A033987 -id:A033987 - cp's OEIS frontend

A046311 Numbers that are divisible by at least 9 primes (counted with multiplicity).

512, 768, 1024, 1152, 1280, 1536, 1728, 1792, 1920, 2048, 2304, 2560, 2592, 2688, 2816, 2880, 3072, 3200, 3328, 3456, 3584, 3840, 3888, 4032, 4096, 4224, 4320, 4352, 4480, 4608, 4800, 4864, 4992, 5120, 5184, 5376, 5632, 5760, 5832, 5888, 6048, 6144

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, and A046309.

Cf. A046312.

Select[Range[6200],PrimeOmega[#]>8&] (* Harvey P. Dale, May 20 2013 *)

is(n)=bigomega(n)>8 \\ Charles R Greathouse IV, Sep 17 2015

from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A046311(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+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(2,9)))
    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

Product p_i^e_i with Sum e_i >= 9.

a(n) = n + O(n (log log n)^7/log n). - Charles R Greathouse IV, Apr 07 2017

A107395 a(n) = binomial(n+4,4)*binomial(n+6,4).

Original entry on oeis.org

15, 175, 1050, 4410, 14700, 41580, 103950, 235950, 495495, 975975, 1821820, 3248700, 5569200, 9224880, 14825700, 23197860, 35441175, 52997175, 77729190, 112015750, 158858700, 222007500, 306101250, 416830050, 561117375, 747325215, 985483800, 1287547800

Offset: 0

Zerinvary Lajos, May 25 2005

			If n=0 then C(0+4,4)*C(0+6,4) = C(4,4)*C(6,4) = 1*15 = 15.
If n=9 then C(9+4,4)*C(9+6,4) = C(13,4)*C(15,4) = 715*1365 = 975975.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A033987, A062196.

A107395:= func< n | Binomial(n+4,4)*Binomial(n+6,4) >;
[A107395(n): n in [0..30]]; // G. C. Greubel, Feb 09 2025

Table[Binomial[n+4,4]Binomial[n+6,4],{n,0,30}] (* Harvey P. Dale, Jun 07 2019 *)

def A107395(n): return binomial(n+4,4)*binomial(n+6,4)
print([A107395(n) for n in range(31)]) # G. C. Greubel, Feb 09 2025

From Amiram Eldar, Sep 06 2022: (Start)
Sum_{n>=0} 1/a(n) = 16*Pi^2 - 3946/25.
Sum_{n>=0} (-1)^n/a(n) = 1776/25 - 512*log(2)/5. (End)
G.f.: 5*(3 + 8*x + 3*x^2)/(1-x)^9. - G. C. Greubel, Feb 09 2025

More terms from Harvey P. Dale, Jun 07 2019

A180150 Numbers n such that n and n+2 are both divisible by exactly 4 primes (counted with multiplicity).

Original entry on oeis.org

54, 88, 150, 196, 232, 248, 294, 306, 328, 340, 342, 348, 460, 488, 490, 568, 570, 664, 712, 738, 774, 850, 856, 858, 868, 870, 948, 1012, 1014, 1060, 1096, 1110, 1148, 1190, 1204, 1206, 1208, 1210, 1218, 1254, 1274, 1276, 1290, 1302, 1314, 1420, 1430, 1448

Offset: 1

Jonathan Vos Post, Aug 12 2010

"Quadruprimes" or "4-almost primes" that keep that property when incremented by 2. This sequence is to 4 as 3 is to A180117, as A092207 is to 2, and as A001359 is to 1. That is, this sequence is the 4th row of the infinite array A[k,n] = n-th natural number m such that m and m+2 are both divisible by exactly k primes (counted with multiplicity). The first row is the lesser of twin primes. The second row is the sequence such that m and m+2 are both semiprimes.

			a(1) = 54 because 54 = 2 * 3^3 is divisible by exactly 4 primes (counted with multiplicity), and so is 54+2 = 56 = 2^3 * 7.

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

Cf. A000040, A001222, A001358, A014614, A033987, A101637, A114106 (number of 4-almost primes <= 10^n).

SequencePosition[PrimeOmega[Range[1500]],{4,,4}][[;;,1]] (* _Harvey P. Dale, Jan 14 2024 *)

is(n)=bigomega(n)==4 && bigomega(n+2)==4 \\ Charles R Greathouse IV, Jan 31 2017

{m in A014613 and m+2 in A014613} = {m such that bigomega(m) = bigomega(m+2) = 4} = {m such that A001222(m) = A001222(m+2) = 4}.

