A046314 -id:A046314 - cp's OEIS frontend

A110893 Numbers with a semiprime number of prime divisors (counted with multiplicity).

16, 24, 36, 40, 54, 56, 60, 64, 81, 84, 88, 90, 96, 100, 104, 126, 132, 135, 136, 140, 144, 150, 152, 156, 160, 184, 189, 196, 198, 204, 210, 216, 220, 224, 225, 228, 232, 234, 240, 248, 250, 260, 276, 294, 296, 297, 306, 308, 315, 324, 328, 330, 336, 340, 342

Offset: 1

Jonathan Vos Post, Sep 20 2005

Below 256 = 2^8 this is identical to A067028 (Numbers with a composite number of prime factors, counted with multiplicity).

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000040, A001222, A001358, A014613, A046306, A046312, A046314, A063989, A067028, A069276.

is(n)=n>9 && bigomega(bigomega(n))==2 \\ Charles R Greathouse IV, Jan 31 2017

a(n) such that A001222(a(n)) is an element of A001358. a(n) such that bigomega(a(n)) is an element of A001358. Union[4-almost primes(A014613), 6-almost primes(A046306), 9-almost primes(A046312), 10-almost primes(A046314), 14-almost primes(A069275), 15-almost primes(A069276), 21-almost primes, 22-almost primes, 25-almost primes, 26-almost primes, ...]

A349029 Lucas-Carmichael numbers with 10 prime factors.

Original entry on oeis.org

989565001538399, 1250312791224959, 1419432982021439, 1518134614712639, 2240225337903839, 2493922560242399, 2708548708646879, 2786001880066559, 2807577905060159, 2808521396058455, 3157015238986895, 3210972445532159, 3221015190555239, 3407706183722399, 3614740529402519

Offset: 1

Tim Johannes Ohrtmann, Nov 06 2021

nonn

			989565001538399 = 11*13*17*19*29*31*41*47*83*149 and 12, 14, 18, 20, 30, 32, 42, 48, 84, and 150 all divide 989565001538400.

Daniel Suteu, Table of n, a(n) for n = 1..5249 (all terms < 10^18)

Intersection of A006972 and A046314.
Cf. A216928 (least Lucas-Carmichael number with n prime factors).
Cf. A216925, A216926, A216927, A217002, A217003, A217091, A349028, A349030 (Lucas-Carmichael numbers with 3-9 and 11 prime factors).

PARI
```
is(n)={omega(n)==10&&is_A006972(n)}
```

A046313 Numbers that are divisible by at least 10 primes (counted with multiplicity).

Original entry on oeis.org

1024, 1536, 2048, 2304, 2560, 3072, 3456, 3584, 3840, 4096, 4608, 5120, 5184, 5376, 5632, 5760, 6144, 6400, 6656, 6912, 7168, 7680, 7776, 8064, 8192, 8448, 8640, 8704, 8960, 9216, 9600, 9728, 9984, 10240, 10368, 10752, 11264, 11520, 11664, 11776

Offset: 1

Patrick De Geest, Jun 15 1998

nonn

John Cerkan, Table of n, a(n) for n = 1..10000

Subsequence of A033987, A046304, A046305, A046307, A046309, and A046311.

Cf. A046314.

Select[Range[12000],PrimeOmega[#]>9&] (* Harvey P. Dale, Dec 17 2018 *)

is(n)=bigomega(n)>9 \\ Charles R Greathouse IV, Sep 17 2015

from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A046313(n):
    def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b,isqrt(x//c)+1),a)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b,integer_nthroot(x//c,m)[0]+1),a) for d in g(x,a2,b2,c*b2,m-1)))
    def f(x): return int(n+primepi(x)+sum(sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,0,1,1,i)) for i in range(2,10)))
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return kmax # Chai Wah Wu, Aug 23 2024

Product p_i^e_i with Sum e_i >= 10.

a(n) = n + O(n (log log n)^8/log n). - Charles R Greathouse IV, Apr 07 2017

A046323 Odd numbers divisible by exactly 10 primes (counted with multiplicity).

