A014613 -id:A014613 - cp's OEIS frontend

A101638 Number of distinct 4-almost primes A014613 which are factors of n.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1

Offset: 1

Jonathan Vos Post, Dec 10 2004

This is the inverse Moebius transform of A101637. If we take the prime factorization of n = (p1^e1)*(p2^e2)* ... * (pj^ej) then a(n) = |{k: ek>=4}| + ((j-1)/2)*|{k: ek>=3}| + C(|{k: ek>=2}|,2) + C(j,4). The first term is the number of distinct 4th powers of primes in the factors of n (the first way of finding a 4-almost prime). The second term is the number of distinct cubes of primes, each of which can be multiplied by any of the other distinct primes, halved to avoid double-counts (the second way of finding a 4-almost prime). The third term is the number of distinct pairs of squares of primes in the factors of n (the third way of finding a 4-almost prime). The 4th term is the number of distinct products of 4 distinct primes, which is the number of combinations of j primes in the factors of n taken 4 at a time, A000332(j), (the 4th way of finding a 4-almost prime).

			a(96) = 2 because 96 = 16 * 6 hence divisible by the 4-almost prime 16 and also 96 = 24 * 4 hence divisible by the 4-almost prime 24.

Hardy, G. H. and Wright, E. M. Section 17.10 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, 1979.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
E. A. Bender and J. R. Goldman, On the Applications of Moebius Inversion in Combinatorial Analysis, Amer. Math. Monthly 82, 789-803, 1975.
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, arXiv:math/0205301 [math.CO], 2002; Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
Eric Weisstein's World of Mathematics, Almost Prime.
Eric Weisstein's World of Mathematics, Moebius Transform.

Cf. A101638, A014613, A000332, A086971, A100605, A000040, A001358, A014612, A014614.

a(n)=my(f=factor(n)[,2], v=apply(k->sum(i=1,#f,f[i]>k), [0..3])); v[4] + v[3]*(v[1]-1) + binomial(v[2],2) + v[2]*binomial(v[1]-1,2) + binomial(v[1],4) \\ Charles R Greathouse IV, Sep 14 2015

A109024 Numbers that have exactly four prime factors counted with multiplicity (A014613) whose digit reversal is different and also has 4 prime factors (with multiplicity).

Original entry on oeis.org

126, 225, 294, 315, 459, 488, 492, 513, 522, 558, 621, 650, 738, 837, 855, 884, 954, 1035, 1062, 1098, 1107, 1197, 1206, 1236, 1287, 1305, 1422, 1518, 1617, 1665, 1917, 1926, 1956, 1962, 1989, 2004, 2034, 2046, 2068, 2104, 2148, 2170, 2180, 2223, 2226

Offset: 1

Jonathan Vos Post, Jun 16 2005

This sequence is the k = 4 instance of the series which begins with k = 1, k = 2, k = 3 (A109023).

			a(1) = 126 is in this sequence because 126 = 2 * 3^2 * 7 is a 4-almost prime and reverse(126) = 621 = 3^3 * 23 is also a 4-almost prime.
a(2) = 225 is in this sequence because 225 = 3^2 * 5^2 is a 4-almost prime and reverse(225) = 522 = 2 * 3^2 * 29 is also a 4-almost prime. (That 225 and 522 are concatenated from entirely prime digits is a coincidence, as with 2223).

David A. Corneth, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Almost Prime.
Eric Weisstein's World of Mathematics, Emirp.
Eric Weisstein's World of Mathematics, Emirpimes.

Cf. A006567, A097393, A109018, A109023, A109025, A109026, A109027, A109028, A109029, A109030, A109031.

Select[Range[2226],PrimeOmega[#]==4 && PrimeOmega[FromDigits[Reverse[IntegerDigits[#]]]]==4 &&!PalindromeQ[#]&] (* James C. McMahon, Mar 07 2024 *)

is(n) = {
	my(r = fromdigits(Vecrev(digits(n))));
	n!=r && bigomega(n) == 4 && bigomega(r) == 4
} \\ David A. Corneth, Mar 07 2024

A124284 Prime(4almostprime(n))-4almostprime(prime(n)). Commutator [A000040,A014613] at n.

