A073919
Smallest prime p with bigomega(p-1)=n, where bigomega(m)=A001222(m) is the number of prime divisors of m (counted with multiplicity).
Original entry on oeis.org
2, 3, 5, 13, 17, 73, 97, 193, 257, 769, 3457, 7681, 15361, 12289, 40961, 114689, 65537, 737281, 1376257, 786433, 5308417, 7340033, 14155777, 28311553, 104857601, 113246209, 167772161, 469762049, 2113929217, 1811939329
Offset: 0
a(2) = 5 = 2*2 + 1. a(5) = 73 = 2*2*2*3*3 + 1.
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ptns[n_, 0] := If[n==0, {{}}, {}]; ptns[n_, k_] := Module[{r}, If[nv, Return[v]]; minp=Min@@Select[l+1, ProvablePrimeQ]; If[minpHarvey P. Dale, Sep 28 2014 *)
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almost_primes(A, B, n) = A=max(A, 2^n); (f(m, p, n) = my(list=List()); if(n==1, forprime(q=max(p, ceil(A/m)), B\m, listput(list, m*q)), forprime(q=p, sqrtnint(B\m, n), list=concat(list, f(m*q, q, n-1)))); list); vecsort(Vec(f(1, 2, n)));
a(n) = if(n==0, return(2)); my(x=2^n, y=2*x); while(1, my(v=almost_primes(x, y, n)); for(k=1, #v, if(isprime(v[k]+1), return(v[k]+1))); x=y+1; y=2*x); \\ Daniel Suteu, Jan 07 2025
A283652
Primes p such that bigomega(p+1) = 20.
Original entry on oeis.org
5505023, 8847359, 13271039, 17915903, 22118399, 24379391, 27131903, 29859839, 31981567, 32440319, 34406399, 36863999, 37486591, 38273023, 42205183, 46448639, 48496639, 54001663, 57016319, 60948479, 61439999, 62128127, 62705663, 67184639
Offset: 1
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p = 4128767; While[p<=57016319, If[PrimeOmega[1 + p] == 20, Print[p,", "]]; p = NextPrime[p + 2]] (* Indranil Ghosh, Mar 13 2017, after the PARI program from the author *)
Select[Prime[Range[4*10^6]],PrimeOmega[#+1]==20&] (* Harvey P. Dale, Jul 05 2020 *)
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{p=4128767; while(p<=57016319, if(bigomega(1+p)==20, print1(p ",")); p=nextprime(p+2))}
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.
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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 *)
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a(n) = forprime(p=2, , if (bigomega(p+2) == n, return (p))); \\ Michel Marcus, Jan 10 2019
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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
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