A014612 -id:A014612 - cp's OEIS frontend

A114403 Triprime gaps. First differences of A014612.

4, 6, 2, 7, 1, 2, 12, 2, 1, 5, 2, 11, 3, 2, 2, 5, 1, 2, 14, 6, 1, 3, 3, 5, 4, 2, 1, 7, 1, 5, 8, 9, 1, 5, 1, 10, 1, 5, 1, 1, 2, 1, 7, 4, 2, 2, 5, 12, 5, 10, 8, 1, 5, 2, 4, 2, 1, 1, 9, 3, 3, 5, 2, 5, 2, 4, 3, 2, 1, 1, 4, 2, 18, 6, 2, 4, 3, 7, 1, 5, 5, 2, 9, 2, 1

Offset: 1

Jonathan Vos Post, Nov 25 2005

			a(1) = 4 = 12-8 where 8 is the first triprime and 12 is the second.
a(2) = 6 = 18-12
a(3) = 2 = 20-18
a(4) = 7 = 27-20

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

Cf. A014612, A065516, A111870, A111871, A114403-A114411, A114412-A114422.

is3Alm := proc(n::integer) local ifa,ex,i ; ifa := op(2,ifactors(n)) ; ex := 0 ; for i from 1 to nops(ifa) do ex := ex+ op(2,op(i,ifa)) ; od : if ex = 3 then RETURN(true) ; else RETURN(false) ; fi ; end: A014612 := proc(n::integer) local resul,i; i :=1; resul := 8 ; while i < n do resul := resul + 1 ; if is3Alm(resul) then i := i+1 ; fi ; od ; RETURN(resul) ; end: A114403 := proc(n::integer) RETURN(A014612(n+1)-A014612(n)) ; end: for n from 1 to 160 do printf("%d,",A114403(n)) ; od: # R. J. Mathar, Apr 25 2006

Differences[Select[Range[425], PrimeOmega[#] == 3 &]] (* Jayanta Basu, Jul 01 2013 *)

a(n) = A014612(n+1) - A014612(n).

Corrected and extended by R. J. Mathar, Apr 25 2006

A109023 3-almost primes (A014612) whose digit reversal is different and also has 3 prime factors (with multiplicity).

Original entry on oeis.org

117, 147, 165, 244, 246, 285, 286, 290, 338, 366, 369, 406, 418, 425, 435, 438, 442, 475, 498, 506, 507, 508, 524, 534, 539, 548, 561, 574, 582, 604, 605, 609, 628, 642, 663, 670, 682, 705, 711, 741, 759, 805, 814, 826, 833, 834, 845, 890, 894, 906, 935

Offset: 1

Jonathan Vos Post, Jun 16 2005

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

			1066 is in this sequence because 1066 = 2 * 13 * 41, making it a 3-almost prime and reverse(1066) = 6601 = 7 * 23 * 41, also a 3-almost prime.
2001 is in this sequence because 2001 = 3 * 23 * 29 and reverse(2001) = 1002 = 2 * 3 * 167.

W. W. R. Ball and H. S. M. Coxeter, Mathematical Recreations and Essays, 13th ed. New York: Dover, pp. 14-15, 1987.
J. Edalj, Problem 1622. L'Intermédiaire des Mathématiciens, 16, 34, 1909.

David A. Corneth, Table of n, a(n) for n = 1..10000
J. Jonesco, Query 1622, L'Intermédiaire des Mathématiciens, 200, Tome VI, 1899.
Eric Weisstein's World of Mathematics, Almost Prime.
Eric Weisstein's World of Mathematics, Emirp.
Eric Weisstein and Jonathan Vos Post, Emirpimes.

Cf. A006567, A097393, A109018, A109024, A109025, A109026, A109027, A109028, A109029, A109030, A109031.

