A046310 -id:A046310 - cp's OEIS frontend

A125149 a(n) is the least k such that the n-almost prime count is positive and equal to the (n-1)-almost prime count. a(0) = 1.

1, 2, 10, 15495, 151165506066

Offset: 0

Jonathan Vos Post and Robert G. Wilson v, Jan 07 2007

Unlike any of the prime number races in which any particular form may lead or trail, this sequence demonstrates that although the count of numbers having k prime factors begins by trailing the count for k-1 prime factors, eventually they exchange positions in the race. This can be seen by looking at A126279 or A126280.
The fundamental theorem of arithmetic, or unique factorization theorem, states that every natural number greater than 1 either is itself a prime number, or can be written as a unique product of prime numbers. It had a proof sketched by Euclid, then corrected and completed in "Disquisitiones Arithmeticae" [Carl Friedrich Gauss, 1801]. It fails in many rings of algebraic integers [Ernst Kummer, 1843], a discovery initiating algebraic number theory. Counting the elements in the unique product of prime numbers classifies natural numbers into primes, semiprimes, 3-almost primes and so on. This sequence quantifies a previously undescribed structure to that classification.
We took the first k where the two relevant counts are the same. If instead we took the least k such that the n-almost prime count from k onwards exceeds the (n-1)-almost prime count, the sequence would begin: 3, 34, 15530, ... [see A180126].
The prime count and the semiprime count are identical for 1, 10, 15, 16, 22, 25, 29, 30, 33.
The semiprime count and the 3-almost prime count are identical for 1, 2, 3, 15495, 15496, 15497, 15498, 15508, 15524, 15525, 15529.
The numbers of 3-almost primes and 4-almost primes are equal at 151165506066 and 731 larger numbers, the last one being 151165607041. See A180126. - T. D. Noe, Aug 11 2010
Landau's asymptotic formula suggests that a(n) is about exp(exp(n-1)). - Charles R Greathouse IV, Mar 14 2011

			a(1) = 2 since 1 has no prime factors and 2 has one prime factor, therefore prime factor counts of 0 and 1 occur equally often in the first 2 integers.
a(2) = 10 since there are 4 primes {2, 3, 5 & 7} and 4 semiprimes {4, 6, 9 & 10} less than or equal to 10.
a(4) = 151165506066 since there are 32437255807 4-almost primes and 3-almost primes <= a(4).

Andrew Granville and Greg Martin, Prime Number Races, arXiv:math/0408319 [math.NT], Aug 24 2004.
Eric Weisstein's World of Mathematics, Fundamental Theorem of Arithmetic.
Eric Weisstein's World of Mathematics, Modular Prime Counting Function.
Eric Weisstein's World of Mathematics, Prime Factor.

Cf. A126279, A126280, A117526.
Sequences listing r-almost primes, that is, k such that A001222(k) = r: 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).
Cf. A180126.

AlmostPrimePi[k_Integer, n_] := Module[{a, i}, a[0] = 1; If[k == 1, PrimePi[n], Sum[PrimePi[n/Times @@ Prime[Array[a, k - 1]]] - a[k - 1] + 1, Evaluate[ Sequence @@ Table[{a[i], a[i - 1], PrimePi[(n/Times @@ Prime[Array[a, i - 1]])^(1/(k - i + 1))]}, {i, k - 1}]]]]]; (* Eric W. Weisstein, Feb 07 2006 *)
f[n_] := Block[{k = 2^n}, While[AlmostPrimePi[n, k] < AlmostPrimePi[n - 1, k], k++ ]; k];

Changed 33 to 34 in a comment. - T. D. Noe, Aug 11 2010

Edited by Peter Munn, Dec 17 2022

A046309 Numbers that are divisible by at least 8 primes (counted with multiplicity).

Original entry on oeis.org

256, 384, 512, 576, 640, 768, 864, 896, 960, 1024, 1152, 1280, 1296, 1344, 1408, 1440, 1536, 1600, 1664, 1728, 1792, 1920, 1944, 2016, 2048, 2112, 2160, 2176, 2240, 2304, 2400, 2432, 2496, 2560, 2592, 2688, 2816, 2880, 2916, 2944, 3024, 3072, 3136

Offset: 1

Patrick De Geest, Jun 15 1998

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A046310.