Original entry on oeis.org

29, 53, 97, 113, 161, 159, 145, 269, 244, 232, 231, 247, 261, 373, 399, 386, 328, 350, 375, 371, 395, 547, 559, 572, 537, 541, 577, 635, 679, 663, 607, 621, 687, 673, 658, 769, 871, 853, 839, 856, 832, 881, 947, 939, 1003, 1007, 955, 915, 907, 889, 941, 989

Offset: 1

Jonathan Vos Post, Oct 24 2006

			a(1) = prime(4almostprime(1)) - 4almostprime(prime(1)) = 53 - 24 = 29.
a(2) = prime(4almostprime(2)) - 4almostprime(prime(2)) = 89 - 36 = 53.
a(3) = prime(4almostprime(3)) - 4almostprime(prime(3)) = 151 - 54 = 97.
It is mere coincidence that the first 4 values are all primes.

Robert G. Wilson v, Table of n, a(n) for n=1..1000

Cf. Primes indexed by 4-almost primes = A124282. 4-almost primes indexed by primes = A124283. Primes indexed by 3-almost primes = A124268. 3-almost primes indexed by primes = A124269. prime(3almostprime(n)) - 3almostprime(prime(n)) = A124270. See also A106349 Primes indexed by semiprimes. See also A106350 Semiprimes indexed by primes. See also A122824 Prime(semiprime(n)) - semiprime(prime(n)).

Cf. A000040, A014613, A114414, A122824, A124269, A124270, A124282, A124283.

FourAlmostPrimePi[n_] := Sum[PrimePi[n/(Prime@i*Prime@j*Prime@k)] - k + 1, {i, PrimePi[n^(1/4)]}, {j, i, PrimePi[(n/Prime@i)^(1/3)]}, {k, j, PrimePi@ Sqrt[n/(Prime@i*Prime@j)]}];
FourAlmostPrime[n_] := Block[{e = Floor[Log[2, n] + 1], a, b}, a = 2^e; Do[b = 2^p; While[ FourAlmostPrimePi@a < n, a = a + b]; a = a - b/2, {p, e, 0, -1}]; a + b/2];
Table[ Prime@ FourAlmostPrime@ n - FourAlmostPrime@ Prime@ n, {n, 52}]

from math import isqrt
from sympy import primepi, primerange, integer_nthroot, prime
def A124284(n):
    def f(x): return int(x-sum(primepi(x//(k*m*r))-c for a,k in enumerate(primerange(integer_nthroot(x,4)[0]+1)) for b,m in enumerate(primerange(k,integer_nthroot(x//k,3)[0]+1),a) for c,r in enumerate(primerange(m,isqrt(x//(k*m))+1),b)))
    m, k = n, f(n)+n
    while m != k:
        m, k = k, f(k)+n
    r, k = (p:=prime(n)), f(p)+p
    while r != k:
        r, k = k, f(k)+p
    return prime(m)-r # Chai Wah Wu, Aug 17 2024

a(n) = prime(4almostprime(n)) - 4almostprime(prime(n)) = A000040(A014613(n)) -A014613(A000040(n)).

More terms from Robert G. Wilson v, Aug 31 2007

A114426 Product of the first n 4-almost primes (A014613).

Original entry on oeis.org

16, 384, 13824, 552960, 29859840, 1672151040, 100329062400, 8126654054400, 682638940569600, 60072226770124800, 5406500409311232000, 540650040931123200000, 56227604256836812800000

Offset: 1

Jonathan Vos Post, Feb 13 2006

4-almost prime analog of primorial (A002110). The semiprime analog of primorial is A112141. Equivalent for product of what A086046 is for sum. Bigomega(a(n)) = the number of not necessarily distinct prime factors of a(n) = A001222(a(n)) = A008586(n) = 4*n.

			a(5) = 29859840 = 16 * 24 * 36 * 40 * 54 = the product of the first 5 values of the 4-almost primes = 2^13 * 3^6 * 5, which has 4*5 = 20 prime factors (with multiplicity).

Cf. A001222, A002110, A008585, A014613, A008586, A086046, A112141.

FoldList[Times,Select[Range[200],PrimeOmega[#]==4&]] (* Harvey P. Dale, Dec 02 2018 *)

a(n) = Prod[from i = 1 to n] A014613(i).

A114404 4-almost prime gaps. First differences of A014613.

