OFFSET
1,1
COMMENTS
Also Heinz numbers of integer partitions with 3 parts, all of which are odd. These partitions are counted by A001399.
EXAMPLE
The sequence of terms together with their prime indices begins:
8: {1,1,1} 268: {1,1,19} 575: {3,3,9}
20: {1,1,3} 275: {3,3,5} 578: {1,7,7}
44: {1,1,5} 292: {1,1,21} 590: {1,3,17}
50: {1,3,3} 310: {1,3,11} 596: {1,1,35}
68: {1,1,7} 332: {1,1,23} 605: {3,5,5}
92: {1,1,9} 374: {1,5,7} 628: {1,1,37}
110: {1,3,5} 388: {1,1,25} 668: {1,1,39}
124: {1,1,11} 410: {1,3,13} 670: {1,3,19}
125: {3,3,3} 412: {1,1,27} 682: {1,5,11}
164: {1,1,13} 425: {3,3,7} 716: {1,1,41}
170: {1,3,7} 436: {1,1,29} 730: {1,3,21}
188: {1,1,15} 470: {1,3,15} 764: {1,1,43}
230: {1,3,9} 506: {1,5,9} 775: {3,3,11}
236: {1,1,17} 508: {1,1,31} 782: {1,7,9}
242: {1,5,5} 548: {1,1,33} 788: {1,1,45}
MATHEMATICA
Select[Range[100], PrimeOmega[#]==3&&OddQ[Times@@PrimePi/@First/@FactorInteger[#]]&]
PROG
(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 sympy import primerange
from itertools import combinations_with_replacement as mc
def aupto(limit):
pois = [p for i, p in enumerate(primerange(2, limit//4+1)) if i%2 == 0]
return sorted(set(a*b*c for a, b, c in mc(pois, 3) if a*b*c <= limit))
print(aupto(971)) # Michael S. Branicky, Aug 20 2021
(Python)
from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A338471(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(integer_nthroot(x, 3)[0]+1), 1)) 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
KEYWORD
nonn,changed
AUTHOR
Gus Wiseman, Nov 08 2020
STATUS
approved