Select[Range[3200],PrimeOmega[#]>7&] (* Harvey P. Dale, May 29 2013 *)

is(n)=bigomega(n)>7 \\ Charles R Greathouse IV, Sep 17 2015

from math import prod, isqrt
from sympy import primerange, integer_nthroot, primepi
def A046309(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(2,8)))
    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 >= 8.

A046334 Palindromes with exactly 8 prime factors (counted with multiplicity).

Original entry on oeis.org

2112, 25152, 29792, 67776, 84448, 219912, 252252, 270072, 272272, 464464, 616616, 631136, 652256, 656656, 696696, 698896, 801108, 802208, 828828, 840048, 848848, 861168, 882288, 886688, 888888, 2110112, 2150512, 2171712, 2175712

Offset: 1

Patrick De Geest, Jun 15 1998

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

Cf. A046310, A046321.

Select[Range[22*10^5],PalindromeQ[#]&&PrimeOmega[#]==8&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 11 2019 *)

A046321 Odd numbers divisible by exactly 8 primes (counted with multiplicity).

Original entry on oeis.org

6561, 10935, 15309, 18225, 24057, 25515, 28431, 30375, 35721, 37179, 40095, 41553, 42525, 47385, 50301, 50625, 56133, 59535, 61965, 63423, 66339, 66825, 67797, 69255, 70875, 78975, 80919, 83349, 83835, 84375, 86751, 88209, 89667

Offset: 1

Patrick De Geest, Jun 15 1998

nonn

Zak Seidov, Table of n, a(n) for n = 1..35737

Cf. A046310.

Select[Range[1,100001,2],PrimeOmega[#]==8&] (* Harvey P. Dale, Apr 28 2018 *)

list(lim)=my(v=List()); forprime(a=3,lim\2187, my(La=lim\a); forprime(b=3,min(La\729,a), my(Lb=La\b); forprime(c=3,min(Lb\243,b), my(Lc=Lb\c); forprime(d=3,min(Lc\81,c), my(Ld=Lc\d); forprime(e=3,min(Ld\27,d), my(Le=Ld\e,E=a*b*c*d*e); forprime(f=3,min(Le\9,e), my(Lf=Le\f,F=E*f); forprime(g=3,min(Lf\3,f), my(Lg=Lf\g,G=F*g); forprime(h=3,min(Lg,g), listput(v,G*h))))))))); Set(v) \\ Charles R Greathouse IV, Aug 23 2024

from math import prod, isqrt
from sympy import primerange, integer_nthroot, primepi
def A046321(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-1+x-sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,1,3,1,8)))
    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

a(n) ~ A046310(n) ~ 5040n log n / (log log n)^7. - Charles R Greathouse IV, Aug 23 2024

Offset changed 0=>1 by Zak Seidov, Feb 08 2016

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

A114636 Numbers k such that the k-th octagonal number is 8-almost prime.

Original entry on oeis.org

22, 70, 80, 84, 102, 108, 118, 126, 134, 160, 174, 184, 200, 230, 240, 250, 252, 262, 264, 272, 318, 330, 334, 336, 350, 368, 378, 400, 408, 420, 430, 434, 444, 450, 454, 459, 462, 464, 484, 494, 500, 502, 510, 518, 520, 522, 540, 560, 564, 566, 570, 574, 582

Offset: 1

Jonathan Vos Post, Feb 18 2006

It is necessary but not sufficient that k must be prime (A000040), semiprime (A001358), 3-almost prime (A014612), 4-almost prime (A014613), 5-almost prime (A014614), 6-almost prime (A046306), or 7-almost prime (A046308).

			a(1) = 22 because OctagonalNumber(22) = Oct(22) = 22*(3*22-2) = 1408 = 2^7 * 11 has exactly 8 prime factors (seven are all equally 2; factors need not be distinct).
a(2) = 70 because Oct(70) = 70*(3*70-2) = 14560 = 2^5 * 5 * 7 * 13 is 8-almost prime.
a(3) = 80 because Oct(80) = 80*(3*80-2) = 19040 = 2^5 * 5 * 7 * 17.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..2000 from Harvey P. Dale)
Eric Weisstein's World of Mathematics, Almost Prime.
Eric Weisstein's World of Mathematics, Octagonal Number.

Cf. A000040, A000567, A001222, A001358, A014612, A014613, A014614, A046306, A046308, A046310.

