 23 2014

with(numtheory): a:=proc(n) if bigomega(n)=3 and nops(factorset(n))=3 then n else fi end: seq(a(n),n=1..450); # Emeric Deutsch
A007304 := proc(n)
    option remember;
    local a;
    if n =1 then
        30;
    else
        for a from procname(n-1)+1 do
            if bigomega(a)=3 and nops(factorset(a))=3 then
                return a;
            end if;
        end do:
    end if;
end proc: # R. J. Mathar, Dec 06 2016
is_a := proc(n) local P; P := NumberTheory:-PrimeFactors(n); nops(P) = 3 and n = mul(P) end:
A007304List := upto -> select(is_a, [seq(1..upto)]):  # Peter Luschny, Apr 14 2025

Union[Flatten[Table[Prime[n]*Prime[m]*Prime[k], {k, 20}, {n, k+1, 20}, {m, n+1, 20}]]]
Take[ Sort@ Flatten@ Table[ Prime@i Prime@j Prime@k, {i, 3, 21}, {j, 2, i - 1}, {k, j - 1}], 53] (* Robert G. Wilson v *)
With[{upto=500},Sort[Select[Times@@@Subsets[Prime[Range[Ceiling[upto/6]]],{3}],#<=upto&]]] (* Harvey P. Dale, Jan 08 2015 *)
Select[Range[100],SquareFreeQ[#]&&PrimeOmega[#]==3&] (* Gus Wiseman, Nov 05 2020 *)

for(n=1,1e4,if(bigomega(n)==3 && omega(n)==3,print1(n", "))) \\ Charles R Greathouse IV, Jun 10 2011

list(lim)=my(v=List(),t);forprime(p=2,(lim)^(1/3),forprime(q=p+1,sqrt(lim\p),t=p*q;forprime(r=q+1,lim\t,listput(v,t*r))));vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 20 2011

list(lim)=my(v=List(), t); forprime(p=2, sqrtnint(lim\=1,3), forprime(q=p+1, sqrtint(lim\p), t=p*q; forprime(r=q+1, lim\t, listput(v, t*r)))); Set(v) \\ Charles R Greathouse IV, Jan 21 2025

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A007304(n):
    def f(x): return int(n+x-sum(primepi(x//(k*m))-b for a,k in enumerate(primerange(integer_nthroot(x,3)[0]+1),1) for b,m in enumerate(primerange(k+1,isqrt(x//k)+1),a+1)))
    kmin, kmax = 0,1
    while f(kmax) > kmax:
        kmax <<= 1
    while kmax-kmin > 1:
        kmid = kmax+kmin>>1
        if f(kmid) <= kmid:
            kmax = kmid
        else:
            kmin = kmid
    return kmax # Chai Wah Wu, Aug 29 2024

def is_a(n):
    P = prime_divisors(n)
    return len(P) == 3 and prod(P) == n
print([n for n in range(1, 439) if is_a(n)]) # Peter Luschny, Apr 14 2025

A008683(a(n)) = -1.
A000005(a(n)) = 8. - R. J. Mathar, Aug 14 2009
A002033(a(n)-1) = 13. - Juri-Stepan Gerasimov, Oct 07 2009, R. J. Mathar, Oct 14 2009
A178254(a(n)) = 36. - Reinhard Zumkeller, May 24 2010
A050326(a(n)) = 5, subsequence of A225228. - Reinhard Zumkeller, May 03 2013
a(n) ~ 2n log n/(log log n)^2. - Charles R Greathouse IV, Sep 14 2015

More terms from Robert G. Wilson v, Jan 04 2006

Comment concerning number of divisors corrected by R. J. Mathar, Aug 14 2009

A332820 Integers in the multiplicative subgroup of positive rationals generated by the products of two consecutive primes and the cubes of primes. Numbers k for which A048675(k) is a multiple of three.

Original entry on oeis.org

1, 6, 8, 14, 15, 20, 26, 27, 33, 35, 36, 38, 44, 48, 50, 51, 58, 63, 64, 65, 68, 69, 74, 77, 84, 86, 90, 92, 93, 95, 106, 110, 112, 117, 119, 120, 122, 123, 124, 125, 141, 142, 143, 145, 147, 156, 158, 160, 161, 162, 164, 170, 171, 177, 178, 185, 188, 196, 198, 201, 202, 208, 209, 210, 214, 215, 216, 217, 219, 221, 225

