A046389 -id:A046389 - cp's OEIS frontend

A338556 Products of three prime numbers of even index.

27, 63, 117, 147, 171, 261, 273, 333, 343, 387, 399, 477, 507, 549, 609, 637, 639, 711, 741, 777, 801, 903, 909, 931, 963, 1017, 1083, 1113, 1131, 1179, 1183, 1251, 1281, 1359, 1421, 1443, 1467, 1491, 1557, 1629, 1653, 1659, 1677, 1729, 1737, 1791, 1813, 1869

Offset: 1

Gus Wiseman, Nov 08 2020

nonn

All terms are odd.

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

			The sequence of terms together with their prime indices begins:
      27: {2,2,2}      637: {4,4,6}     1183: {4,6,6}
      63: {2,2,4}      639: {2,2,20}    1251: {2,2,34}
     117: {2,2,6}      711: {2,2,22}    1281: {2,4,18}
     147: {2,4,4}      741: {2,6,8}     1359: {2,2,36}
     171: {2,2,8}      777: {2,4,12}    1421: {4,4,10}
     261: {2,2,10}     801: {2,2,24}    1443: {2,6,12}
     273: {2,4,6}      903: {2,4,14}    1467: {2,2,38}
     333: {2,2,12}     909: {2,2,26}    1491: {2,4,20}
     343: {4,4,4}      931: {4,4,8}     1557: {2,2,40}
     387: {2,2,14}     963: {2,2,28}    1629: {2,2,42}
     399: {2,4,8}     1017: {2,2,30}    1653: {2,8,10}
     477: {2,2,16}    1083: {2,8,8}     1659: {2,4,22}
     507: {2,6,6}     1113: {2,4,16}    1677: {2,6,14}
     549: {2,2,18}    1131: {2,6,10}    1729: {4,6,8}
     609: {2,4,10}    1179: {2,2,32}    1737: {2,2,44}

A014612 allows all prime indices (not just even) (strict: A007304).
A066207 allows products of any length (strict: A258117).
A338471 is the version for odds instead of evens (strict: A307534).
A338557 is the strict case.
A014311 is a ranking of ordered triples (strict: A337453).
A001399(n-3) counts 3-part partitions (strict: A001399(n-6)).
A005117 lists squarefree numbers, with even case A039956.
A008284 counts partitions by sum and length (strict: A008289).
A023023 counts 3-part relatively prime partitions (strict: A101271).
A046316 lists products of exactly three odd primes (strict: A046389).
A066208 lists numbers with all odd prime indices (strict: A258116).
A075818 lists even Heinz numbers of 3-part partitions (strict: A075819).
A307719 counts 3-part pairwise coprime partitions (strict: A220377).
A285508 lists Heinz numbers of non-strict triples.
Cf. A000217, A001221, A001222, A037144, A056239, A112798, A337599, A337600.
Subsequence of A332820.

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

isok(m) = my(f=factor(m)); (bigomega(f)==3) && (#select(x->(x%2), apply(primepi, f[,1]~)) == 0); \\ Michel Marcus, Nov 10 2020

from itertools import filterfalse
from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A338556(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)+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(k,isqrt(x//k)+1),a))))
    return bisection(f,n,n) # Chai Wah Wu, Oct 18 2024

A234099 Integers of the form (pqr - 1)/2, where p, q, r are distinct primes.

Original entry on oeis.org

52, 82, 97, 115, 127, 136, 142, 172, 178, 192, 199, 214, 217, 227, 232, 241, 277, 280, 297, 304, 307, 313, 322, 325, 331, 332, 352, 357, 370, 379, 388, 397, 402, 430, 442, 448, 451, 457, 467, 478, 484, 493, 500, 502, 507, 511, 522, 532, 542, 547, 552, 556

Offset: 1

Clark Kimberling, Dec 27 2013

			52 = (3*5*7 - 1)/2.

