A014612 -id:A014612 - cp's OEIS frontend

A146483 Decimal expansion of Product_{q in A014612} (1-1/(q*(q-1))).

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

Offset: 0

R. J. Mathar, Feb 13 2009

3-almost prime analog of A005596.

			0.9587521164357309277147402... = (1-1/56)*(1-1/132)*(1-1/306)*(1-1/380)*..

R. J. Mathar, Hardy-Littlewood constants embedded into infinite products over all positive integers, arXiv:0903.2514 [math.NT], table 3 fourth line. [From _R. J. Mathar_, Mar 28 2009]

The logarithm is -sum_{s>=2} sum_{j=1..floor[s/(1+r)]} binomial(s-r*j-1,j-1)*P_3(s)/j at r=1, where P_k(s) are the k-almost prime zeta functions of arXiv:0803.0900.

A146487 Decimal expansion of Product_{q in A014612} (1-1/(q^2*(q-1))).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Feb 13 2009

3-almost prime analog of A065414.

			0.99659892748024127.. = (1-1/448)*(1-1/1584)*(1-1/5508)*(1-1/7600)*..

R. J. Mathar, Hardy-Littlewood constants embedded into infinite products over all positive integers, arXiv:0903.2514 [math.NT], table 3 at r=2, k=3. [From _R. J. Mathar_, Mar 28 2009]

The logarithm is -sum_{s>=2} sum_{j=1..floor[s/(1+r)]} binomial(s-r*j-1,j-1)*P_3(s)/j at r=2, where P_k(s) are the k-almost prime zeta functions of arXiv:0803.0900.

A185445 Smallest number having exactly t divisors, where t is the n-th triprime (A014612).

Original entry on oeis.org

24, 60, 180, 240, 900, 960, 720, 2880, 15360, 3600, 6480, 61440, 14400, 46080, 983040, 25920, 32400, 3932160, 184320, 62914560, 233280, 230400, 2949120, 129600, 414720, 11796480, 4026531840, 921600, 16106127360, 810000, 1658880, 188743680, 1166400, 1030792151040, 14745600, 3732480

Offset: 1

Jonathan Vos Post, Feb 03 2011

This is the 3rd row of an infinite array A[k,n] = smallest number having exactly j divisors where j is the n-th natural number with exactly k prime factors (with multiplicity).

The first row is A061286, the second row is A096932.

			a(10) is 3600 because the 10th triprime is 45, and the smallest number with exactly 45 factors is 3600 = 2^4 * 3^2 * 5^2.
a(20) is 62914560 because the 10th triprime is 92, and the smallest number with exactly 92 factors is 62914560 = 2^22 * 3 * 5.

Cf. A005179, A014612, A061286, A096932.

from math import isqrt, prod
from sympy import isprime, primepi, primerange, integer_nthroot, prime, divisors
def A185445(n):
    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(A014612(n)).

A211410 Chen triprimes, triprimes (A014612) m such that m+2 is either prime or semiprime.

Original entry on oeis.org

8, 12, 20, 27, 44, 45, 63, 75, 92, 99, 105, 116, 117, 125, 147, 153, 164, 165, 171, 175, 195, 207, 212, 231, 245, 255, 261, 275, 279, 285, 325, 332, 333, 345, 356, 357, 363, 369, 387, 399, 425, 429, 435, 452, 455, 465, 477, 483, 507, 524

Offset: 1

Jonathan Vos Post, Feb 09 2013

			27=3^3 and 45=3^2*9 are in the sequence because 27+2 = 29 and 45+2 = 47 are primes.
8=2^3, 12=2^2*3, and 20=2^2*5 are in the sequence because 8+2=10=2*5, 12+2=14=2*7, and 20+2=22=2*11 are semiprimes (A001358).

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

Cf. A001358, A014612, A109611, A175634.