Select[Range[1000],PrimeOmega[#]==3&&PrimeOmega[FromDigits[Reverse[IntegerDigits[#]]]]==3&&!PalindromeQ[#]&] (* James C. McMahon, Mar 06 2024 *)

is(n) = {
	my(r = fromdigits(Vecrev(digits(n))));
	n!=r && bigomega(n) == 3 && bigomega(r) == 3
} \\ David A. Corneth, Mar 07 2024

1002 replaced by 935 - R. J. Mathar, Dec 14 2009

A124270 a(n) = prime(A014612(n)) - A014612(prime(n)). Commutator [A000040,A014612] at n.

Original entry on oeis.org

7, 19, 34, 41, 53, 44, 38, 103, 91, 73, 99, 75, 135, 142, 147, 118, 133, 125, 118, 193, 229, 191, 212, 202, 197, 201, 216, 213, 248, 239, 209, 248, 279, 279, 277, 277, 333, 325, 350, 327, 299, 308, 264, 309, 314, 322, 297, 281, 363, 374, 461, 488, 484, 482

Offset: 1

Jonathan Vos Post, Oct 23 2006

			a(1) = prime(3almostprime(1)) - 3almostprime(prime(1)) = prime(8) - 3almostprime(2) = 19 - 12 = 7.
a(2) = prime(3almostprime(2)) - 3almostprime(prime(2)) = prime(12) - 3almostprime(3) = 37 - 18 = 19.
a(3) = prime(3almostprime(3)) - 3almostprime(prime(3)) = prime(18) - 3almostprime(5) = 61 - 27 = 34.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A000040 (primes), A014612 (3-almost primes).
Cf. A124268 (prime(3-almost prime(n))), A124269 (3-almost prime(prime(n))).
Cf. A106349 (prime(semiprime(n))), A106350 (semiprime(prime(n))), A122824 (prime(semiprime(n)) - semiprime(prime(n))).

lista(nn) = {p = primes(nn); pp = select(x->bigomega(x)==3, vector(nn, n, n)); for (n=1, nn, print1(p[pp[n]] - pp[p[n]], ", "););} \\ Michel Marcus, Oct 15 2014

a(n) = A000040(A014612(n)) - A014612(A000040(n)).

a(n) = A124268(n) - A124269(n).

A130650 a(n) = smallest k such that A014612(n+1) = A014612(n) + (A014612(n) mod k), or 0 if no such k exists.

Original entry on oeis.org

0, 0, 4, 13, 2, 13, 18, 4, 43, 8, 3, 41, 4, 4, 3, 13, 2, 37, 16, 43, 97, 4, 9, 10, 53, 4, 5, 10, 3, 6, 61, 43, 2, 11, 2, 12, 163, 8, 13, 2, 5, 173, 8, 89, 4, 3, 37, 61, 101, 101, 107, 229, 113

Offset: 1

Rémi Eismann, Aug 16 2007 - Jan 21 2011

nonn

a(n) is the "weight" of 3-almost primes.

The decomposition of 3-almost primes into weight * level + gap is A014612(n) = a(n) * A184753(n) + A114403(n) if a(n) > 0.

			For n = 1 we have A014612(1) = 8, A014612(2) = 12; there is no k such that 12 - 8 = 4 = (8 mod k), hence a(1) = 0.
For n = 3 we have A014612(3) = 18, A014612(4) = 20; 4 is the smallest k such that 20 - 18 = 2 = (18 mod k), hence a(3) = 4.
For n = 21 we have A014612(21) = 98, A014612(22) = 99; 97 is the smallest k such that 99 - 98 = 1 = (97 mod k), hence a(21) = 97.

Remi Eismann, Table of n, a(n) for n=1..9999

Cf. A014612, A114403, A184753, A184752, A117078, A117563, A001223, A118534.

A072114 Number of 3-almost primes (A014612) <= n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 4, 4, 4, 5, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 9, 10, 10, 10, 10, 10, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 14, 14, 15, 15, 16, 16, 16, 16, 16, 17, 18, 18, 19, 19, 19, 19, 19, 19, 19

Offset: 0

Benoit Cloitre, Jun 19 2002

Number of k <= n such that bigomega(k) = 3.
Let A be a positive integer then card{ x <= n : bigomega(x) = A } ~ (n/Log(n))*Log(Log(n))^(A-1)/(A-1)!. For which n, card{ x <= n : bigomega(x) = 3 } >= card{ x <= n : bigomega(x) = 2 } ?
15530 is the first number for which there are more 3-almost primes than 2-almost primes. See A125149.

