A338557 - OEIS

%I #17 Oct 18 2024 18:07:52

%S 273,399,609,741,777,903,1113,1131,1281,1443,1491,1653,1659,1677,1729,

%T 1869,2067,2109,2121,2247,2373,2379,2451,2639,2751,2769,2919,3021,

%U 3081,3171,3219,3367,3423,3471,3477,3633,3741,3801,3857,3913,3939,4047,4053,4173

%N Products of three distinct prime numbers of even index.

%C All terms are odd.

%C Also sphenic numbers (A007304) with all even prime indices (A031215).

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

%e The sequence of terms together with their prime indices begins:

%e      273: {2,4,6}     1869: {2,4,24}    3219: {2,10,12}

%e      399: {2,4,8}     2067: {2,6,16}    3367: {4,6,12}

%e      609: {2,4,10}    2109: {2,8,12}    3423: {2,4,38}

%e      741: {2,6,8}     2121: {2,4,26}    3471: {2,6,24}

%e      777: {2,4,12}    2247: {2,4,28}    3477: {2,8,18}

%e      903: {2,4,14}    2373: {2,4,30}    3633: {2,4,40}

%e     1113: {2,4,16}    2379: {2,6,18}    3741: {2,10,14}

%e     1131: {2,6,10}    2451: {2,8,14}    3801: {2,4,42}

%e     1281: {2,4,18}    2639: {4,6,10}    3857: {4,8,10}

%e     1443: {2,6,12}    2751: {2,4,32}    3913: {4,6,14}

%e     1491: {2,4,20}    2769: {2,6,20}    3939: {2,6,26}

%e     1653: {2,8,10}    2919: {2,4,34}    4047: {2,8,20}

%e     1659: {2,4,22}    3021: {2,8,16}    4053: {2,4,44}

%e     1677: {2,6,14}    3081: {2,6,22}    4173: {2,6,28}

%e     1729: {4,6,8}     3171: {2,4,36}    4179: {2,4,46}

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

%o (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

%o (Python)

%o from itertools import filterfalse

%o from math import isqrt

%o from sympy import primepi, primerange, nextprime, integer_nthroot

%o def A338557(n):

%o     def bisection(f,kmin=0,kmax=1):

%o         while f(kmax) > kmax: kmax <<= 1

%o         while kmax-kmin > 1:

%o             kmid = kmax+kmin>>1

%o             if f(kmid) <= kmid:

%o                 kmax = kmid

%o             else:

%o                 kmin = kmid

%o         return kmax

%o     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))))

%o     return bisection(f,n,n) # _Chai Wah Wu_, Oct 18 2024

%Y For the following, NNS means "not necessarily strict".

%Y A007304 allows all prime indices (not just even) (NNS: A014612).

%Y A046389 allows all odd primes (NNS: A046316).

%Y A258117 allows products of any length (NNS: A066207).

%Y A307534 is the version for odds instead of evens (NNS: A338471).

%Y A337453 is a different ranking of ordered triples (NNS: A014311).

%Y A338556 is the NNS version.

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

%Y A005117 lists squarefree numbers, with even case A039956.

%Y A078374 counts 3-part relatively prime strict partitions (NNS: A023023).

%Y A075819 lists even Heinz numbers of strict triples (NNS: A075818).

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

%Y A258116 lists squarefree numbers with all odd prime indices (NNS: A066208).

%Y A285508 lists Heinz numbers of non-strict triples.

%Y Cf. A000217, A001221, A001222, A037144, A056239, A112798, A337605.

%K nonn

%O 1,1

%A _Gus Wiseman_, Nov 08 2020