Original entry on oeis.org

8, 12, 4, 14, 2, 4, 21, 3, 4, 2, 10, 4, 22, 6, 3, 1, 4, 10, 2, 4, 28, 5, 7, 2, 6, 6, 10, 5, 3, 4, 2, 14, 2, 10, 16, 18, 2, 1, 9, 2, 7, 13, 2, 10, 2, 2, 4, 2, 1, 13, 8, 3, 1, 4, 10, 24, 10, 17, 3, 15, 1, 2, 10, 4, 8, 4, 2, 2, 3, 15, 3, 3, 6, 3, 7, 4, 10, 4, 8, 6, 4, 2, 2, 8, 4, 1, 35, 1, 4, 7, 4, 8, 6

Offset: 1

Jonathan Vos Post, Nov 25 2005

			a(1) = 8 = 24-16 where 16 is the first 4-almost prime and 24 is the second.
a(2) = 12 = 36-24.
a(3) = 4 = 40-36.
a(4) = 14 = 54-40.
a(5) = 2 = 56-54.
a(6) = 4 = 60-56.
a(7) = 21 = 81-60.
a(13) = 22 = 126-104.
a(21) = 28 = 184-156.

T. D. Noe, Table of n, a(n) for n=1..10000

Cf. A014613, A065516, A111870, A111871, A114403-A114411, A114412-A114422.

A114404 := proc(nmax) local a,i,a014613 ; a := [] ; i := 1 ; a014613 := -1 ; while nops(a) < nmax do if numtheory[bigomega](i) = 4 then if a014613 > 0 then a := [op(a),i-a014613] ; fi ; a014613 := i ; fi ; i := i+1 ; end: a ; end: A114404(200) ; # R. J. Mathar, May 10 2007

Differences[Select[Range[800],Total[FactorInteger[#][[All,2]]]==4&]] (* Harvey P. Dale, Feb 14 2017 *)
Select[Range[1000],PrimeOmega[#]==4&]//Differences (* Harvey P. Dale, May 12 2018 *)

a(n) = A014613(n+1) - A014613(n).

Corrected and extended by R. J. Mathar, May 10 2007

A061218 Least number whose number of divisors is n-th term from A014613 (numbers of form pqr*s, products of exactly 4 primes, counted with multiplicity).

Original entry on oeis.org

120, 360, 1260, 1680, 6300, 6720, 5040, 44100, 20160, 107520, 25200, 45360, 430080, 100800, 322560, 176400, 6881280, 181440, 226800, 27525120, 1290240, 440401920, 705600, 1632960, 1612800, 20643840, 907200, 2903040, 1587600, 82575360, 28185722880, 6451200, 112742891520

Offset: 1

Labos Elemer, Jun 06 2001

nonn

			p*q*r*s = 210 is the 27th term in A014613; the smallest number with 210 divisors is 907200 = 2*2*2*2*2*2*3*3*3*3*5*5*7.

Cf. A000005, A005179, A014613, A061148, A061149, A061299.

from math import prod, isqrt
from sympy import primepi, primerange, integer_nthroot, isprime, divisors, prime
def A061218(n):
    def f(x): return int(n+x-sum(primepi(x//(k*m*r))-c for a,k in enumerate(primerange(integer_nthroot(x,4)[0]+1)) for b,m in enumerate(primerange(k,integer_nthroot(x//k,3)[0]+1),a) for c,r in enumerate(primerange(m,isqrt(x//(k*m))+1),b)))
    def mult_factors(n):
        if isprime(n):
            return [(n,)]
        c = []
        for d in divisors(n,generator=True):
            if 1Chai Wah Wu, Aug 17 2024

a(n) = A005179(A014613(n)).

Corrected and extended by Michel Marcus, Sep 05 2017

A109989 4-almost primes (A014613) ordered alphabetically by where they occur in A000052.

Original entry on oeis.org

88, 84, 81, 54, 56, 40, 90

Offset: 1

Jonathan Vos Post, Jul 07 2005

A109986 is primes ordered alphabetically by where they occur in A000052. A109987 is semiprimes ordered alphabetically by where they occur in A000052. A109988 is 3-almost primes ordered alphabetically by where they occur in A000052.

			a(1) = 88 because eighty-eight is the first 4-almost prime in alphabetical order, there being no 1-digit 4-almost primes.
a(2) = 84 because eighty-four is the second 2-digit 4-almost prime in alphabetical order.

Cf. A000052, A014613, A109986, A109987, A109988.

A114432 Primes of the form 1 + product of the first k 4-almost primes A014613.