E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, vol. 1, Teubner, Leipzig; third edition : Chelsea, New York (1974).
G. Tenenbaum, Introduction à la théorie analytique et probabiliste des nombres, p. 203, Publications de l'Institut Cartan, 1990.

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000
E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, vol. 1 and vol. 2, Leipzig, Berlin, B. G. Teubner, 1909.

Cf. A014612, A109251, A001358, A072000.
Partial sums of A101605.
Cf. A125149.

Table[Sum[KroneckerDelta[PrimeOmega[i], 3], {i, n}], {n, 0, 50}] (* Wesley Ivan Hurt, Oct 07 2014 *)

for(n=1,100,print1(sum(i=1,n,bigomega(i)==3),","))

a(n)=my(j,s);forprime(p=2,(n+.5)^(1/3),j=primepi(p)-2;forprime(q=p,sqrtint(n\p),s+=primepi(n\(p*q))-j++));s \\ Charles R Greathouse IV, Mar 21 2012

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A072114(n): return int(sum(primepi(n//(k*m))-b for a,k in enumerate(primerange(integer_nthroot(n,3)[0]+1)) for b,m in enumerate(primerange(k,isqrt(n//k)+1),a))) # Chai Wah Wu, Aug 17 2024

a(n) = card{ x <= n : bigomega(x) = 3 }, asymptotically : a(n) ~ (n/log(n))*log(log(n))^2/2 [Landau, p. 211].

A162361 Central prime factor of A014612(n).

Original entry on oeis.org

2, 2, 3, 2, 3, 2, 3, 3, 2, 3, 5, 2, 3, 3, 2, 5, 5, 2, 3, 2, 7, 3, 3, 5, 5, 3, 2, 3, 2, 5, 5, 3, 7, 2, 3, 7, 2, 5, 5, 3, 2, 3, 5, 7, 3, 2, 5, 5, 3, 2, 3, 5, 7, 2, 7, 11, 2, 7, 3, 5, 3, 3, 7, 2, 7, 5, 3, 3, 2, 5, 11, 5, 2, 5, 2, 3, 7, 5, 2, 3, 13, 7, 5, 3, 2, 7, 11, 3, 3, 5, 11, 7, 3, 2, 7, 3, 2, 7, 5, 2, 11, 3, 5

Offset: 1

Juri-Stepan Gerasimov, Jul 02 2009

nonn

			a(1)=2 since A014612(1) =  8 = 2*2*2.
a(2)=2 since A014612(2) = 12 = 2*2*3.
a(3)=3 since A014612(3) = 18 = 2*3*3.
a(4)=2 since A014612(4) = 20 = 2*2*5.
a(5)=3 since A014612(5) = 27 = 3*3*3.
a(6)=2 since A014612(6) = 28 = 2*2*7.
a(7)=3 since A014612(7) = 30 = 2*3*5.

Michel Marcus, Table of n, a(n) for n = 1..10000

Cf. A014612, A020639 (Lpf), A006530 (Gpf).

A014612 := proc(n) option remember ; if n = 1 then 8 ; else for a from procname(n-1)+1 do if numtheory[bigomega](a) = 3 then RETURN(a) ; fi; od: fi; end:
A162361 := proc(n) tpr := A014612(n) ; pf := sort(convert(numtheory[factorset](tpr),list)) ; tpr/op(1,pf)/op(-1,pf) ; end:
seq(A162361(n),n=1..120) ; # R. J. Mathar, Jul 06 2009

f[n_] := With[{fi = FactorInteger[n][[All, 1]]}, n/(fi[[1]] fi[[-1]])];
f /@ Select[Range[500], PrimeOmega[#] == 3&] (* Jean-François Alcover, Aug 05 2022 *)

isok(n) = bigomega(n)==3;
lista(nn) = {for (n=1, nn, if (isok(n), my(f=factor(n)[,1]); print1(n/(vecmin(f)*vecmax(f)), ", ")););} \\ Michel Marcus, Feb 25 2019

a(n) = m/(A020639(m)*A006530(m)) where m = A014612(n). - Michel Marcus, Feb 25 2019