Cf. A046389, A234100, A234101, A234093, A234102.

t = Select[Range[1, 10000, 2], Map[Last, FactorInteger[#]] == Table[1, {3}] &]; Take[(t - 1)/2, 120] (* A234099 *)
v = Flatten[Position[PrimeQ[(t - 1)/2], True]] ; w = Table[t[[v[[n]]]], {n, 1, Length[v]}]  (* A234100 *)
(w - 1)/2 (* A234101 *)    (* Peter J. C. Moses, Dec 23 2013 *)

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A234099(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))-b for a,k in enumerate(primerange(3,integer_nthroot(x,3)[0]+1),2) for b,m in enumerate(primerange(k+1,isqrt(x//k)+1),a+1)))
    return bisection(f,n,n)>>1 # Chai Wah Wu, Oct 18 2024

-1 + A234102.

a(n) = (A046389(n)-1)/2. - Chai Wah Wu, Oct 18 2024

A338471 Products of three prime numbers of odd index.

Original entry on oeis.org

8, 20, 44, 50, 68, 92, 110, 124, 125, 164, 170, 188, 230, 236, 242, 268, 275, 292, 310, 332, 374, 388, 410, 412, 425, 436, 470, 506, 508, 548, 575, 578, 590, 596, 605, 628, 668, 670, 682, 716, 730, 764, 775, 782, 788, 830, 844, 902, 908, 932, 935, 964, 970

Offset: 1

Gus Wiseman, Nov 08 2020

nonn

Also Heinz numbers of integer partitions with 3 parts, all of which are odd. These partitions are counted by A001399.

			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}

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

A066208 allows products of any length (strict: A258116).
A307534 is the squarefree case.
A338469 is the restriction to odds.
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)).
A008284 counts partitions by sum and length.
A014311 is a ranking of ordered triples (strict: A337453).
A014612 lists Heinz numbers of all triples (strict: A007304).
A023023 counts 3-part relatively prime partitions (strict: A101271).
A023023 counts 3-part relatively prime partitions (strict: A078374).
A046316 lists products of exactly three odd primes (strict: A046389).
A066207 lists numbers with all even prime indices (strict: A258117).
A075818 lists even Heinz numbers of 3-part partitions (strict: A075819).
A285508 lists Heinz numbers of non-strict triples.
A307719 counts 3-part pairwise coprime partitions (strict: A220377).
Cf. A000217, A001221, A001222, A005117, A037144, A056239, A112798.
Subsequence of A332820.

N:= 1000: # for terms <= N
R:= NULL:
for i from 1 by 2 do
  p:= ithprime(i);
  if p^3 >= N then break fi;
  for j from i by 2 do
    q:= ithprime(j);
    if p*q^2 >= N then break fi;
    for k from j by 2 do
      x:= p*q*ithprime(k);
      if x > N then break fi;
      R:= R,x;
od od od:
sort([R]); # Robert Israel, Jun 11 2025

Select[Range[100],PrimeOmega[#]==3&&OddQ[Times@@PrimePi/@First/@FactorInteger[#]]&]

isok(m) = my(f=factor(m)); (bigomega(f)==3) && (#select(x->!(x%2), apply(primepi, f[,1]~)) == 0); \\ Michel Marcus, Nov 10 2020

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

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

A338557 Products of three distinct prime numbers of even index.

Original entry on oeis.org

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

Gus Wiseman, Nov 08 2020

nonn

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.

			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}

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.

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

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

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

A128284 Numbers of the form m = p1 * p2 * p3 where for each d|m we have (d+m/d)/2 prime and p1 < p2 < p3 each prime.