More terms from R. J. Mathar, Aug 13 2010

A340140 a(1) = -1, for n > 1, a(n) = Sum_{d|n, dA340197(n/d) * a(d).

Original entry on oeis.org

-1, 0, 0, -2, 0, -7, 0, -6, -6, -13, 0, -13, 0, -19, -22, -22, 0, -19, 0, -23, -32, -31, 0, -81, -20, -37, -24, -33, 0, -21, 0, -78, -52, -49, -58, -183, 0, -55, -62, -147, 0, -29, 0, -53, -52, -67, 0, -321, -42, -53, -82, -63, 0, -223, -94, -213, -92, -85, 0, -591, 0, -91, -74, -278, -112, -45, 0, -83, -112, -45, 0, -733

Offset: 1

Antti Karttunen, Jan 05 2021

This seems to differ from 1-A318833(n) at the points given by A033987.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A023900, A033987, A318833, A340197, A340367.

A023900(n) = factorback(apply(p -> 1-p, factor(n)[, 1]));
A318833(n) = (n+A023900(n));
A340197(n) = if(1==n,1,sumdiv(n,d,if(dA318833(n/d)-1)*A340197(d),0)));
A340140(n) = if(1==n, -1, sumdiv(n, d, if(dA340197(n/d)*A340140(d), 0)));

a(1) = -1, for n > 1, a(n) = Sum_{d|n, dA340197(n/d) * a(d).

A340090 Dirichlet inverse of A219428, n - phi(n) - 1.

Original entry on oeis.org

-1, 0, 0, -1, 0, -3, 0, -3, -2, -5, 0, -7, 0, -7, -6, -8, 0, -11, 0, -11, -8, -11, 0, -21, -4, -13, -8, -15, 0, -21, 0, -21, -12, -17, -10, -36, 0, -19, -14, -33, 0, -29, 0, -23, -20, -23, 0, -63, -6, -29, -18, -27, 0, -47, -14, -45, -20, -29, 0, -85, 0, -31, -26, -55, -16, -45, 0, -35, -24, -45, 0, -123, 0, -37

Offset: 1

Antti Karttunen, Jan 05 2021

sign

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000010, A033987, A051953, A219428, A340094, A340140, A340197, A340198.

up_to = 2^14;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA219428(n) = (n - 1 - eulerphi(n));
v340090 = DirInverseCorrect(vector(up_to, n, A219428(n)));
A340090(n) = v340090[n];
\\ Or as:
A340090(n) = if(1==n, -1, sumdiv(n, d, if(dA219428(n/d)*A340090(d), 0)));

a(1) = -1, for n > 1, a(n) = Sum_{d|n, dA219428(n/d) * a(d).

A380476 Numbers k with at least 4 prime factors such that A380459(k) is in A048103, i.e., has no divisors of the form p^p.

Original entry on oeis.org

4686, 32406, 184866, 209166, 388086, 1099626, 1714866, 2111406, 2166846, 2356206, 3081606, 3303366, 6445806, 11366106, 21621606, 23022366, 39824466, 39826986, 42882846, 43197846, 46043826, 58216686, 61265886, 63603546, 66496506, 66611166, 87941706, 88968246, 92086746, 97117026, 101108706, 103367886, 118743306, 119658066

Offset: 1

Antti Karttunen, Feb 04 2025

nonn

Numbers m with four or more distinct prime factors such that their arithmetic derivative (A003415) can be formed as a carryless (or "carry-free") sum (in the primorial base, A049345) of the respective summands. See the example.
The terms are all squarefree and even (see A380468 and A380478 to find out why). Moreover, they are all multiples of six, because A380459(n) = Product_{d|n} A276086(n/d)^A349394(d) applied to a product of 2*p*q*r, with p, q, r three odd primes > 3 would yield three subproducts which would be multiples of 3 (consider A047247), so the 3-adic valuation of the whole product would be >= 3; hence the second smallest prime factor must be 3. For a similar reason, with terms that are product of four primes, the two remaining prime factors are either both of the form 6m+1 (A002476), or they are both of the form 6m-1 (A007528).
It is conjectured that there are no terms with more than four prime factors. See A380475 and A380528, A380530, also A380526.

