A338469 Products of three odd prime numbers of odd index.

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

Robert Israel, Table of n, a(n) for n = 1..10000

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.

Cf. A001221, A001222, A002620, A005117, A037144, A056239, A112798.

Sequence in context: A260604 A023079 A172460 * A253226 A223182 A256362

Adjacent sequences: A338466 A338467 A338468 * A338470 A338471 A338472

Keyword

nonn

Author

Gus Wiseman, Nov 08 2020

Status

approved