Offset: 1

Antti Karttunen and Peter Munn, Feb 25 2020

nonn

The positive integers are partitioned between this sequence, A332821 and A332822, which list the integers in respective cosets of the subgroup.
As the sequence lists the integers in a multiplicative subgroup of the positive rationals, the sequence is closed under multiplication and, provided the result is an integer, under division.
It follows that for any n in this sequence, all powers n^k are present (k >= 0), as are all cubes.
If we take each odd term of this sequence and replace each prime in its factorization by the next smaller prime, the resulting numbers are a permutation of the full sequence; and if we take the square root of each square term we get the full sequence.
There are no primes in the sequence, therefore if k is present and p is a prime, k*p and k/p are absent (noting that k/p might not be an integer). This property extends from primes to all terms of A050376 (often called Fermi-Dirac primes), therefore to squares of primes, 4th powers of primes etc.
The terms are the even numbers in A332821 halved. The terms are also the numbers m such that 5m is in A332821, and so on for alternate primes: 11, 17, 23 etc. Likewise, the terms are the numbers m such that 3m is in A332822, and so on for alternate primes: 7, 13, 19 etc.
The numbers that are half of the even terms of this sequence are in A332822, which consists exactly of those numbers. The numbers that are one third of the terms that are multiples of 3 are in A332821, which consists exactly of those numbers. These properties extend in a pattern of alternating primes as described in the previous paragraph.
If k is an even number, exactly one of {k/2, k, 2k} is in the sequence (cf. A191257 / A067368 / A213258); and generally if k is a multiple of a prime p, exactly one of {k/p, k, k*p} is in the sequence.
If m and n are in this sequence then so is m*n (the definition of "multiplicative semigroup"), while if n is in this sequence, and x is in the complement A359830, then n*x is in A359830. This essentially follows from the fact that A048675 is totally additive sequence. Compare to A329609. - Antti Karttunen, Jan 17 2023

Positions of zeros in A332823; equivalently, numbers in row 3k of A277905 for some k >= 0.
Cf. A048675, A195017, A332821, A332822, A353350 (characteristic function), A353348 (its Dirichlet inverse), A359830 (complement).
Comparable 2 or 3-way classifications: A000379/A000028, A001969/A000069, A003159/A036554, A005843/A005408, A028260/A026424, A191257/A067368/A213258, A325431/A325432, A329609/A329604/A332812.
Subsequences: A000578\{0}, A006094, A090090, A099788, A245630 (A191002 in ascending order), A244726\{0}, A325698, A338471, A338556, A338907.
Subsequence of {1} U A268388.

Select[Range@ 225, Or[Mod[Total@ #, 3] == 0 &@ Map[#[[-1]]*2^(PrimePi@ #[[1]] - 1) &, FactorInteger[#]], # == 1] &] (* Michael De Vlieger, Mar 15 2020 *)

isA332820(n) =  { my(f = factor(n)); !((sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2)%3); };

{a(n) : n >= 1} = {1} U {2 * A332822(k) : k >= 1} U {A003961(a(k)) : k >= 1}.
{a(n) : n >= 1} = {1} U {a(k)^2 : k >= 1} U {A331590(2, A332822(k)) : k >= 1}.
From Peter Munn, Mar 17 2021: (Start)
{a(n) : n >= 1} = {k : k >= 1, 3|A048675(k)}.
{a(n) : n >= 1} = {k : k >= 1, 3|A195017(k)}.
{a(n) : n >= 1} = {A332821(k)/2 : k >= 1, 2|A332821(k)}.
{a(n) : n >= 1} = {A332822(k)/3 : k >= 1, 3|A332822(k)}.
(End)

New name from Peter Munn, Mar 08 2021

A046316 Numbers of the form pqr where p,q,r are (not necessarily distinct) odd primes.

Original entry on oeis.org

27, 45, 63, 75, 99, 105, 117, 125, 147, 153, 165, 171, 175, 195, 207, 231, 245, 255, 261, 273, 275, 279, 285, 325, 333, 343, 345, 357, 363, 369, 385, 387, 399, 423, 425, 429, 435, 455, 465, 475, 477, 483, 507, 531, 539, 549, 555, 561, 575, 595, 603, 605

Offset: 1

Patrick De Geest, Jun 15 1998

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

A369979 sorted into ascending order.
Subsequence of A014612 and of A046340.
Cf. A255646 (final digits), A369054, A369058 (characteristic function), A369252 [= A003415(a(n))].
Subsequences: A046389, A046373, A046405, A075814, A338469, A338556, A338557, A369246.

a046316 n = a046316_list !! (n-1)
a046316_list = filter ((== 3) . a001222) [1, 3 ..]
-- Reinhard Zumkeller, May 05 2015

list(lim)=my(v=List(),pq); forprime(p=3,lim\9, forprime(q=3,min(lim\3\p,p), pq=p*q; forprime(r=3,lim\pq, listput(v, pq*r)))); Set(v) \\ Charles R Greathouse IV, Aug 23 2017

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A046316(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return int(n+x-sum(primepi(x//(k*m))-b+1 for a,k in enumerate(primerange(3,integer_nthroot(x,3)[0]+1),2) for b,m in enumerate(primerange(k,isqrt(x//k)+1),a)))
    return bisection(f,n,n) # Chai Wah Wu, Oct 18 2024

Definition clarified by N. J. A. Sloane, Dec 19 2017

A338471 Products of three prime numbers of odd index.