Original entry on oeis.org

59049, 98415, 137781, 164025, 216513, 229635, 255879, 273375, 321489, 334611, 360855, 373977, 382725, 426465, 452709, 455625, 505197, 535815, 557685, 570807, 597051, 601425, 610173, 623295, 637875, 710775, 728271, 750141, 754515, 759375

Offset: 1

Patrick De Geest, Jun 15 1998

nonn

John Cerkan, Table of n, a(n) for n = 1..10000

Cf. A046314.

Select[Range[9,800001,2],PrimeOmega[#]==10&] (* Harvey P. Dale, May 26 2013 *)

from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A046323(n):
    def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b,isqrt(x//c)+1),a)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b,integer_nthroot(x//c,m)[0]+1),a) for d in g(x,a2,b2,c*b2,m-1)))
    def f(x): return int(n+x-sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,1,3,1,10)))
    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
    return bisection(f,n,n) # Chai Wah Wu, Sep 09 2024

A101696 a(n) = sum(i=1,n)(i-th i-almost prime). Cumulative sums of A101695.

Original entry on oeis.org

2, 8, 26, 66, 174, 398, 878, 2174, 4862, 10494, 22014, 45054, 98302, 222718, 480766, 1021438, 2127358, 4355582, 8943102, 18773502, 38696446, 79590910, 175142398, 368080382, 764442110, 1586525694, 3247470078, 6644856318, 13489960446

Offset: 1

Jonathan Vos Post, Dec 12 2004, Sep 28 2006

nonn

It seems that this sum can never be a prime after a(1) = 2, since the n-th n-almost prime is always even. The number of prime factors (with multiplicity) of a(n) is 1, 3, 2, 3, 3, 2, 2, 2, 4, 5, 4, 4, 3, 3, 5, 4, 3, 4, 7, 4, 2, 5, 5, 2, 3, 7, 4, 3, 4.

This is the diagonalization of the set of sequences {j-almost prime(k)}. The cumulative sums of this sequence are in A101696. a(1)=2 is prime. a(2)=8 is a 3-almost prime. a(3)=26 is a semiprime. a(4)=66 is a 3-almost prime. a(5)= 174 is a 3-almost prime. a(6)=398 is a semiprime. a(7)=878 is a semiprime. a(8)=2174 is a semiprime. a(9)=4862 is a 4-almost prime. a(10)=10494 is a 5-almost prime. a(11)=22014 is a 4-almost prime. a(12)=45054 is a 3-almost prime. a(13)=98302 is a 3-almost prime. a(14)=222718 is a 3-almost prime. a(15)=480766 is a 5-almost prime. a(16)=1021438 is a 4-almost prime. a(17)=2127358 is a 3-almost prime. a(18)=4355582 is a 4-almost prime. a(19)=8943102 is a 7-almost prime. a(20)=18773502 is a 4-almost prime. 21-almost numbers are not yet listed in the OEIS.

			a(1) = first 1-almost prime = first prime = A000040(1) = 2.
a(2) = 2 + 2nd 2-almost prime = 2 + A001358(2) = 2+ 6 = 8.
a(3) = a(2) + 3rd 3-almost prime = 8+A014612(3) = 8+18 = 26.
a(4) = a(3) + 4th 4-almost prime = 26+A014613(4) = 26+40 = 66.
a(5) = a(4) + 5th 5-almost prime = 66+A014614(5) = 66+108=174.
...
a(12) = a(11) + 12th 12-almost prime = 22014 + 23040 = 45054 (the first nontrivial palindrome in the sequence).

Eric Weisstein's World of Mathematics, Almost Prime.

Cf. A000040, A001358, A014612, A014613, A046314, A046306, A046308, A046310, A046312, A046314, A069272, A069273, A069274, A069275, A069276, A069277, A069278, A069279, A069280, A069281, A101637, A101638, A101605, A101606, A101695.

a(1) = first 1-almost prime = first prime = A000040(1). a(2) = a(1) + 2nd 2-almost prime = a(1) + 2nd semiprime = A000040(1)+A001358(2). a(3) = a(2) + 3rd 3-almost prime = a(2) + A014612(3). a(4) = a(3) + 4th 4-almost prime = a(3) + A014613(4)... a(n) = a(n-1) + n-th n-almost prime.