A211410 := proc(n)
    option remember;
    local a;
    if n = 1 then
        8;
    else
        for a from procname(n-1)+1 do
            if numtheory[bigomega](a) = 3 then
                if isprime(a+2) or numtheory[bigomega](a+2) = 2 then
                    return a;
                end if;
            end if;
        end do:
    end if;
end proc:
seq(A211410(n),n=1..80) ; # R. J. Mathar, Feb 10 2013

Select[Range[600],PrimeOmega[#]==3&&PrimeOmega[#+2]<3&] (* Harvey P. Dale, Jul 15 2019 *)

issemi(n)=bigomega(n)==2
list(lim)=my(v=List(),pq); forprime(p=2,lim\4, forprime(q=2,min(lim\2\p,p), pq=p*q; forprime(r=2,min(lim\pq,q), if(isprime(pq*r+2) || issemi(pq*r+2), listput(v,pq*r))))); Set(v) \\ Charles R Greathouse IV, Aug 23 2017

A211872 For each triprime (A014612) less than or equal to n, sum the positive integers less than or equal to the number of divisors of the triprime.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 10, 10, 10, 10, 31, 31, 31, 31, 31, 31, 52, 52, 73, 73, 73, 73, 73, 73, 73, 83, 104, 104, 140, 140, 140, 140, 140, 140, 140, 140, 140, 140, 140, 140, 176, 176, 197, 218, 218, 218, 218, 218, 239, 239, 260, 260, 260, 260, 260, 260, 260, 260

Offset: 1

Wesley Ivan Hurt, Feb 12 2013

nonn

The largest difference between any pair of consecutive numbers in the sequence = 36, The second largest difference = 21, the third largest = 10, and the fourth (and last) possible difference is 0.

			a(1) = a(2) = a(3) = a(4) = a(5) = a(6) = a(7) = 0. a(8) = 10 since 8 has 4 divisors, and the sum of all the numbers up to 4 is 1 + 2 + 3 + 4 = 10.  The next triprime is 12, so a(8) = a(9) = a(10) = a(11) = 10. Since there are two triprimes less than or equal to 12, we sum the numbers from 1 to d(8) and 1 to d(12), then take the sum total. Thus, a(12) = 10 + 21 = 31.

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A000005, A001221, A001222, A014612, A209323.

sm = 0; Table[If[Total[Transpose[FactorInteger[n]][[2]]] == 3, d = DivisorSigma[0, n]; sm = sm + d (d + 1)/2]; sm, {n, 100}] (* T. D. Noe, Feb 14 2013 *)
Table[Sum[KroneckerDelta[PrimeOmega[i], 3]*Sum[j, {j, DivisorSigma[0, i]}], {i, n}], {n, 50}] (* Wesley Ivan Hurt, Oct 07 2014 *)

a(n) = Sum_{i=1..n} [Omega(i) = 3] * Sum_{j = 1..d(i)} j.
a(n) = Sum_{i=1..n} [Omega(i) = 3] * (omega(i) + 1) * (d(i) + 1).
a(n) = Sum_{i=1..n} [Omega(i) = 3] * (2*omega(i)^2 + 5*omega(i) + 3), where [ ] is the Iverson bracket.

A364442 a(n) is the smallest number > a(n-1) such that a(n-1) + a(n) is a triprime (A014612), with a(1) = 1.

Original entry on oeis.org

1, 7, 11, 16, 26, 37, 38, 40, 52, 53, 57, 59, 65, 73, 74, 79, 85, 86, 88, 94, 96, 99, 108, 114, 116, 120, 122, 123, 132, 134, 139, 140, 142, 143, 147, 163, 169, 174, 180, 183, 186, 188, 197, 202, 204, 206, 212, 213, 215, 219, 223, 229, 236, 238, 239, 244, 250, 256, 262, 268, 271, 277, 278, 283

Offset: 1

Zak Seidov and Robert Israel, Jul 25 2023

nonn

For n > 1, a(n) is the least number > a(n-1) such that A001222(a(n) + a(n-1)) = 3.

a(n-1) + a(n) is the least triprime > 2*a(n-1).

			a(3) = 11 because a(2) = 7, none of 7 + 8 = 15, 7 + 9 = 16 and 7 + 10 = 17 is a triprime, but 7 + 11 = 18 = 2*3^2 is a triprime.