Original entry on oeis.org

8, 20, 44, 50, 68, 92, 110, 124, 125, 164, 170, 188, 230, 236, 242, 268, 275, 292, 310, 332, 374, 388, 410, 412, 425, 436, 470, 506, 508, 548, 575, 578, 590, 596, 605, 628, 668, 670, 682, 716, 730, 764, 775, 782, 788, 830, 844, 902, 908, 932, 935, 964, 970

Offset: 1

Gus Wiseman, Nov 08 2020

nonn

Also Heinz numbers of integer partitions with 3 parts, all of which are odd. These partitions are counted by A001399.

			The sequence of terms together with their prime indices begins:
       8: {1,1,1}      268: {1,1,19}     575: {3,3,9}
      20: {1,1,3}      275: {3,3,5}      578: {1,7,7}
      44: {1,1,5}      292: {1,1,21}     590: {1,3,17}
      50: {1,3,3}      310: {1,3,11}     596: {1,1,35}
      68: {1,1,7}      332: {1,1,23}     605: {3,5,5}
      92: {1,1,9}      374: {1,5,7}      628: {1,1,37}
     110: {1,3,5}      388: {1,1,25}     668: {1,1,39}
     124: {1,1,11}     410: {1,3,13}     670: {1,3,19}
     125: {3,3,3}      412: {1,1,27}     682: {1,5,11}
     164: {1,1,13}     425: {3,3,7}      716: {1,1,41}
     170: {1,3,7}      436: {1,1,29}     730: {1,3,21}
     188: {1,1,15}     470: {1,3,15}     764: {1,1,43}
     230: {1,3,9}      506: {1,5,9}      775: {3,3,11}
     236: {1,1,17}     508: {1,1,31}     782: {1,7,9}
     242: {1,5,5}      548: {1,1,33}     788: {1,1,45}

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

A066208 allows products of any length (strict: A258116).
A307534 is the squarefree case.
A338469 is the restriction to odds.
A338556 is the version for evens (strict: A338557).
A000009 counts partitions into odd parts (strict: A000700).
A001399(n-3) counts 3-part partitions (strict: A001399(n-6)).
A008284 counts partitions by sum and length.
A014311 is a ranking of ordered triples (strict: A337453).
A014612 lists Heinz numbers of all triples (strict: A007304).
A023023 counts 3-part relatively prime partitions (strict: A101271).
A023023 counts 3-part relatively prime partitions (strict: A078374).
A046316 lists products of exactly three odd primes (strict: A046389).
A066207 lists numbers with all even prime indices (strict: A258117).
A075818 lists even Heinz numbers of 3-part partitions (strict: A075819).
A285508 lists Heinz numbers of non-strict triples.
A307719 counts 3-part pairwise coprime partitions (strict: A220377).
Cf. A000217, A001221, A001222, A005117, A037144, A056239, A112798.
Subsequence of A332820.

N:= 1000: # for terms <= N
R:= NULL:
for i from 1 by 2 do
  p:= ithprime(i);
  if p^3 >= N then break fi;
  for j from i by 2 do
    q:= ithprime(j);
    if p*q^2 >= N then break fi;
    for k from j by 2 do
      x:= p*q*ithprime(k);
      if x > N then break fi;
      R:= R,x;
od od od:
sort([R]); # Robert Israel, Jun 11 2025

Select[Range[100],PrimeOmega[#]==3&&OddQ[Times@@PrimePi/@First/@FactorInteger[#]]&]

isok(m) = my(f=factor(m)); (bigomega(f)==3) && (#select(x->!(x%2), apply(primepi, f[,1]~)) == 0); \\ Michel Marcus, Nov 10 2020

from sympy import primerange
from itertools import combinations_with_replacement as mc
def aupto(limit):
    pois = [p for i, p in enumerate(primerange(2, limit//4+1)) if i%2 == 0]
    return sorted(set(a*b*c for a, b, c in mc(pois, 3) if a*b*c <= limit))
print(aupto(971)) # Michael S. Branicky, Aug 20 2021

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A338471(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return int(n+x-sum((primepi(x//(k*m))+1>>1)-(b+1>>1)+1 for a,k in filter(lambda x:x[0]&1,enumerate(primerange(integer_nthroot(x,3)[0]+1),1)) for b,m in filter(lambda x:x[0]&1,enumerate(primerange(k,isqrt(x//k)+1),a))))
    return bisection(f,n,n) # Chai Wah Wu, Oct 18 2024

A338557 Products of three distinct prime numbers of even index.

