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A338469
Products of three odd prime numbers of odd index.
3
125, 275, 425, 575, 605, 775, 935, 1025, 1175, 1265, 1331, 1445, 1475, 1675, 1705, 1825, 1955, 2057, 2075, 2255, 2425, 2575, 2585, 2635, 2645, 2725, 2783, 3175, 3179, 3245, 3425, 3485, 3565, 3685, 3725, 3751, 3925, 3995, 4015, 4175, 4301, 4475, 4565, 4715
OFFSET
1,1
COMMENTS
Also Heinz numbers of integer partitions with 3 parts, all of which are odd and > 1. These partitions are counted by A001399.
LINKS
EXAMPLE
The sequence of terms together with their prime indices begins:
125: {3,3,3} 1825: {3,3,21} 3425: {3,3,33}
275: {3,3,5} 1955: {3,7,9} 3485: {3,7,13}
425: {3,3,7} 2057: {5,5,7} 3565: {3,9,11}
575: {3,3,9} 2075: {3,3,23} 3685: {3,5,19}
605: {3,5,5} 2255: {3,5,13} 3725: {3,3,35}
775: {3,3,11} 2425: {3,3,25} 3751: {5,5,11}
935: {3,5,7} 2575: {3,3,27} 3925: {3,3,37}
1025: {3,3,13} 2585: {3,5,15} 3995: {3,7,15}
1175: {3,3,15} 2635: {3,7,11} 4015: {3,5,21}
1265: {3,5,9} 2645: {3,9,9} 4175: {3,3,39}
1331: {5,5,5} 2725: {3,3,29} 4301: {5,7,9}
1445: {3,7,7} 2783: {5,5,9} 4475: {3,3,41}
1475: {3,3,17} 3175: {3,3,31} 4565: {3,5,23}
1675: {3,3,19} 3179: {5,7,7} 4715: {3,9,13}
1705: {3,5,11} 3245: {3,5,17} 4775: {3,3,43}
MAPLE
N:= 10000: # for terms <= N
P0:= [seq(ithprime(i), i=3..numtheory:-pi(floor(N/25)), 2)]:
sort(select(`<=`, [seq(seq(seq(P0[i]*P0[j]*P0[k], k=1..j), j=1..i), i=1..nops(P0))], N)); # Robert Israel, Nov 12 2020
MATHEMATICA
Select[Range[1, 1000, 2], PrimeOmega[#]==3&&OddQ[Times@@PrimePi/@First/@FactorInteger[#]]&]
PROG
(PARI) isok(m) = my(f=factor(m)); (m%2) && (bigomega(f)==3) && (#select(x->!(x%2), apply(primepi, f[, 1]~)) == 0); \\ Michel Marcus, Nov 10 2020
(Python)
from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A338469(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(5, integer_nthroot(x, 3)[0]+1), 3)) 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
CROSSREFS
A046316 allows all primes (strict: A046389).
A338471 allows all odd primes (strict: A307534).
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)).
A005408 lists odds (strict: A056911).
A008284 counts partitions by sum and length.
A014311 is a ranking of 3-part compositions (strict: A337453).
A014612 lists Heinz numbers of 3-part partitions (strict: A007304).
A023023 counts 3-part relatively prime partitions (strict: A101271).
A066207 lists numbers with all even prime indices (strict: A258117).
A066208 lists numbers with all odd prime indices (strict: A258116).
A075818 lists even Heinz numbers of 3-part partitions (strict: A075819).
A285508 lists Heinz numbers of non-strict 3-part partitions.
Sequence in context: A260604 A023079 A172460 * A253226 A223182 A256362
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 08 2020
STATUS
approved