A069272 -id:A069272 - cp's OEIS frontend

A288517 Least integer k such that A001358(k) + A001358(k+1) is the product of exactly n prime factors (counting multiplicity).

3, 1, 28, 4, 19, 39, 48, 89, 120, 551, 447, 589, 3707, 10137, 21644, 28456, 22998, 44494, 86132, 166930, 703448, 628371, 1220814, 1608668, 11153853, 6091437, 56676014, 268389220, 146153797, 193010987, 916382785, 738246947, 4702317172, 2830095027, 12627951809

Offset: 1

Zak Seidov, Jun 10 2017

nonn

			n=1: k=3, A001358(3) + A001358(4) = 9 + 10 = 19 = A000040(8) (8th prime),
n=2: k=1, A001358(1)+A001358(2) = 4+6 = 10 = 2*5 = A001358(4) (4th semiprime),
n=11: k=447, A001358(447)+A001358(448) = 1535+1537 = 3072 = 2^10*3 = A069272(2) (2nd 11-almost prime).

Charles R Greathouse IV, Table of n, a(n) for n = 1..50

Cf. A000040, A001358, A014612, A014614, A046306, A046308, A046310, A046312, A046314, A069272.

a(21)-a(35) from Charles R Greathouse IV, Jun 10 2017

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

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A288517 Least integer k such that A001358(k) + A001358(k+1) is the product of exactly n prime factors (counting multiplicity).

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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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