Original entry on oeis.org

105, 165, 273, 345, 357, 385, 777, 897, 1045, 1173, 1353, 1653, 1677, 1705, 2193, 2233, 2373, 2905, 3157, 3237, 3333, 3417, 3445, 3553, 3565, 3945, 4053, 4585, 4953, 5665, 5817, 6097, 6513, 6693, 7077, 7833, 8437, 8565, 8845, 10153, 11005, 11433

Offset: 1

Kok Seng Chua (chuakokseng(AT)hotmail.com), Mar 05 2007

nonn

The symmetric representation of sigma (cf. A237593), SRS(a(n)), of any number in this sequence has between 4 and 8 regions with 3 regions impossible because p1 < p2 < p3 implies 2*p3 < p1*p2. When there are 8 regions they all have width 1 and their areas are the prime numbers (d+a(n)/d)/2 for the 4 respective pairs of divisors of a(n). In general, the areas of the regions in SRS(a(n)) need not be prime, except for the two symmetric outer regions (n+1)/2. - Hartmut F. W. Hoft, Jan 09 2021

			165=3*5*11 and (3*5*11+1)/2=83, (3+5*7)/2=19, (5+3*7)/2=13, (7+3*5)/2=11 are all primes, so 165 is a term.
From _Hartmut F. W. Hoft_, Jan 09 2021: (Start)
a(1) = 105 = 3*5*7 and SRS(a(1)) consists of four regions with areas ( 53, 43, 43, 53 ); the center areas have maximum width 2 and represent the sum of primes (3+35)/2 + (5+21)/2 + (7+15)/2 = 43.
a(17) = 2373 = 3*7*17 is the first number in the sequence whose symmetric representation of sigma consists of 8 regions, all of width 1 and the respective symmetric regions have areas:  (2373 + 1)/2 = 1187, (791 + 3)/2 = 397, (339 + 7)/2 = 173, (21 + 113)/2 = 67. (End)

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A128281, A005383, A128283, A128285, A128286.
Cf. A237048, A237591, A237593, A249223, A262045.
Subsequence of A046389.

(* function goodL[] is defined in A128283 *)
a128284[n_] := goodL[{1, n}, 3]
a128284[11433] (* Hartmut F. W. Hoft, Jan 09 2021 *)

A046405 Numbers of the form pqr where p,q,r are distinct odd palindromic primes (odd terms from A002385).

Original entry on oeis.org

105, 165, 231, 385, 1515, 1965, 2121, 2265, 2715, 2751, 2865, 3171, 3333, 3535, 3801, 4011, 4323, 4585, 4695, 4983, 5285, 5295, 5555, 5595, 5745, 5973, 6303, 6335, 6573, 6685, 7205, 7413, 7777, 7833, 8043, 8305, 9955, 10087, 10329, 10505, 10905

Offset: 1

Patrick De Geest, Jun 15 1998

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Intersection of A033620 and A046389.

Cf. A002385, A046373.

Definition clarified by N. J. A. Sloane, Dec 19 2017 at the suggestion of Harvey P. Dale

Offset changed by Andrew Howroyd, Aug 14 2024

A069158 a(n) = Product{d|n} mu(d), product over positive divisors, d, of n, where mu(d) = Moebius function (A008683).

Original entry on oeis.org

1, -1, -1, 0, -1, 1, -1, 0, 0, 1, -1, 0, -1, 1, 1, 0, -1, 0, -1, 0, 1, 1, -1, 0, 0, 1, 0, 0, -1, 1, -1, 0, 1, 1, 1, 0, -1, 1, 1, 0, -1, 1, -1, 0, 0, 1, -1, 0, 0, 0, 1, 0, -1, 0, 1, 0, 1, 1, -1, 0, -1, 1, 0, 0, 1, 1, -1, 0, 1, 1, -1, 0, -1, 1, 0, 0, 1, 1, -1, 0, 0, 1, -1, 0, 1, 1, 1, 0, -1, 0, 1, 0, 1, 1, 1, 0, -1, 0, 0, 0, -1, 1, -1, 0, 1, 1

