A155800
a(n) = smallest prime p such that p-1 and p+1 together have n prime divisors, or a(n) = 0 if no such prime exists.
Original entry on oeis.org
2, 0, 3, 5, 7, 19, 17, 31, 97, 191, 127, 449, 769, 3329, 6143, 7937, 12799, 51199, 8191, 165887, 65537, 131071, 1179649, 2654209, 7995391, 524287, 10616831, 12910591, 167772161, 113246209, 169869311, 155189249, 1887436799, 3221225473
Offset: 1
For p=2, the adjacent numbers 1 and 3 together have one prime divisor, hence a(1) = 2. For p=3, the adjacent numbers 2 and 4 together have three prime divisors, hence a(3) = 3. For primes greater than 3, the adjacent numbers are composite and therefore together have at least four prime divisors, so no prime exists whose neighbors together have two prime divisors. Hence a(2) = 0.
For p = 19, p-1 = 18 = 2*3*3 and p+1 = 20 = 2*2*5 together have six prime divisors. All smaller primes are surrounded by numbers which together have fewer or more than six (for 17 there are seven) prime divisors, hence a(6) = 19.
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Join[{2,0},With[{m=SortBy[{PrimeOmega[#-1]+PrimeOmega[#+1],#}&/@Prime[ Range[200000]],First]},Transpose[Flatten[Table[Select[m,#[[1]]==n&,1],{n,3,24}],1]][[2]]]] (* Harvey P. Dale, Sep 24 2013 *)
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N=10^7; default(primelimit, N); M=26;
/* M must be determined empirically for each N. Program-generated zeros other than a(2) appearing as terms when N and M are increased must be regarded as provisional */
for(m=1,M,t=0;forprime(p=1,N,if(bigomega(p-1)+bigomega(p+1)==m,t=1;print1(p", ");break));if(t==0,print1("0, "))) \\ Chris Boyd, Mar 24 2014
Edited, 1151 replaced by 769, 3457 replaced by 3329, extended beyond a(14) by
Klaus Brockhaus, Jan 28 2009
A371651
a(n) is the first prime p such that p - 2 and p + 2 both have exactly n prime factors, counted with multiplicity.
Original entry on oeis.org
5, 23, 173, 2693, 32587, 495637, 4447627, 35303123, 717591877, 928090627, 69692326373, 745041171877, 5012236328123, 64215009765623, 945336806640623, 8885812685546873
Offset: 1
a(3) = 173 because 173 is prime, 173 - 2 = 171 = 3^2 * 19 and 173 + 2 = 175 = 5^2 * 7 are both products of 3 primes with multiplicity, and no smaller number works.
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V:= Vector(8):
p:= 3: count:= 0:
while count < 8 do
p:= nextprime(p);
i:= numtheory:-bigomega(p-2);
if i <= 8 and V[i] = 0 and numtheory:-bigomega(p+2) = i
then V[i]:= p; count:= count+1
fi
od:
convert(V,list);
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generate(A, B, n) = A=max(A, 2^n); (f(m, p, j) = my(list=List()); if(j==1, forprime(q=max(p,ceil(A/m)), B\m, my(t=m*q); if(isprime(t-2) && bigomega(t-4) == n, listput(list, t-2))), forprime(q = p, sqrtnint(B\m, j), list=concat(list, f(m*q, q, j-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)); if(#v >= 1, return(v[1])); x=y+1; y=2*x); \\ Daniel Suteu, Apr 13 2024
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from sympy import primeomega, nextprime
def A371651(n):
p = 3
while True:
if n == primeomega(p-2) == primeomega(p+2):
return p
p = nextprime(p) # Chai Wah Wu, Apr 02 2024
A155850
a(n) = smallest k > 1 such that k-1 and k+1 together have n prime divisors.
Original entry on oeis.org
2, 4, 3, 5, 7, 15, 17, 31, 65, 129, 127, 449, 511, 2561, 1025, 7937, 12799, 20481, 8191, 28673, 65537, 131071, 458751, 360449, 966655, 524287, 4194303, 2097151, 29360129, 34865153, 67108865, 134217729, 33554431, 608174081, 268435457, 536870911, 4831838207
Offset: 1
Adjacent to 2 are the numbers 1 and 3 which together have one prime divisor, hence a(1) = 2. Adjacent to 3 are 2 and 4; together they have three prime divisors, hence a(3) = 3. Adjacent to 4 are the primes 3 and 5, each having one prime divisor; hence a(2) = 4.
For k = 129, k-1 = 128 = 2*2*2*2*2*2*2 and k+1 = 130 = 2*5*13 together have ten prime divisors. For all numbers k < 129 the adjacent numbers k-1 and k+1 together have fewer or more than ten (for 127 there are eleven) prime divisors, hence a(10) = 129.
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
A384876
Smallest number m such that both m-1 and m+1 are products of at least n (not necessarily distinct) primes.
Original entry on oeis.org
3, 5, 17, 55, 161, 1457, 2431, 13121, 101249, 153089, 2086399, 7991297, 65071999, 72630271, 2829746177, 2975006719, 68278476799, 75389157377, 159703334911, 1570258288639, 9714181341185, 91845775327231, 551785225781249, 2123044908695551, 4560483868737535, 4560483868737535, 424428773098651649
Offset: 1
The smallest number surrounded by semiprime numbers is 5 (between 4 and 6).
And 17 lies between 16 = 2^4 and 18 = 2*3^2.
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F:= proc(n) local pq,t,x,y,z,p,i,m;
uses priqueue;
initialize(pq);
insert([-2^n, 2$n], pq);
y:= -infinity; z:= -infinity;
do
t:= extract(pq);
x:= -t[1];
if x-y=2 or x-z=2 then return x-1 fi;
z:= y; y:= x; m:= nops(t);
if t[-1] = 2 then insert([2*t[1],2$m],pq) fi;
p:= nextprime(t[-1]);
for i from m to 2 by -1 while t[i] = t[-1] do
insert([t[1]*(p/t[-1])^(m+1-i), op(t[2..i-1]), p$(m+1-i)], pq)
od;
od
end proc:
seq(F(i),i=1..20); # Robert Israel, Jun 12 2025
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a(n) = my(m=2); while((bigomega(m-1)Michel Marcus, Jun 13 2025
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