Edited (but not checked) by N. J. A. Sloane, Jul 05 2009

Corrected and extended by R. J. Mathar and Ray Chandler, Jul 06 2009

A184752 a(n) = largest k such that A014612(n+1) = A014612(n) + (A014612(n) mod k), or 0 if no such k exists.

Original entry on oeis.org

0, 0, 16, 13, 26, 26, 18, 40, 43, 40, 48, 41, 60, 64, 66, 65, 74, 74, 64, 86, 97, 96, 99, 100, 106, 112, 115, 110, 123, 120, 122, 129, 146, 143, 152, 144, 163, 160, 169, 170, 170, 173, 168, 178, 184, 186, 185, 183, 202, 202, 214

Offset: 1

Rémi Eismann, Jan 21 2011

nonn

From the definition, a(n) = A014612(n) - A114403(n) if A014612(n) - A114403(n) > A114403(n), 0 otherwise where A014612 are the 3-almost primes and A114403 are the gaps between 3-almost primes.

			For n = 1 we have A014612(1) = 8, A014612(2) = 12; there is no k such that 12 - 8 = 4 = (8 mod k), hence a(1) = 0.
For n = 3 we have A014612(3) = 18, A014612(4) = 20; 16 is the largest k such that 20 - 18 = 2 = (18 mod k), hence a(3) = 16.
For n = 21 we have A014612(21) = 98, A014612(22) = 99; 97 is the largest k such that 99 - 98 = 1 = (97 mod k), hence a(21) = 97.

Rémi Eismann, Table of n, a(n) for n = 1..10000

Cf. A014612, A114403, A130650, A184753, A117078, A117563, A001223, A118534.

A109988 3-almost primes (A014612) ordered alphabetically by where they occur in A000052.

Original entry on oeis.org

8, 18, 50, 52, 45, 44, 42, 98, 99, 92, 70, 78, 75, 76

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. A109989 is 4-almost primes ordered alphabetically by where they occur in A000052.

			a(1) = 8 because eight is the first 3-almost prime in alphabetical order.
a(4) = 18 because eighteen is the first 2-digit 3-almost prime in alphabetical order.

Cf. A000052, A014612, A109986, A109987.

A123073 Number of ordered triples of primes (p,q,r) such that pqr = n-th 3-almost prime A014612(n).

Original entry on oeis.org

1, 3, 3, 3, 1, 3, 6, 6, 3, 3, 3, 3, 3, 6, 3, 6, 3, 3, 6, 3, 3, 3, 6, 6, 6, 6, 3, 3, 3, 1, 6, 6, 3, 3, 3, 6, 3, 6, 6, 3, 3, 6, 3, 6, 6, 3, 6, 6, 3, 3, 6, 6, 6, 3, 6, 3, 3, 3, 6, 6, 6, 3, 6, 3, 6, 3, 3, 6, 3, 6, 6, 6, 3, 6, 3, 6, 6, 3, 3, 3, 3, 1, 6, 6, 3, 6, 3, 6, 3, 6, 6, 6, 3, 3, 6, 6, 3, 6, 6, 3, 6, 3, 3, 6, 3

Offset: 1

N. J. A. Sloane and T. D. Noe, Sep 29 2006

nonn

The nonzero subsequence of A123074.

Cf. A123074, A014612.

from math import isqrt
from sympy import primepi, primerange, integer_nthroot, primefactors
def A123073(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(integer_nthroot(x,3)[0]+1)) for b,m in enumerate(primerange(k,isqrt(x//k)+1),a)))
    return (1,3,6)[len(primefactors(bisection(f,n,n)))-1] # Chai Wah Wu, Oct 20 2024

More terms from T. D. Noe, Sep 29 2006

A124319 Semiprime(3almostprime(n))-3almostprime(semiprime(n)). Commutator[A001358, A014612] at n.

