A338556 Products of three prime numbers of even index.

27, 63, 117, 147, 171, 261, 273, 333, 343, 387, 399, 477, 507, 549, 609, 637, 639, 711, 741, 777, 801, 903, 909, 931, 963, 1017, 1083, 1113, 1131, 1179, 1183, 1251, 1281, 1359, 1421, 1443, 1467, 1491, 1557, 1629, 1653, 1659, 1677, 1729, 1737, 1791, 1813, 1869

Offset: 1

Author

Gus Wiseman, Nov 08 2020

Keywords

• nonn

Comments

All terms are odd.

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

Examples

			The sequence of terms together with their prime indices begins:
      27: {2,2,2}      637: {4,4,6}     1183: {4,6,6}
      63: {2,2,4}      639: {2,2,20}    1251: {2,2,34}
     117: {2,2,6}      711: {2,2,22}    1281: {2,4,18}
     147: {2,4,4}      741: {2,6,8}     1359: {2,2,36}
     171: {2,2,8}      777: {2,4,12}    1421: {4,4,10}
     261: {2,2,10}     801: {2,2,24}    1443: {2,6,12}
     273: {2,4,6}      903: {2,4,14}    1467: {2,2,38}
     333: {2,2,12}     909: {2,2,26}    1491: {2,4,20}
     343: {4,4,4}      931: {4,4,8}     1557: {2,2,40}
     387: {2,2,14}     963: {2,2,28}    1629: {2,2,42}
     399: {2,4,8}     1017: {2,2,30}    1653: {2,8,10}
     477: {2,2,16}    1083: {2,8,8}     1659: {2,4,22}
     507: {2,6,6}     1113: {2,4,16}    1677: {2,6,14}
     549: {2,2,18}    1131: {2,6,10}    1729: {4,6,8}
     609: {2,4,10}    1179: {2,2,32}    1737: {2,2,44}
		

Crossrefs

A014612 allows all prime indices (not just even) (strict: A007304).

A066207 allows products of any length (strict: A258117).

A338471 is the version for odds instead of evens (strict: A307534).

A338557 is the strict case.

A014311 is a ranking of ordered triples (strict: A337453).

A001399(n-3) counts 3-part partitions (strict: A001399(n-6)).

A005117 lists squarefree numbers, with even case A039956.

A008284 counts partitions by sum and length (strict: A008289).

A023023 counts 3-part relatively prime partitions (strict: A101271).

A046316 lists products of exactly three odd primes (strict: A046389).

A066208 lists numbers with all odd prime indices (strict: A258116).

A075818 lists even Heinz numbers of 3-part partitions (strict: A075819).

A307719 counts 3-part pairwise coprime partitions (strict: A220377).

A285508 lists Heinz numbers of non-strict triples.

Cf. A000217, A001221, A001222, A037144, A056239, A112798, A337599, A337600.

Subsequence of A332820.

Programs

Mathematica

Select[Range[1000],PrimeOmega[#]==3&&OddQ[Times@@(1+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 itertools import filterfalse
from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A338556(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)+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(k,isqrt(x//k)+1),a))))
    return bisection(f,n,n) # Chai Wah Wu, Oct 18 2024