			4686 = 2*3*11*71 and taking subproducts of three primes at time, we obtain 2*3*11 = 66, 2*3*71 = 426, 2*11*71 = 1562, 3*11*71 = 2343. Then A380459(4686) = A276086(66) * A276086(426) * A276086(1562) * A276086(2343) = 1622849599205985150 = 2^1 * 3^2 * 5^2 * 7^6 * 11^9 * 13^1, and because all the exponents are less than the corresponding primes, the product is in A048103.
Considering the primorial base expansions of the same summands (subproducts), we obtain
    2100  = A049345(66)
   20100  = A049345(426)
   73010  = A049345(1562)
  101011  = A049345(2343)
  ------
  196221  = A049345(A003415(4686)), with the summands adding together cleanly without any carries.
Note how the primorial base digits at the bottom are the exponents in the product A380459(4686) given above, read from the largest to the smallest prime factor

Antti Karttunen, Table of n, a(n) for n = 1..433; all terms <= 2^35
Index entries for sequences related to primorial base.

Intersection of A033987 and A380468.
Subsequence of A005117, A358673, A380478.
Conjectured to be a subsequence of A046386.
Cf. A003415, A001222, A048103, A276086, A349394, A380459, A380467.
Cf. A002476, A007528, A047247, A047257.
Cf. also A317836, A358235, A380475, A380526, A380528, A380530.

is_A380476(n) = (issquarefree(n) && (omega(n)>=4) && A380467(n)); \\ Note that issquarefree here is just an optimization as A380467(n) = 1 implies squarefreeness of 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

A180151 Numbers k such that k and k + 2 are both divisible by exactly five primes (counted with multiplicity).

Original entry on oeis.org

270, 592, 700, 750, 918, 1168, 1240, 1638, 1648, 1672, 1710, 1750, 2070, 2310, 2392, 2548, 2550, 2608, 2728, 2860, 2862, 2896, 2898, 3184, 3330, 3568, 3630, 3822, 3848, 3850, 3942, 3976, 4230, 4264, 4648, 4662, 5070, 5080, 5236, 5238, 5390, 5550, 5560

Offset: 1

Jonathan Vos Post, Aug 12 2010

"5-almost primes" that keep that property when incremented by 2. This sequence is to 5 as 4 is to A180150, as 3 is to A180117, as A092207 is to 2, and as A001359 is to 1. That is, this sequence is the 5th row of the infinite array A[k,n] = n-th natural number m such that m and m+2 are both divisible by exactly k primes (counted with multiplicity). The first row is the lesser of twin primes. The second row is the sequence such that m and m+2 are both semiprimes.

			a(1) = 270 because 270 = 2 * 3^3 * 5 is divisible by exactly 5 primes (counted with multiplicity), and so is 270+2 = 272 = 2^4 * 17.

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

Cf. A000040, A001222, A001358, A001359, A014614, A033987, A101637, A180117, A180150.

f[n_] := Plus @@ (Last@# & /@ FactorInteger@n); fQ[n_] := f[n] == 5 == f[n + 2]; Select[ Range@ 10000, fQ] (* Robert G. Wilson v, Aug 15 2010 *)

for(x=2,10^4,if(bigomega(x)==5&&bigomega(x+2)==5,print1(x", "))) \\ Zak Seidov, Aug 12 2010

{m in A014614 and m+2 in A014614} = {m such that bigomega(m) = bigomega(m+2) = 5} = {m such that A001222(m) = A001222(m+2) = 5}.

Corrected and extended by Zak Seidov and R. J. Mathar, Aug 12 2010

A346483 Sum of A005171 (characteristic function of nonprimes) and its Dirichlet inverse.

Original entry on oeis.org

2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 3, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 4, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 4, 1, 0, 0, 4, 0, 0, 0, 2, 0, 4, 0, 0, 0, 0, 0, 3, 0, 0, 0, 3, 0, 0, 0, 2, 0

Offset: 1

Mats Granvik and Antti Karttunen, Aug 17 2021

sign

The first negative term is a(192) = -1.