Original entry on oeis.org

2, 6, 7, 12, 16, 17, -11, 24, 23, 20, -1, 10, 48, 40, 39, 26, 14, 4, -1, 51, 60, 48, 48, 43, 31, 39, 22, 15, 37, 32, 39, 60, 90, 82, 68, 63, 64, 58, 66, 51, 53, 48, 28, 34, 42, 24, 28, 39, 87, 96, 106, 124, 124, 135, 131, 131, 88, 91, 72, 96, 103, 83, 83, 81, 91

Offset: 1

Jonathan Vos Post, Oct 26 2006

			a(1) = semiprime(3almostprime(1)) - 3almostprime(semiprime(1)) = 22 - 20 = 2.
a(2) = semiprime(3almostprime(2)) - 3almostprime(semiprime(2)) = 34 - 28 = 6.
a(3) = semiprime(3almostprime(3)) - 3almostprime(semiprime(3)) = 51 - 44 = 7.
a(4) = semiprime(3almostprime(4)) - 3almostprime(semiprime(4)) = 57 - 45 = 12.
a(7) = semiprime(3almostprime(7)) - 3almostprime(semiprime(7)) = 87 - 98 = -11, which is the first negative value in the commutators we have seen in these related set of sequences, exposing an incorrect assumption.

Cf. A124317 Semiprimes indexed by 3-almost primes. A124318 3-almost primes indexed by semiprimes. A124319 semiprime(3almostprime(n)) - 3almostprime(semiprime(n)). A124308 Primes indexed by 5-almost primes. A124309 5-almost primes indexed by primes. A124310 prime(5almostprime(n)) - 5almostprime(prime(n)). 4-almost primes indexed by primes = A124283. prime(4almostprime(n)) - 4almostprime(prime(n)) = A124284. 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)). Commutator [A000040, A001358] at n.

Cf. A000040, A001358, A014612, A065516, A114403, A122824, A124269, A124270, A124282-A124284, A124309, A124310, A124317, A124318.

p[k_] := p[k] = Select[Range[1000], PrimeOmega[#] == k &]; p[2][[ Take[p[3], 70]]] - p[3][[Take[p[2], 70]]] (* Giovanni Resta, Jun 13 2016 *)

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A124319(n):
    def f(x): return int(x-sum(primepi(x//(k*m))-b for a,k in enumerate(primerange(integer_nthroot(x,3)[0]+1)) for b,m in enumerate(primerange(k,isqrt(x//k)+1),a)))
    def g(x): return int(x+((t:=primepi(s:=isqrt(x)))*(t-1)>>1)-sum(primepi(x//k) for k in primerange(1, s+1)))
    def A001358(n):
        m, k = n, g(n)+n
        while m != k:
            m, k = k, g(k)+n
        return m
    m, k = n, f(n)+n
    while m != k:
        m, k = k, f(k)+n
    r, k = (p:=A001358(n)), f(p)+p
    while r != k:
        r, k = k, f(k)+p
    return A001358(m)-r # Chai Wah Wu, Aug 17 2024

a(18) corrected and a(22)-a(65) from Giovanni Resta, Jun 13 2016

cp's OEIS Frontend

A114403 Triprime gaps. First differences of A014612.

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A109023 3-almost primes (A014612) whose digit reversal is different and also has 3 prime factors (with multiplicity).

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A124270 a(n) = prime(A014612(n)) - A014612(prime(n)). Commutator [A000040,A014612] at n.

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A130650 a(n) = smallest k such that A014612(n+1) = A014612(n) + (A014612(n) mod k), or 0 if no such k exists.

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A072114 Number of 3-almost primes (A014612) <= n.

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A162361 Central prime factor of A014612(n).

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A184752 a(n) = largest k such that A014612(n+1) = A014612(n) + (A014612(n) mod k), or 0 if no such k exists.

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A109988 3-almost primes (A014612) ordered alphabetically by where they occur in A000052.