Offset: 1

Leroy Quet, Apr 08 2002

sign

Absolute value of a(n) = absolute value of mu(n).
Differs from A080323 at n=2, 105, 165, 195, 231, ..., 15015,..., 19635,.. (cf. A046389, A046391, ...) - R. J. Mathar, Dec 15 2008
Not multiplicative: For example a(2)*a(15) <> a(30). - R. J. Mathar, Mar 31 2012
Row products of table A225817. - Reinhard Zumkeller, Jul 30 2013

			a(6) = mu(1)*mu(2)*mu(3)*mu(6) = 1*(-1)*(-1)*1 = 1.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A008683, A008966, A010051, A013929, A046389, A046391, A080323, A225817.

a069158 = product . a225817_row  -- Reinhard Zumkeller, Jul 30 2013

f := function(n); t1 := &*[MoebiusMu(d) : d in Divisors(n) ]; return t1; end function;

A069158 := proc(n)
    mul(numtheory[mobius](d),d=numtheory[divisors](n)) ;
end proc: # R. J. Mathar, May 28 2016

a[n_] := Product[MoebiusMu[d], {d, Divisors[n]}]; Array[a, 106] (* Jean-François Alcover, Feb 22 2018 *)

a(n) = vecprod(apply(moebius, divisors(n))); \\ Amiram Eldar, Feb 10 2025

a(n) = 0 if mu(n) = 0 (A013929); a(n) = -1 if n = prime; a(n) = 1 if n = squarefree composite (A120944) or 1.

a(n) = A008966(n) - 2*A010051(n). - Amiram Eldar, Feb 10 2025

A160355 Odd indices pqr of flat cyclotomic polynomials of order 3 which are not of the form r = +/-1 (mod pq).

Original entry on oeis.org

231, 399, 483, 651, 663, 741, 1113, 1173, 1209, 1281, 1311, 1353, 1443, 1479, 1533, 1581, 1599, 1653, 1833, 1947, 2163, 2247, 2301, 2337, 2379, 2409, 2829, 2877, 2915, 3129, 3297, 3363, 3441, 3531, 3621, 3723, 3759, 3783, 3813, 4011, 4029, 4071, 4161

Offset: 1

M. F. Hasler, May 11 2009

nonn

This is in some sense the nontrivial part of A160350: Indeed, Kaplan (2007) has shown that Phi[pqr] has coefficients in {0,1,-1} if r = +-1 (mod pq), where pA160350 (i.e. of A117223) which do not satisfy this equality (i.e. which are not in A160353).

See A160350 for further details and references.

			a(1)=231=3*7*11 is the smallest "nontrivial" element of A160350 in the sense that it is neither of the form 2pq, and that its largest factor (11) is not congruent to +- 1 modulo the product of the smaller factors (3*7).

Robin Visser, Table of n, a(n) for n = 1..10000

Cf. A046389, A117223, A160350, A160353, A160354.

forstep( pqr=1,5999,2, my(f=factor(pqr)); #f~==3 & vecmax(f[,2])==1 & abs((f[3,1]+1)%(f[1,1]*f[2,1])-1)!=1 & vecmax(abs(Vec(polcyclo(pqr))))==1 & print1(pqr","))

Equals A117223 \ A160353 = A160354 intersect A046389.

A180197 a(n) is 2^(p*q) mod r for the n-th odd number with exactly 3 distinct prime factors p < q < r.

Original entry on oeis.org

1, 10, 8, 2, 9, 5, 12, 16, 15, 10, 8, 5, 27, 7, 1, 12, 23, 2, 8, 17, 9, 12, 2, 2, 9, 10, 9, 11, 8, 1, 29, 14, 4, 2, 23, 18, 42, 11, 9, 3, 12, 12, 6, 5, 12, 8, 2, 37, 1, 64, 2, 48, 18, 13, 62, 16, 14, 15, 56, 66, 1, 33, 19, 15, 4, 16, 33, 52, 32, 9

Offset: 1

Juri-Stepan Gerasimov, Jan 16 2011

The lines with negative slope appearing to rise out of plots is caused by pq = 15 and large r. - T. D. Noe, Jul 28 2011

			a(1) = 2^(3*5) mod 7 = 32768 mod 7 = 1 because A046389(1) = 105 = 3*5*7.

