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

1,1

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.

Links

Table of n, a(n) for n=1..44.

Example

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}

Mathematica

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

Prog

(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

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.

Sequence in context: A043821 A217004 A256638 * A347882 A347888 A306812

Adjacent sequences: A338554 A338555 A338556 * A338558 A338559 A338560

Keyword

nonn

Author

Gus Wiseman, Nov 08 2020

Status

approved