A338557 Products of three distinct prime numbers of even index.

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

Author

Gus Wiseman, Nov 08 2020

Keywords

• nonn

Comments

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.

Examples

			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}
		

Crossrefs

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.

Programs

Mathematica

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

PARI

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

Python

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