A338471 Products of three prime numbers of odd index.

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

Author

Gus Wiseman, Nov 08 2020

Keywords

• nonn

Comments

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

Examples

			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}
		

Links

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

Crossrefs

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.

Programs

Maple

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

Mathematica

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

PARI

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

Python

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

Python

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