Edited by N. J. A. Sloane, Jul 03 2008 at the suggestion of R. J. Mathar

A101744 Triangular numbers which are 10-almost primes.

Original entry on oeis.org

32640, 73920, 130816, 165600, 204480, 265356, 294528, 401856, 592416, 839160, 947376, 990528, 1279200, 1445850, 1492128, 1606528, 1842240, 1844160, 2031120, 2049300, 2821500, 2956096, 3571128, 3963520, 4148640, 4250070, 4335040

Offset: 1

Jonathan Vos Post, Dec 14 2004

A101745 contains the indices of this sequence, i.e. T(n) for what values of n are these 10-almost primes.

			a(1) = 32640 because that is the smallest triangular number which is also a 10-almost prime; specifically T(255) = 255*(255+1)/2 = 32640 = 2^7 * 3 * 5 * 17.

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, p. 59, 1987.
Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 33-38, 1996.
Dudeney, H. E. Amusements in Mathematics. New York: Dover, pp. 67 and 167, 1970.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Triangular Number.
Eric Weisstein's World of Mathematics, Almost Prime.

Cf. A000217, A046314, A068443, A075875, A076578-A076583, A075088, A101745.

BigOmega[n_Integer]:=Plus@@Last[Transpose[FactorInteger[n]]]; Select[Table[n*(n+1)/2, {n, 2, 5000}], BigOmega[ # ]==10&] (* Ray Chandler, Dec 14 2004 *)

list(lim)=my(v=List(),cur,last=3,n=256,t); while((t=n*(n-1)/2) <= lim, cur=bigomega(n); if(cur+old==11, listput(v,t)); old=cur; n++); Vec(v) \\ Charles R Greathouse IV, Feb 05 2017

a(n) is in the intersection of {A000217} and {A046314}. Integers of the form k*(k+1)/2 which have exactly 10 prime factors.

More terms from Ray Chandler, Dec 14 2004

A337112 Smallest term of A337081 that has exactly n prime factors, or 0 if no such term exists.

Original entry on oeis.org

0, 4, 0, 90, 675, 1134, 6318, 4374, 32805, 255879, 1003833, 531441, 327544803, 20751953125, 225830078125, 91552734375, 1068115234375, 23651123046875, 316619873046875, 1697540283203125, 13256072998046875, 85353851318359375, 541210174560546875, 4518032073974609375, 58233737945556640625

Offset: 1

Matej Veselovac, Aug 16 2020

nonn

a(n) is the smallest product of n primes that has unordered factorizations whose sums of factors are the same (is a term of A337080) and all of whose proper divisors have the complementary property: that every unordered factorization has a distinct sum of factors (i.e., all proper divisors are terms of A337037).

Cf. A337113 (factors of terms).
Cf. A337037, A337080, A337081.
Cf. A056472 (all factorizations of n).
Cf. r-almost primes: A000040 (r = 1), A001358 (r = 2), A014612 (r = 3), A014613 (r = 4), A014614 (r = 5), A046306 (r = 6), A046308 (r = 7), A046310 (r = 8), A046312 (r = 9), A046314 (r = 10), A069272 (r = 11), A069273 (r = 12), A069274 (r = 13), A069275 (r = 14), A069276 (r = 15), A069277 (r = 16), A069278 (r = 17), A069279 (r = 18), A069280 (r = 19), A069281 (r = 20).

a(14) onward from David A. Corneth, Aug 26 2020

A338442 Carmichael numbers with 10 prime factors.

Original entry on oeis.org

1436697831295441, 1493812621027441, 2094319836529921, 2349991949342401, 2842648863161185, 2859959706040801, 3455134500424321, 3871703982953521, 4177950872896801, 4289150794129201, 4937378437571041, 5071419883911745, 5778659093725441, 6665161459969441, 6682056104892961

Offset: 1

Tim Johannes Ohrtmann, Oct 28 2020

nonn

			1436697831295441 = 11*13*19*29*31*37*41*43*71*127 and 10, 12, 18, 28, 30, 36, 40, 42, 70, 126 all divide 1436697831295440.