Select[Range[400],PrimeOmega[PolygonalNumber[8,#]]==8&] (* Harvey P. Dale, Aug 31 2020 *)

Numbers k such that k*(3*k-2) has exactly eight prime factors (with multiplicity).
Numbers k such that A000567(k) is a term of A046310.
Numbers k such that A001222(A000567(k)) = 8.
Numbers k such that A001222(k) + A001222(3*k-2) = 8.
Numbers k such that [(3*k-2)*(3*k-1)*(3*k)]/[(3*k-2)+(3*k-1)+(3*k)] is a term of A046310.

A240528 Indices of 8-almost prime triangular numbers.

Original entry on oeis.org

63, 80, 128, 256, 287, 288, 319, 320, 324, 383, 399, 440, 447, 480, 495, 539, 560, 567, 576, 608, 640, 648, 671, 675, 703, 720, 729, 799, 831, 863, 927, 935, 972, 975, 1000, 1007, 1055, 1056, 1071, 1079, 1080, 1104, 1119, 1120, 1160, 1175, 1183, 1184

Offset: 1

Vincenzo Librandi, Apr 07 2014

nonn

			a(1) = 63 because A000217(63) = 63*(63+1)/2 = 2016 = 2^5 * 3^2 * 7 is an 8-almost prime.

Muniru A Asiru, Table of n, a(n) for n = 1..10000 (first 1000 terms from Vincenzo Librandi)

Cf. A000217, A069904, A108815, A114435, A114436, A114437, A164977.

Cf. A046310 (8-almost primes).

F:=List([1..1200],n->Length(Factors(n*(n+1)/2)));; a:=Filtered([1..Length(F)],i->F[i]=8); # Muniru A Asiru, Dec 22 2018

[n: n in [2..1200] | &+[d[2]: d in Factorization((n*(n+1)))] eq 9]; // Vincenzo Librandi, Dec 22 2018

Flatten[Position[Accumulate[Range[1500]], _?(PrimeOmega[#]== 8 &)]]

{ m : A069904(m) = 8 }. - Alois P. Heinz, Aug 05 2019

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

A112446 Pseudoprimes (base-2) equal to the product of 8 primes not necessarily distinct.

Original entry on oeis.org

38504389105, 79753114441, 86757431761, 95665544065, 101135235565, 160727267701, 184395152305, 232250619601, 241543943641, 267149008945, 281825521705, 298045705321, 306177962545, 360805362841, 432207073585, 435343749841, 473056144561, 503876371285, 518896999621

Offset: 1

Shyam Sunder Gupta, Dec 11 2005

			a(1) = 38504389105 = 5*7*13*17*19*37*73*97.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Intersection of A046310 and A001567.

s={}; Do[If[PrimeOmega[n] == 8 && PowerMod[2, n - 1, n] == 1, AppendTo[s, n]], {n, 1, 10^11}]; s (* Amiram Eldar, Nov 10 2019 *)

More terms from Amiram Eldar, Nov 10 2019

A114418 Records in 8-almost prime gaps ordered by merit.

Original entry on oeis.org

128, 192, 224, 336, 420, 560

Offset: 1

Jonathan Vos Post, Dec 03 2005

			Records defined in terms of A114408 and A046310:
n A114418(n) A114418(n)/log(A046310(n)).
1 128 128/log 256 = 53.1508495
2 192 192/log 384 = 74.2938824
3 64 64/log 576 = 23.1848568
4 224 224/log 640 = 79.8238182
5 32 32/log 864 = 10.8972758
6 64 64/log 896 = 21.6779549
7 336 336/log 960 = 112.665809
8 48 48/log 1296 = 15.4211665
22 420 420/log 2496 = 123.629603

Cf. A046310, A065516, A111870, A111871, A114403-A114411, A114412-A114422.

a(n) = records in A114418(n)/log(A046310(n)) = records in (A046310(n+1) - A046310(n))/log(A046310(n)).

Offset corrected and a(6) from Donovan Johnson, Feb 17 2010

cp's OEIS Frontend

A125149 a(n) is the least k such that the n-almost prime count is positive and equal to the (n-1)-almost prime count. a(0) = 1.

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A046309 Numbers that are divisible by at least 8 primes (counted with multiplicity).

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A046334 Palindromes with exactly 8 prime factors (counted with multiplicity).

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A046321 Odd numbers divisible by exactly 8 primes (counted with multiplicity).

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

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A114636 Numbers k such that the k-th octagonal number is 8-almost prime.

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A240528 Indices of 8-almost prime triangular numbers.

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

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