Original entry on oeis.org

273, 399, 609, 741, 777, 903, 1113, 1131, 1281, 1443, 1491, 1653, 1659, 1677, 1729, 1869, 2067, 2109, 2121, 2247, 2373, 2379, 2451, 2639, 2751, 2769, 2919, 3021, 3081, 3171, 3219, 3367, 3423, 3471, 3477, 3633, 3741, 3801, 3857, 3913, 3939, 4047, 4053, 4173

Offset: 1

Gus Wiseman, Nov 08 2020

nonn

All terms are odd.
Also sphenic numbers (A007304) with all even prime indices (A031215).
Also Heinz numbers of strict integer partitions with 3 parts, all of which are even. These partitions are counted by A001399.

			The sequence of terms together with their prime indices begins:
     273: {2,4,6}     1869: {2,4,24}    3219: {2,10,12}
     399: {2,4,8}     2067: {2,6,16}    3367: {4,6,12}
     609: {2,4,10}    2109: {2,8,12}    3423: {2,4,38}
     741: {2,6,8}     2121: {2,4,26}    3471: {2,6,24}
     777: {2,4,12}    2247: {2,4,28}    3477: {2,8,18}
     903: {2,4,14}    2373: {2,4,30}    3633: {2,4,40}
    1113: {2,4,16}    2379: {2,6,18}    3741: {2,10,14}
    1131: {2,6,10}    2451: {2,8,14}    3801: {2,4,42}
    1281: {2,4,18}    2639: {4,6,10}    3857: {4,8,10}
    1443: {2,6,12}    2751: {2,4,32}    3913: {4,6,14}
    1491: {2,4,20}    2769: {2,6,20}    3939: {2,6,26}
    1653: {2,8,10}    2919: {2,4,34}    4047: {2,8,20}
    1659: {2,4,22}    3021: {2,8,16}    4053: {2,4,44}
    1677: {2,6,14}    3081: {2,6,22}    4173: {2,6,28}
    1729: {4,6,8}     3171: {2,4,36}    4179: {2,4,46}

For the following, NNS means "not necessarily strict".
A007304 allows all prime indices (not just even) (NNS: A014612).
A046389 allows all odd primes (NNS: A046316).
A258117 allows products of any length (NNS: A066207).
A307534 is the version for odds instead of evens (NNS: A338471).
A337453 is a different ranking of ordered triples (NNS: A014311).
A338556 is the NNS version.
A001399(n-6) counts strict 3-part partitions (NNS: A001399(n-3)).
A005117 lists squarefree numbers, with even case A039956.
A078374 counts 3-part relatively prime strict partitions (NNS: A023023).
A075819 lists even Heinz numbers of strict triples (NNS: A075818).
A220377 counts 3-part pairwise coprime strict partitions (NNS: A307719).
A258116 lists squarefree numbers with all odd prime indices (NNS: A066208).
A285508 lists Heinz numbers of non-strict triples.
Cf. A000217, A001221, A001222, A037144, A056239, A112798, A337605.

Select[Range[1000],SquareFreeQ[#]&&PrimeOmega[#]==3&&OddQ[Times@@(1+PrimePi/@First/@FactorInteger[#])]&]

isok(m) = my(f=factor(m)); (bigomega(f)==3) && (omega(f)==3) && (#select(x->(x%2), apply(primepi, f[,1]~)) == 0); \\ Michel Marcus, Nov 10 2020

from itertools import filterfalse
from math import isqrt
from sympy import primepi, primerange, nextprime, integer_nthroot
def A338557(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return int(n+x-sum((primepi(x//(k*m))>>1)-(b>>1) for a,k in filterfalse(lambda x:x[0]&1,enumerate(primerange(3,integer_nthroot(x,3)[0]+1),2)) for b,m in filterfalse(lambda x:x[0]&1,enumerate(primerange(nextprime(k)+1,isqrt(x//k)+1),a+2))))
    return bisection(f,n,n) # Chai Wah Wu, Oct 18 2024

A338469 Products of three odd prime numbers of odd index.