Original entry on oeis.org

17, 8126654054401

Offset: 1

Jonathan Vos Post, Feb 13 2006

nonn

The next term is too large to display here. - N. J. A. Sloane, Jul 30 2009
4-almost prime analog of primorial primes A005234 as indexed by A014545. In that sense, this sequence is indexed by (1, 8, ...). No more through product of first 16 of the 4-almost primes.
Terms are one more than the products of 4-almost primes up to 16, 81, 294, 513, 825, 1356, 1612, 2004, 2756, 7714, ... - Charles R Greathouse IV, Jul 28 2009

			a(1) = 17 because 1 + 16 = 1 + A014613(1) = 1 more than the first 4-almost prime is itself prime.
a(2) = 8126654054401 = 1 + (16 * 24 * 36 * 40 * 54 * 56 * 60 * 81) = 1 more than the product of the first 8 of the 4-almost primes and is prime.

Amiram Eldar, Table of n, a(n) for n = 1..8

Cf. A000040, A014613, A005234, A014545.

Select[FoldList[Times, Select[Range[200], PrimeOmega[#] == 4 &]] + 1, PrimeQ] (* Amiram Eldar, Jul 20 2025 *)

{1 + Product_{i=1..k} A014613(i)} INTERSECTION A000040.

A146484 Decimal expansion of Product_{q in A014613} (1-1/(q*(q-1))).

Original entry on oeis.org

9, 8, 9, 6, 2, 8, 8, 6, 7, 1, 6, 6, 4, 2, 7, 6, 6, 5, 5, 0, 4, 3, 2, 2, 8, 3, 7, 4, 5, 7, 9, 2, 4, 3, 0, 8, 0, 5, 7, 5, 5, 7, 5, 8, 9, 3, 5, 0, 2, 9, 6, 5, 3, 4, 8, 4, 4

Offset: 0

R. J. Mathar, Feb 13 2009

4-almost prime analog of A005596.

			0.989628867166427665504.. = (1-1/240)*(1-1/552)*...

The logarithm is -sum_{s>=2} sum_{j=1..floor[s/(1+r)]} binomial(s-r*j-1,j-1)*P_4(s)/j at r=1, where P_k(s) are the k-almost prime zeta functions of arXiv:0803.0900.

A146488 Decimal expansion of Product_{q in A014613} (1-1/(q^2*(q-1))).

Original entry on oeis.org

9, 9, 9, 5, 9, 5, 2, 7, 8, 5, 8, 6, 5, 3, 5, 5, 3, 5, 6, 3, 7, 4, 5, 2, 4, 9, 3, 2, 4, 8, 3, 3, 6, 4, 5, 3, 0, 8, 3, 6, 5, 0, 6, 3, 2, 4, 1, 2, 6, 7, 4, 0, 4, 9, 8, 8, 7

Offset: 0

R. J. Mathar, Feb 13 2009

4-almost prime analog of A065414.

			0.99959527858653553563... = (1-1/3840)*(1-1/13248)*(1-1/45360)*(1-1/62400)*..

The logarithm is -sum_{s>=2} sum_{j=1..floor[s/(1+r)]} binomial(s-r*j-1,j-1)*P_4(s)/j at r=2, where P_k(s) are the k-almost prime zeta functions of arXiv:0803.0900.

cp's OEIS Frontend

A101638 Number of distinct 4-almost primes A014613 which are factors of n.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

PARI

A109024 Numbers that have exactly four prime factors counted with multiplicity (A014613) whose digit reversal is different and also has 4 prime factors (with multiplicity).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A124284 Prime(4almostprime(n))-4almostprime(prime(n)). Commutator [A000040,A014613] at n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Python

Formula

Extensions

A114426 Product of the first n 4-almost primes (A014613).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A114404 4-almost prime gaps. First differences of A014613.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A061218 Least number whose number of divisors is n-th term from A014613 (numbers of form p*q*r*s, products of exactly 4 primes, counted with multiplicity).

Views

Author

Keywords

Examples

Crossrefs

Programs

Python

Formula

Extensions

A109989 4-almost primes (A014613) ordered alphabetically by where they occur in A000052.

Views

Author

Keywords

Comments

Examples

Crossrefs

A114432 Primes of the form 1 + product of the first k 4-almost primes A014613.

Views

Author

Keywords

Comments

Examples

A061218 Least number whose number of divisors is n-th term from A014613 (numbers of form pqr*s, products of exactly 4 primes, counted with multiplicity).