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

Cf. A001222, A014612, A073627, A263349, A357713.

R:= 1: x:= 1:
for i from 1 to 100 do
   for y from x+1 while numtheory:-bigomega(x+y) <> 3 do od:
   R:= R,y;
   x:= y
od:
R;

s = {p = 1}; Do[q = p + 1; While[3 != PrimeOmega[p + q],
q++];  AppendTo[s, p = q], {100}]; s

A365145 Lexicographically least increasing sequence of triprimes (A014612) whose first differences are triprimes.

Original entry on oeis.org

8, 20, 28, 70, 78, 98, 110, 130, 138, 165, 195, 207, 273, 285, 363, 426, 434, 442, 470, 498, 506, 518, 530, 548, 556, 574, 582, 590, 598, 606, 618, 638, 646, 654, 682, 710, 722, 730, 742, 754, 762, 782, 790, 834, 854, 874, 892, 942, 962, 970, 978, 986, 994, 1002, 1010, 1022, 1030, 1038, 1058

Offset: 1

Zak Seidov and Robert Israel, Aug 23 2023

nonn

a(n) - a(n-1) >= 8. If a(n-1) = 4*p where p is in A001359 then a(n) - a(n-1) = 8.

			a(2) = 20, a(3) = 28 = 2^2 * 7 is a triprime and 28 - 20 = 8 = 2^3 is a triprime, and this is the least number > 20 that works, so a(4) = 28.

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

Cf. A001359, A014612, A175588.

R:= 8: m:= 8: count:= 1:
for t from 9 while count < 100 do
  if numtheory:-bigomega(t) = 3 and numtheory:-bigomega(t-m) = 3 then
    R:= R,t; m:= t; count:= count+1
  fi
od:
R;

Do[n = m + 8; While[{3, 3} != PrimeOmega[{n, n - m}],
n++]; AppendTo[s, m = n], {100}]; s

A367841 Numbers k such that k, k + 2, k + 4, k + 6, k + 8, k + 10, and k + 12 are all triprimes (A014612).

Original entry on oeis.org

151401, 151403, 151405, 151407, 179535, 201085, 247349, 248411, 250933, 250935, 292407, 298433, 322215, 379761, 441327, 482691, 482693, 499907, 508671, 517427, 584219, 584221, 586257, 586259, 605207, 705055, 705057, 705059, 718193, 726563, 727639, 728815, 812601, 814247, 814249, 814251, 831385

Offset: 1

Zak Seidov and Robert Israel, Dec 31 2023

nonn

All terms are odd, because if k is even, at least one of k, k + 2, k + 4 and k + 6 is divisible by 8.
In the case of a(1) = 151401, k + 14, k + 16 and k + 18 are also triprimes.
In the case of a(143) = 2560187, k + 14, k + 16, k + 18 and k + 20 are also triprimes.

			a(5) = 179535 is a term because
179535 = 3 * 5 * 11969
179535 + 2 = 179537 = 17 * 59 * 179
179535 + 4 = 179539 = 29 * 41 * 151
179535 + 6 = 179541 = 3 * 3 * 19949
179535 + 8 = 179543 = 7 * 13 * 1973
179535 + 10 = 179545 = 5 * 149 * 241
179535 + 12 = 179547 = 3 * 97 * 617
are all triprimes.

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

Cf. A014612.

filter:= (t -> andmap(x -> numtheory:-bigomega(x)=3, [t,t+2,t+4,t+6,t+8,
t+10,t+12])):
select(filter, [seq(i,i=1 .. 10^6, 2)]);

A114430 Primes of the form 1 + product of the first n 3-almost primes A014612.

Original entry on oeis.org

97, 32920473601, 1448500838401, 65182537728001, 1491301685600774317670400000001, 48235157779343672198731287466250036763794299837586774072944798728192000000000000000001

Offset: 1

Jonathan Vos Post, Feb 13 2006

3-almost prime analog of primorial primes A005234 (primes p such that 1 + product of primes up to p is prime) as indexed by A014545 (n such that n-th Euclid number (A006862(n)) = 1 + (Product of first n primes) is prime). In that sense, this sequence is indexed by (2, 8, 9, 10, 19, ...).

			a(1) = 97 = 96 + 1 = 1 + (8 * 12) = 1 + A014612(1)*A014612(2) = 1 more than the product of the first 2 of the 3-almost primes and is prime.
a(2) = 32920473601 = 1 + (8 * 12 * 18 * 20 * 27 * 28 * 30 * 42) = 1 more than the product of the first 8 of the 3-almost primes and is prime.
a(3) = 1 more than the product of the first 9 of the 3-almost primes and is prime.
a(4) = 1 more than the product of the first 10 of the 3-almost primes and is prime.
a(5) = 1 more than the product of the first 19 of the 3-almost primes and is prime.

Cf. A000040, A001358, A014612, A002110, A005234, A014545, A112141.

Select[Rest[FoldList[Times,1,Select[Range[250],PrimeOmega[#]==3&]]]+1,PrimeQ] (* Harvey P. Dale, Dec 21 2013 *)

{a(n)} = {1 + Prod[from i = 1 to n] A014612(i)} INTERSECTION {A000040}.

One more term (a(6)) from Harvey P. Dale, Dec 21 2013

A121061 Numbers k such that k-th partition number A000041(k) is a 3-almost prime (A014612).

Original entry on oeis.org

9, 10, 16, 18, 20, 22, 25, 29, 31, 34, 37, 42, 45, 48, 50, 56, 57, 60, 63, 69, 71, 72, 73, 83, 85, 91, 101, 102, 112, 113, 114, 119, 139, 144, 148, 151, 155, 156, 172, 175, 183, 185, 190, 192, 195, 202, 208, 217, 238, 242, 244, 245, 247, 256, 257, 272, 275, 291, 293

Offset: 1

Jonathan Vos Post, Aug 09 2006

			a(1) = 9 because P(9) = 30 = 2 * 3 * 5.
a(2) = 10 because P(10) = 42 = 2 * 3 * 7.
a(3) = 16 because P(16) = 231 = 3 * 7 * 11.

Cf. A000041, A046063, A065728, A065729.

Select[Range[300],PrimeOmega[PartitionsP[#]]==3&] (* James C. McMahon, Oct 12 2024 *)

A000041(a(n)) in A014612.

112 and 113 inserted by R. J. Mathar, Dec 22 2010

cp's OEIS Frontend

A146483 Decimal expansion of Product_{q in A014612} (1-1/(q*(q-1))).

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A146487 Decimal expansion of Product_{q in A014612} (1-1/(q^2*(q-1))).

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A185445 Smallest number having exactly t divisors, where t is the n-th triprime (A014612).

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A211410 Chen triprimes, triprimes (A014612) m such that m+2 is either prime or semiprime.

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A211872 For each triprime (A014612) less than or equal to n, sum the positive integers less than or equal to the number of divisors of the triprime.

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A364442 a(n) is the smallest number > a(n-1) such that a(n-1) + a(n) is a triprime (A014612), with a(1) = 1.

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A365145 Lexicographically least increasing sequence of triprimes (A014612) whose first differences are triprimes.

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A367841 Numbers k such that k, k + 2, k + 4, k + 6, k + 8, k + 10, and k + 12 are all triprimes (A014612).

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