Original entry on oeis.org

125, 275, 425, 575, 605, 775, 935, 1025, 1175, 1265, 1331, 1445, 1475, 1675, 1705, 1825, 1955, 2057, 2075, 2255, 2425, 2575, 2585, 2635, 2645, 2725, 2783, 3175, 3179, 3245, 3425, 3485, 3565, 3685, 3725, 3751, 3925, 3995, 4015, 4175, 4301, 4475, 4565, 4715

Offset: 1

Gus Wiseman, Nov 08 2020

nonn

Also Heinz numbers of integer partitions with 3 parts, all of which are odd and > 1. These partitions are counted by A001399.

			The sequence of terms together with their prime indices begins:
     125: {3,3,3}     1825: {3,3,21}    3425: {3,3,33}
     275: {3,3,5}     1955: {3,7,9}     3485: {3,7,13}
     425: {3,3,7}     2057: {5,5,7}     3565: {3,9,11}
     575: {3,3,9}     2075: {3,3,23}    3685: {3,5,19}
     605: {3,5,5}     2255: {3,5,13}    3725: {3,3,35}
     775: {3,3,11}    2425: {3,3,25}    3751: {5,5,11}
     935: {3,5,7}     2575: {3,3,27}    3925: {3,3,37}
    1025: {3,3,13}    2585: {3,5,15}    3995: {3,7,15}
    1175: {3,3,15}    2635: {3,7,11}    4015: {3,5,21}
    1265: {3,5,9}     2645: {3,9,9}     4175: {3,3,39}
    1331: {5,5,5}     2725: {3,3,29}    4301: {5,7,9}
    1445: {3,7,7}     2783: {5,5,9}     4475: {3,3,41}
    1475: {3,3,17}    3175: {3,3,31}    4565: {3,5,23}
    1675: {3,3,19}    3179: {5,7,7}     4715: {3,9,13}
    1705: {3,5,11}    3245: {3,5,17}    4775: {3,3,43}

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

A046316 allows all primes (strict: A046389).
A338471 allows all odd primes (strict: A307534).
A338556 is the version for evens (strict: A338557).
A000009 counts partitions into odd parts (strict: A000700).
A001399(n-3) counts 3-part partitions (strict: A001399(n-6)).
A005408 lists odds (strict: A056911).
A008284 counts partitions by sum and length.
A014311 is a ranking of 3-part compositions (strict: A337453).
A014612 lists Heinz numbers of 3-part partitions (strict: A007304).
A023023 counts 3-part relatively prime partitions (strict: A101271).
A066207 lists numbers with all even prime indices (strict: A258117).
A066208 lists numbers with all odd prime indices (strict: A258116).
A075818 lists even Heinz numbers of 3-part partitions (strict: A075819).
A285508 lists Heinz numbers of non-strict 3-part partitions.
Cf. A001221, A001222, A002620, A005117, A037144, A056239, A112798.

N:= 10000: # for terms <= N
P0:= [seq(ithprime(i),i=3..numtheory:-pi(floor(N/25)),2)]:
sort(select(`<=`,[seq(seq(seq(P0[i]*P0[j]*P0[k],k=1..j),j=1..i),i=1..nops(P0))], N)); # Robert Israel, Nov 12 2020

Select[Range[1,1000,2],PrimeOmega[#]==3&&OddQ[Times@@PrimePi/@First/@FactorInteger[#]]&]

isok(m) = my(f=factor(m)); (m%2) && (bigomega(f)==3) && (#select(x->!(x%2), apply(primepi, f[,1]~)) == 0); \\ Michel Marcus, Nov 10 2020

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A338469(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return int(n+x-sum((primepi(x//(k*m))+1>>1)-(b+1>>1)+1 for a,k in filter(lambda x:x[0]&1,enumerate(primerange(5,integer_nthroot(x,3)[0]+1),3)) for b,m in filter(lambda x:x[0]&1,enumerate(primerange(k,isqrt(x//k)+1),a))))
    return bisection(f,n,n) # Chai Wah Wu, Oct 18 2024

cp's OEIS Frontend

A007304 Sphenic numbers: products of 3 distinct primes.

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A332820 Integers in the multiplicative subgroup of positive rationals generated by the products of two consecutive primes and the cubes of primes. Numbers k for which A048675(k) is a multiple of three.

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A046316 Numbers of the form p*q*r where p,q,r are (not necessarily distinct) odd primes.

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A338471 Products of three prime numbers of odd index.

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A338557 Products of three distinct prime numbers of even index.

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A338469 Products of three odd prime numbers of odd index.

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A046316 Numbers of the form pqr where p,q,r are (not necessarily distinct) odd primes.