Daniel Suteu, Table of n, a(n) for n = 1..10000
Richard Pinch, Tables relating to Carmichael numbers.

Cf. A002997 (Carmichael numbers).
Cf. A006931 (Least Carmichael number with n prime factors).
Cf. A299710 (Number of terms less than 10^n).
Cf. A087788, A074379, A112428, A112429, A112430, A112431, A112432, A338443 (Carmichael numbers with 3-9 and 11 prime factors).

PARI
```
is(n)={omega(n)==10&&is_A002997(n)}
```

Equals A002997 intersect A046314.

A369897 Numbers k such that k and k + 1 each have 10 prime divisors, counted with multiplicity.

Original entry on oeis.org

3290624, 4122495, 4402431, 5675264, 6608384, 6890624, 7914752, 8614592, 9454400, 11553920, 12613887, 13466816, 14493248, 14853375, 15473024, 16719615, 17494784, 18272384, 18309375, 22784895, 24890624, 25200800, 25869375, 25957503, 26903744, 26921727, 27510272, 28350080, 29761424, 31802624

Offset: 1

Zak Seidov and Robert Israel, Feb 04 2024

nonn

Numbers k such that k and k + 1 are in A046314.

If a and b are coprime terms of A046312, one of them even, then Dickson's conjecture implies there are infinitely many terms k where k/a and (k+1)/b are primes.

			a(5) = 6608384 is a term because 6608384 = 2^9 * 12907 and 6608385 = 3^6 * 5 * 7^2 * 37 each have 10 prime divisors, counted with multiplicity.

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

Cf. A001222, A046312, A046314, A115186.

with(priqueue):
R:= NULL:  count:= 0:
initialize(Q); r:= 0:
insert([-2^10, [2$10]],Q);
while count < 30 do
  T:= extract(Q);
  if -T[1] = r + 1 then
    R:= R, r; count:= count+1;
  fi;
  r:= -T[1];
  p:= T[2][-1];
  q:= nextprime(p);
  for i from 10 to 1 by -1 while T[2][i] = p do
    insert([-r*(q/p)^(11-i), [op(T[2][1..i-1]),q$(11-i)]],Q);
  od
od:
R;

A247114 Primes sandwiched between 4-almost primes (A014613).

Original entry on oeis.org

89, 151, 197, 233, 307, 349, 461, 491, 569, 571, 739, 857, 859, 1013, 1061, 1097, 1277, 1291, 1303, 1483, 1667, 1747, 1831, 1913, 1973, 2003, 2131, 2357, 2503, 2531, 2621, 2683, 3011, 3067, 3163, 3209, 3229, 3259, 3271, 3581, 3797, 3929, 4013, 4027, 4073, 4219, 4327, 4597, 4793, 4877, 4903

Offset: 1

Zak Seidov, Jan 10 2015

nonn

Primes p such that p - 1 and p + 1 are 4-almost primes.

			89 - 1 = 2^3*11, 89 + 1 = 2*3^2*5.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A014612, A014613, A046314, A154598, A063644, A247088.

Select[Prime[Range[1000]], 4 == PrimeOmega[# - 1] == PrimeOmega[# + 1] &]

forprime(p= 1,5000, if(4==bigomega(p-1)&&4==bigomega(p+1), print1(p", ")))

is(n)=bigomega(n-1)==4 && bigomega(n+1)==4 && isprime(n) \\ Charles R Greathouse IV, Apr 27 2015

cp's OEIS Frontend

A110893 Numbers with a semiprime number of prime divisors (counted with multiplicity).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A349029 Lucas-Carmichael numbers with 10 prime factors.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

A046313 Numbers that are divisible by at least 10 primes (counted with multiplicity).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A046323 Odd numbers divisible by exactly 10 primes (counted with multiplicity).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Python

A101696 a(n) = sum(i=1,n)(i-th i-almost prime). Cumulative sums of A101695.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

A101744 Triangular numbers which are 10-almost primes.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A337112 Smallest term of A337081 that has exactly n prime factors, or 0 if no such term exists.

Views

Author

Keywords

Comments

Crossrefs

Extensions

A338442 Carmichael numbers with 10 prime factors.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI