A046312 -id:A046312 - cp's OEIS frontend

A369898 Numbers k such that k and k + 1 each have 9 prime divisors, counted with multiplicity.

203391, 698624, 1245375, 1942784, 2176064, 2282175, 2536191, 2858624, 2953664, 3282687, 3560192, 3655935, 3914000, 4068224, 4135616, 4205600, 4244967, 4586624, 4695488, 4744575, 4991679, 5055615, 5450624, 5475519, 5519744, 6141824, 6246800, 6410096, 6655040, 6660224, 6753375, 6816879, 6862400

Offset: 1

Robert Israel, Feb 04 2024

nonn

Numbers k such that k and k + 1 are in A046312.

If a and b are coprime terms of A046310, one of them even, then Dickson's conjecture implies there are infinitely many terms k where k/a and (k+1)/b are primes.

			a(3) = 1245375 is a term because 1245375 = 3^5 * 5^3 * 41 and 1245376 = 2^6 * 11 * 29 * 61 each have 9 prime factors, counted with multiplicity.

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

Cf. A001222, A046310, A046312, A115186, A369897.

with(priqueue):
R:= NULL:  count:= 0:
initialize(Q); r:= 0:
insert([-2^9, [2$9]], Q);
while count < 40 do
  T:= extract(Q);
  if -T[1] = r + 1 then
    R:= R, r; count:= count+1;
  fi;
  r:= -T[1];
  p:= T[2][-1];
  q:= nextprime(p);
  for i from 9 to 1 by -1 while T[2][i] = p do
    insert([-r*(q/p)^(10-i), [op(T[2][1..i-1]), q$(10-i)]], Q);
  od
od:
R;

A028261 Numbers whose total number of prime factors (counting multiplicity) is squarefree.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74

Offset: 1

Daniel Asimov (dan(AT)research.att.com)

nonn

The complement is 1, 16, 24, 36, 40, 54, 56, 60, 81, 84, 88, 90, 100.. and contains all entries of A014613, all of A046310, all of A046312 etc. - R. J. Mathar, Dec 15 2015

Cf. A001222, A005117.

isok(n) = issquarefree(bigomega(n)); \\ Michel Marcus, Aug 31 2013

A123118 Partial products of A101695.

Original entry on oeis.org

2, 12, 216, 8640, 933120, 209018880, 100329062400, 130026464870400, 349511137571635200, 1968446726803449446400, 22676506292775737622528000, 522466704985552994823045120000, 27820307107070725868337506549760000

Offset: 1

Jonathan Vos Post, Sep 28 2006

nonn

The number of prime factors (with multiplicity) of a(n) is T(n) = A000217(n) = n*(n+1)/2.

			a(1) = 2 = prime(1).
a(2) = 12 = 2 * 6 = prime(1) * semiprime(2) = 2^2 * 3.
a(3) = 216 = 2 * 6 * 18 = prime(1) * semiprime(2) * 3-almostprime(3) = 2^3 * 3^3.
a(4) = 8640 = 2 * 6 * 18 * 40 = prime(1) * semiprime(2) * 3-almostprime(3) * 4-almostprime(4) = 2^6 * 3^3 * 5.
a(15) = 893179304874387947794472921245209518407680000 = 2 * 6 * 18 * 40 * 108 * 224 * 480 * 1296 * 2688 * 5632 * 11520 * 23040 * 53248 * 124416 * 258048 = 2^88 * 3^23 * 5^4 * 7^3 * 11 * 13.

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(n) = Prod[i=1..n] i-th i-almost prime = Prod[i=1..n] A101695(i).

A321590 Smallest number m that is a product of exactly n primes and is such that m-1 and m+1 are products of exactly n-1 primes.

Original entry on oeis.org

4, 50, 189, 1863, 10449, 447849, 4449249, 5745249, 3606422049, 16554218751, 105265530369, 1957645712385

Offset: 2

Zak Seidov, Nov 13 2018

From Jon E. Schoenfield, Nov 15 2018: (Start)
If a(11) is odd, it is 16554218751.
If a(12) is odd, it is 105265530369.
If a(13) is odd, it is 1957645712385. (End)
a(11), a(12), and a(13) are indeed odd. - Giovanni Resta, Jan 04 2019
10^13 < a(14) <= 240455334218751, a(15) <= 2992278212890624. - Giovanni Resta, Jan 06 2019

			For n = 3, 50 = 2*5*5, and the numbers before and after 50 are 49 = 7*7 and 51 = 3*17.

Cf. A078840.

Sequences listing r-almost primes, that is, the n such that A001222(n) = 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).

a[n_] := Module[{o={0,0,0}, k=1}, While[o!={n-1,n,n-1}, o=Rest[AppendTo[o,PrimeOmega[k]]]; k++]; k-2]; Array[a,7,2] (* Amiram Eldar, Nov 14 2018 *)

{for(n=2,10,for(k=2^n,10^12,if(n==bigomega(k) &&
n-1==bigomega(k-1) && n-1==bigomega(k+1),print1(k", ");break())))}

a(10) from Jon E. Schoenfield, Nov 14 2018

a(11)-a(13) from Giovanni Resta, Jan 04 2019

A374231 a(n) is the minimum number of distinct numbers with exactly n prime factors (counted with multiplicity) whose sum of reciprocals exceeds 1.

Original entry on oeis.org

3, 13, 96, 1772, 108336, 35181993

Offset: 1

Amiram Eldar, Jul 01 2024

			a(1) = 3 since Sum_{k=1..2} 1/prime(k) = 1/2 + 1/3 = 5/6 < 1 and Sum_{k=1..3} 1/prime(k) = 1/2 + 1/3 + 1/5 = 31/30 > 1.
a(2) = 13 since Sum_{k=1..12} 1/A001358(k) = 1/4 + 1/6 + 1/9 + 1/10 + 1/14 + 1/15 + 1/21 + 1/22 + 1/25 + 1/26 + 1/33 + 1/34 = 15271237/15315300 < 1 and Sum_{k=1..13} 1/A001358(k) = 1/4 + 1/6 + ... + 1/35 = 15708817/15315300 > 1.

Cf. A078840.

Cf. A000040, A001358, A014612, A014613, A014614, A046306, A046308, A046310, A046312, A046314.

next[p_, n_] := Module[{k = p + 1}, While[PrimeOmega[k] != n, k++]; k]; a[n_] := Module[{k = 0, sum = 0, p = 0}, While[sum <= 1, p = next[p, n]; sum += 1/p; k++]; k]; Array[a, 5]

nextnum(p, n) = {my(k = p + 1); while(bigomega(k) != n, k++); k;}
a(n) = {my(k = 0, sum = 0, p = 0); while(sum <= 1, p = nextnum(p, n); sum += 1/p; k++); k;}

cp's OEIS Frontend

A369898 Numbers k such that k and k + 1 each have 9 prime divisors, counted with multiplicity.

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A028261 Numbers whose total number of prime factors (counting multiplicity) is squarefree.

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A123118 Partial products of A101695.

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A321590 Smallest number m that is a product of exactly n primes and is such that m-1 and m+1 are products of exactly n-1 primes.

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A374231 a(n) is the minimum number of distinct numbers with exactly n prime factors (counted with multiplicity) whose sum of reciprocals exceeds 1.

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