Original entry on oeis.org
2, 12, 216, 8640, 933120, 209018880, 100329062400, 130026464870400, 349511137571635200, 1968446726803449446400, 22676506292775737622528000, 522466704985552994823045120000, 27820307107070725868337506549760000
Offset: 1
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.
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
For n = 3, 50 = 2*5*5, and the numbers before and after 50 are 49 = 7*7 and 51 = 3*17.
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())))}
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
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.
-
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;}
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