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Cf. A046389.

Cf. A027746.

import Data.List (nub)
a180197 n = a180197_list !! (n-1)
a180197_list = f 1 where
   f x = if length ps == 3 && nub ps == ps
         then (2 ^ (ps!!0 * ps!!1) `mod` ps!!2) : f (x+2) else f (x+2)
         where ps = a027746_row x
-- Reinhard Zumkeller, Jan 29 2014

b:= proc(n) option remember; local i, k, l;
       if n=1 then 3,5,7
       else for k from mul(i, i=b(n-1)) +2 by 2
              do l:= ifactors(k)[2];
                 if nops(l) = 3 and add(i[2], i=l) = 3
                    then break fi
              od; sort(map(i-> i[1], l))[]
       fi
    end:
a:= proc(n) option remember; local p, q, r;
      p,q,r:= b(n);
      2 &^ (p*q) mod r
    end:
seq(a(n), n=1..70);  # Alois P. Heinz, Jan 17 2011

Reap[For[n = 105, n < 2000, n += 2, f = FactorInteger[n] // Transpose; If[f[[2]] == {1, 1, 1}, {p, q, r} = f[[1]]; Sow[Mod[2^(p*q), r]]]]][[2, 1]] (* Jean-François Alcover, Oct 24 2016 *)

More terms from Alois P. Heinz, Jan 17 2011

A278569 Numbers of the form p^iq^jr^k where p,q,r are distinct odd primes and i,j,k >= 1.

Original entry on oeis.org

105, 165, 195, 231, 255, 273, 285, 315, 345, 357, 385, 399, 429, 435, 455, 465, 483, 495, 525, 555, 561, 585, 595, 609, 615, 627, 645, 651, 663, 665, 693, 705, 715, 735, 741, 759, 765, 777, 795, 805, 819, 825, 855, 861, 885, 897, 903, 915, 935, 945, 957, 969, 975, 987, 1001, 1005, 1015, 1023, 1035, 1045, 1065, 1071, 1085, 1095, 1105, 1113, 1131, 1173, 1185, 1197, 1209

Offset: 1

N. J. A. Sloane, Nov 27 2016

nonn

More than the usual number of terms are included to show the difference from A216918 (the latter includes 3*5*7*11 = 1155 and all terms of A046390, A046391 etc).

If i,j,k are all equal to 1 we get A046389.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Includes A046389, subsequence of A216918.

Select[Range@ 1500, PrimeNu@ # == 3 && OddQ@ # &] (* Michael De Vlieger, Dec 05 2016 *)

A033992 INTERSECT A005408. - R. J. Mathar, Dec 05 2016

cp's OEIS Frontend

A338556 Products of three prime numbers of even index.

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A234099 Integers of the form (p*q*r - 1)/2, where p, q, r are distinct primes.

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A338471 Products of three prime numbers of odd index.

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A338557 Products of three distinct prime numbers of even index.

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A128284 Numbers of the form m = p1 * p2 * p3 where for each d|m we have (d+m/d)/2 prime and p1 < p2 < p3 each prime.

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A046405 Numbers of the form p*q*r where p,q,r are distinct odd palindromic primes (odd terms from A002385).

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A069158 a(n) = Product{d|n} mu(d), product over positive divisors, d, of n, where mu(d) = Moebius function (A008683).

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A160355 Odd indices pqr of flat cyclotomic polynomials of order 3 which are not of the form r = +/-1 (mod pq).

A234099 Integers of the form (pqr - 1)/2, where p, q, r are distinct primes.

A046405 Numbers of the form pqr where p,q,r are distinct odd palindromic primes (odd terms from A002385).

A278569 Numbers of the form p^iq^jr^k where p,q,r are distinct odd primes and i,j,k >= 1.