A088075
Smallest number sandwiched between two numbers having exactly n prime divisors.
Original entry on oeis.org
3, 11, 131, 1429, 77141, 1456729, 117048931, 10326137821, 1110819807371, 140734085123059, 11639258217451019
Offset: 1
n = 3: a(3) = 131, 130 = 2*5*13 and 132 = 2^2*3*11 both have three prime divisors.
n = 6: a(6) = 1456729 = 103*14143 is between 1456728 = 8*3*7*13*23*29 and 1456730 = 2*5*11*17*19*41.
- Jean-Marie De Koninck, "Those Fascinating Numbers", American Mathematical Society, 2009, entry 1428. (This gives the lower of the two numbers.)
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isok(k, n) = (omega(k-1)==n) && (omega(k+1)==n);
a(n) = my(k=2); while (!isok(k, n), k++); k; \\ Michel Marcus, Sep 03 2023
A335667
a(n) is the smallest even number k such that k-1 and k+1 are both n-almost primes.
Original entry on oeis.org
4, 34, 274, 2276, 8126, 184876, 446876, 18671876, 95234374, 1144976876, 6018359374, 281025390626, 2068291015624, 6254345703124, 181171630859374, 337813720703126, 31079046044921876, 205337027587890626
Offset: 1
a(1) = 4 since 4 - 1 and 4 + 1 are both primes.
a(2) = 34 since 34 - 1 = 33 = 3*11 and 34 + 1 = 35 = 5*7 are both semiprimes.
a(3) = 274 since 274 - 1 = 273 = 3*7*13 and 274 + 1 = 275 = 5^2 * 11 are both 3-almost primes.
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m = 8; v = Table[0, {m}]; c = 0; o1 = 1; n = 4; While[c < m, o2 = PrimeOmega[n + 1]; If[o1 == o2 && v[[o1]] == 0, c++; v[[o1]] = n]; o1 = o2; n += 2]; v
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generate(A, B, n, k) = A=max(A, 2^n); (f(m, p, n) = my(list=List()); if(n==1, forprime(q=max(p, ceil(A/m)), B\m, if(bigomega(m*q+2) == k, listput(list, m*q+1))), forprime(q=p, sqrtnint(B\m, n), list=concat(list, f(m*q, q, n-1)))); list); vecsort(Vec(f(1, 3, n)));
a(n) = my(x=2^n, y=2*x); while(1, my(v=generate(x, y, n, n)); if(#v >= 1, return(v[1])); x=y+1; y=2*x); \\ Daniel Suteu, Jul 10 2023
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