A106325 -id:A106325 - cp's OEIS frontend

A106326 Smallest odd semiprime not less than n.

9, 9, 9, 9, 9, 9, 9, 9, 9, 15, 15, 15, 15, 15, 15, 21, 21, 21, 21, 21, 21, 25, 25, 25, 25, 33, 33, 33, 33, 33, 33, 33, 33, 35, 35, 39, 39, 39, 39, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 51, 51, 55, 55, 55, 55, 57, 57, 65, 65, 65, 65, 65, 65, 65, 65, 69, 69, 69, 69, 77, 77, 77

Offset: 1

Reinhard Zumkeller, Apr 29 2005

nonn

a(A046315(n)) = A046315(n).

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A106325.

sos[n_]:=Module[{s=If[OddQ[n],n,n+1]},While[PrimeOmega[s]!=2,s=s+2];s]; Table[sos[n],{n,80}] (* Harvey P. Dale, Jun 14 2015 *)

A173717 Sum of n mod m, summed over semiprimes m = 4, 6, 9, ..., smallest semiprime >= n.

Original entry on oeis.org

1, 2, 3, 0, 6, 2, 11, 10, 4, 7, 22, 17, 22, 13, 18, 36, 43, 35, 42, 35, 21, 28, 59, 58, 42, 51, 79, 72, 83, 63, 74, 81, 59, 70, 82, 112, 126, 102, 116, 157, 173, 148, 164, 154, 146, 116, 179, 186, 154, 186, 153, 193, 212, 216, 180, 237, 200, 220, 300, 287, 309, 269, 324, 343, 301, 329, 353, 339

Offset: 1

Juri-Stepan Gerasimov, Nov 25 2010

Cf. A106325.

A173717 := proc(n) a := 0 ; for i from 1 do s := A001358(i) ; a := a + (n mod s) ;  if s >= n then return a; end if; end do: end proc: # R. J. Mathar, Nov 26 2010

Corrected by R. J. Mathar, Nov 26 2010

A179464 a(n) = min(nextprime(n),nextsemiprime(n)).

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 9, 9, 10, 11, 13, 13, 14, 15, 17, 17, 19, 19, 21, 21, 22, 23, 25, 25, 26, 29, 29, 29, 31, 31, 33, 33, 34, 35, 37, 37, 38, 39, 41, 41, 43, 43, 46, 46, 46, 47, 49, 49, 51, 51, 53, 53, 55, 55, 57, 57, 58, 59, 61, 61, 62, 65, 65, 65, 67, 67, 69, 69, 71, 71, 73, 73, 74, 77, 77, 77, 79, 79, 82, 82, 82, 83, 85, 85, 86, 87, 89, 89, 91, 91, 93, 93, 94

Offset: 1

Zak Seidov, Jan 08 2011

nonn

			n=1: nextprime(1)=2, nextsemiprime(1)=4, hence a(1)=2,
n=2: nextprime(2)=3, nextsemiprime(2)=4, hence a(2)=3,
n=3: nextprime(3)=5, nextsemiprime(3)=4, hence a(3)=4.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000040 The prime numbers, A001358 Semiprimes.

Cf. A106325, A151800.

PS:= select(t -> numtheory:-bigomega(t)<=2, [$2..500]):
Res:= NULL:
k:= 1;
for n from 2 to max(PS) do
  if n > PS[k] then k:= k+1 fi;
  Res:= Res, PS[k];
od:
Res; # Robert Israel, Oct 25 2017

Table[m=n+1;While[2!= Plus@@Last/@FactorInteger[m],m++];Min[NextPrime[n],m],{n,200}]
mnp[n_]:=Module[{s=n+1},While[PrimeOmega[s]!=2,s++];Min[NextPrime[n],s]]; Array[mnp,100] (* Harvey P. Dale, Apr 23 2019 *)

{for(n=1,200,m=n+1;while(2<>bigomega(m),m++);print(min(nextprime(n+1),m)))}

a(n) = min(A106325(n+1), A151800(n)). - Robert Israel, Oct 25 2017

A200927 Difference between (least semiprime >= n) and (largest semiprime <= n).

Original entry on oeis.org

0, 2, 0, 3, 3, 0, 0, 4, 4, 4, 0, 0, 6, 6, 6, 6, 6, 0, 0, 3, 3, 0, 0, 7, 7, 7, 7, 7, 7, 0, 0, 0, 3, 3, 0, 0, 7, 7, 7, 7, 7, 7, 0, 3, 3, 0, 2, 0, 4, 4, 4, 0, 2, 0, 0, 4, 4, 4, 0, 3, 3, 0, 4, 4, 4, 0, 5, 5, 5, 5, 0, 3, 3, 0, 5, 5, 5, 5, 0, 3, 3, 0, 0, 0, 4, 4, 4

Offset: 4

Arkadiusz Wesolowski, Nov 24 2011

nonn

a(n) = 0 if and only if n is semiprime.

Arkadiusz Wesolowski, Table of n, a(n) for n = 4..10000

Cf. A001358, A065516.

A106325 := proc(n)
    for a from n do
        if numtheory[bigomega](a) = 2 then
            return a;
        end if;
    end do:
end proc;
prevSpr := proc(n)
    for a from n by -1 do
        if numtheory[bigomega](a) = 2 then
            return a;
        end if;
    end do:
end proc;
A200927 := proc(n)
    A106325(n)-prevSpr(n) ;
end proc:
seq(A200927(n),n=4..80) ; # R. J. Mathar, Nov 26 2011

Table[a = b = 0; While[! PrimeOmega[n - a] == 2, a++]; While[! PrimeOmega[n + b] == 2, b++]; a + b, {n, 4, 100}]

A379407 a(n) is the smallest semiprime > primorial(n).

Original entry on oeis.org

4, 9, 33, 213, 2315, 30031, 510515, 9699691, 223092871, 6469693233, 200560490134, 7420738134814, 304250263527221, 13082761331670031, 614889782588491414, 32589158477190044737, 1922760350154212639074, 117288381359406970983271, 7858321551080267055879091

Offset: 1

Alexandre Herrera, Dec 22 2024

nonn

			primorial(2) = 2*3 = 6 so a(2) = 9 because 9 = 3*3 is next semiprime > 6.

Cf. A001358, A002110, A089539, A106325.

a[n_] := Module[{m = Times @@ Prime[Range[n]] + 1}, While[PrimeOmega[m] != 2, m++]; m]; Array[a, 20] (* Amiram Eldar, Jan 01 2025 *)

import sympy
def ok(n): return sum(sympy.factorint(n).values()) == 2
primorial = 1
l = []
for i in range(1,20):
    primorial *= sympy.prime(i)
    next_sp = primorial + 1
    while not(ok(next_sp)):
        next_sp += 1
    l.append(next_sp)
print(l)

a(n) = A106325(A002110(n)+1).

cp's OEIS Frontend

A106326 Smallest odd semiprime not less than n.

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A173717 Sum of n mod m, summed over semiprimes m = 4, 6, 9, ..., smallest semiprime >= n.

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A179464 a(n) = min(nextprime(n),nextsemiprime(n)).

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A200927 Difference between (least semiprime >= n) and (largest semiprime <= n).

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A379407 a(n) is the smallest semiprime > primorial(n).

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