A288517
Least integer k such that A001358(k) + A001358(k+1) is the product of exactly n prime factors (counting multiplicity).
Original entry on oeis.org
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
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).
A321169
a(n) is the smallest prime p such that p + 2 is a product of n primes (counted with multiplicity).
Original entry on oeis.org
3, 2, 43, 79, 241, 727, 3643, 15307, 19681, 164023, 1673053, 885733, 2657203, 18600433, 23914843, 100442347, 358722673, 645700813, 4519905703, 18983603959, 48427561123, 31381059607, 261508830073, 1307544150373, 3295011258943, 24006510600883, 12709329141643, 53379182394907, 190639937124673, 2615579937350539
Offset: 1
a(1) = 3 as 3 + 2 = 5 (prime),
a(2) = 2 as 2 + 2 = 4 = 2*2 (semiprime),
a(3) = 43 as 43 + 2 = 45 = 3*3*5 (3-almost prime),
a(4) = 79 as 79 + 2 = 81 = 3*3*3*3 (4-almost prime).
Cf.
A001222,
A001358,
A014612,
A014613,
A014614,
A046306,
A046308,
A046310,
A046312,
A046314,
A073919,
A118883.
-
ptns[n_, 0] := If[n == 0, {{}}, {}]; ptns[n_, k_] := Module[{r}, If[n < k, Return[{}]]; ptns[n, k] = 1 + Union @@ Table[PadRight[#, k] & /@ ptns[n - k, r], {r, 0, k}]]; a[n_] := Module[{i, l, v}, v = Infinity; For[i = n, True, i++, l = (Times @@ Prime /@ # &) /@ ptns[i, n]; If[Min @@ l > v, Return[v]]; minp = Min @@ Select[l - 2, PrimeQ]; If[minp < v, v = minp]]] ; Array[a, 10] (* after Amarnath Murthy at A073919 *)
-
a(n) = forprime(p=2, , if (bigomega(p+2) == n, return (p))); \\ Michel Marcus, Jan 10 2019
-
a(n) = {my(p3 = 3^n, u, c); if(n <= 2, return(4 - n)); if(isprime(p3 - 2), return(p3 - 2)); forprime(p = 5, oo, if(isprime(p3 / 3 * p - 2), u = p3 / 3 * p - 2; break ) ); for(i = 2, n, if(p3 * (5/3)^i > u, return(u)); for(j = 1, oo, if(p3 * j \ 3^i > u, next(2)); if(bigomega(j) == i, if(isprime(p3 / 3^(i) * j - 2), u = p3 / 3^(i) * j - 2; next(2) ) ) ) ); return(u) } \\ David A. Corneth, Jan 11 2019
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.
A281927
Numbers that are the product of exactly 10 primes and are of the form prime(n) + prime(n + 1).
Original entry on oeis.org
2304, 3456, 5184, 5376, 8448, 9600, 14400, 14976, 18816, 19008, 19440, 21888, 29440, 30208, 31488, 34048, 36096, 36608, 43264, 43904, 46848, 47040, 47232, 55552, 59520, 60000, 60160, 63936, 69696
Offset: 1
2304 = 2^8 * 3^2 = 1151 + 1153, 3456 = 2^7 * 3^3 = 1723 + 1733, 5184 = 2^6 * 3^4 = 2591 + 2593.
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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