Positions of nonzero terms are given by A033987, except for positions n = 256, 512, 6561, 16384, 19683, 32768, 390625, 1048576, ..., at which a(n) = 0 also.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A005171, A010051, A033987, A346482.

nn = 87; b = Table[If[PrimeQ[n], 1, 0], {n, nn}]; a = 1 - b; A = Table[Table[If[Mod[n, k] == 0, a[[n/k]], 0], {k, 1, nn}], {n, 1, nn}]; B = Inverse[A]; S = A[[Range[nn]]] + B[[Range[nn]]]; S[[All, 1]]

up_to = 65537;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA005171(n) = (1-isprime(n));
v346482 = DirInverseCorrect(vector(up_to,n,A005171(n)));
A346482(n) = v346482[n];
A346483(n) = (A005171(n)+A346482(n));

a(n) = A005171(n) + A346482(n).

For n > 1, a(n) = -Sum_{d|n, 1A005171(d) * A346482(n/d).

A376271 Numbers k such that there exists at least one proper divisor that is neither squarefree nor a prime power, i.e., m is in A126706.

Original entry on oeis.org

24, 36, 40, 48, 54, 56, 60, 72, 80, 84, 88, 90, 96, 100, 104, 108, 112, 120, 126, 132, 135, 136, 140, 144, 150, 152, 156, 160, 162, 168, 176, 180, 184, 189, 192, 196, 198, 200, 204, 208, 216, 220, 224, 225, 228, 232, 234, 240, 248, 250, 252, 260, 264, 270, 272

Offset: 1

Michael De Vlieger, Sep 28 2024

Numbers k such that A376514(k) > 1. A376514(k) >= 1 for all k in A126706.
Numbers k such that the cardinality of the intersection of row n of A027750 and A126706 exceeds 1.
a(n) is not in A366825, since for k in A366825, there is only one divisor that is in A126706, and that is k itself.

			4 is not in the sequence since 4 is a prime power, and all divisors d | k of prime power k = p^e are also prime powers.
6 is not in the sequence since 6 is squarefree, and all divisors d | k of squarefree k are also squarefree.
12 is not in the sequence since 12 is in A366825, and there is only 1 divisor in A126706, which is 12 itself.
24 is in the sequence since the intersection of A126706 and row 24 of A027750, indicated by bracketed numbers, is {1, 2, 3, 4, 6, [12, 24]}, etc.
Table listing the intersection of A126706 and row a(n) of A027750 for n <= 12:
  24: {12, 24}
  36: {12, 18, 36}
  40: {20, 40}
  48: {12, 24, 48}
  54: {18, 54}
  56: {28, 56}
  60: {12, 20, 60}
  72: {12, 18, 24, 36, 72}
  80: {20, 40, 80}
  84: {12, 28, 84}
  88: {44, 88}
  90: {18, 45, 90}

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

Cf. A007947, A027750, A033987, A126706, A366825, A376514.

Select[Range[300], Function[k, DivisorSum[k, 1 &, Nor[PrimePowerQ[#], SquareFreeQ[#]] &] > 1]]
(* Second program *)
Select[Range[300], And[#2 > #1 > 1, #2 > 3] & @@ {PrimeNu[#], PrimeOmega[#]} &] (* Michael De Vlieger, Dec 24 2024 *)

list(lim)=my(v=List()); forfactored(k=24,lim\1, my(e=k[2][,2]); if(#e>1 && vecmax(e)>1 && (#e>2 || vecsum(e)>3), listput(v,k[1]))); Vec(v) \\ Charles R Greathouse IV, Oct 01 2024

Intersection of A033987 and A126706, i.e., { k : bigomega(k) > omega(k) > 1, bigomega(k) > 3 }, where bigomega = A001222 and omega(k) = A001221. - Michael De Vlieger, Dec 24 2024

cp's OEIS Frontend

A046311 Numbers that are divisible by at least 9 primes (counted with multiplicity).

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A107395 a(n) = binomial(n+4,4)*binomial(n+6,4).

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A180150 Numbers n such that n and n+2 are both divisible by exactly 4 primes (counted with multiplicity).

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A340140 a(1) = -1, for n > 1, a(n) = Sum_{d|n, dA340197(n/d) * a(d).

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A340090 Dirichlet inverse of A219428, n - phi(n) - 1.

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A380476 Numbers k with at least 4 prime factors such that A380459(k) is in A048103, i.e., has no divisors of the form p^p.

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

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A180151 Numbers k such that k and k + 2 are both divisible by exactly five primes (counted with multiplicity).

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