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).
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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 *)
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{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())))}
A335737
a(n) is the smallest prime p such that 2p+3q and 3p+2q are n-almost primes, where q is next prime after p.
Original entry on oeis.org
5, 47, 139, 2521, 77269, 631459, 6758117, 33059357, 7607209367, 173030234371, 152129921851
Offset: 1
Both 2*5+3*7 = 31 and 3*5 +2*7 = 29 are primes.
Both 2*47+3*53 = 253 = 11*23 and 3*47+2*53 = 247 = 13*19 are semiprimes.
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m = 6; s = Table[0, {m}]; p = 2; c = 0; While[c < m, q = NextPrime[p]; If[(o = PrimeOmega[2 p + 3 q]) == PrimeOmega[ 3 p + 2 q] && o <= m && s[[o]] == 0, c++; s[[o]] = p]; p = q]; s (* Amiram Eldar, Jun 23 2020 *)
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for(n=1,8,my(p=2); forprime(q=3, oo, my(pq=2*p+3*q, qp=3*p+2*q); if(bigomega(pq)==n, if(bigomega(qp)==n, print1(p, ", "); break));p=q)) \\ Hugo Pfoertner, Jun 24 2020
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.
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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]
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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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