A064910 -id:A064910 - cp's OEIS frontend

A064911 If n is semiprime (or 2-almost prime) then 1 else 0.

0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Patrick De Geest, Oct 13 2001

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65545 (first 10000 terms from Reinhard Zumkeller)
Eric Weisstein's World of Mathematics, Semiprime
Eric Weisstein's World of Mathematics, Prime zeta function primezeta(s).
Index entries for characteristic functions
Index entries for sequences computed from exponents in factorization of n

Cf. A010051, A064899-A064910, A053409, A046413, A101040, A105700, A280710, A302047, A302048, A302049, A353475, A353476, A353477, A353478, A353479, A353480, A353481.

a064911 = a010051 . a032742 -- Reinhard Zumkeller, Mar 13 2011

with(numtheory):
a:= n-> `if`(bigomega(n)=2, 1, 0):
seq(a(n), n=1..120);  # Alois P. Heinz, Mar 16 2011

Table[If[PrimeOmega[n] == 2, 1, 0], {n, 105}] (* Jayanta Basu, May 25 2013 *)

a(n)=bigomega(n)==2 \\ Charles R Greathouse IV, Mar 13 2011

a(n) = 1 iff n is in A001358 (semiprimes), a(n) = 0 iff n is in A100959 (non-semiprimes). - Reinhard Zumkeller, Nov 24 2004
Dirichlet g.f.: (primezeta(2s) + primezeta(s)^2)/2. - Franklin T. Adams-Watters, Jun 09 2006
a(n) = A057427(A174956(n)); a(n)*A072000(n) = A174956(n). - Reinhard Zumkeller, Apr 03 2010
a(n) = A010051(A032742(n)) (i.e., largest proper divisor is prime). - Reinhard Zumkeller, Mar 13 2011
From Antti Karttunen, Apr 24 2018 & Apr 22 2022: (Start)
a(n) = A280710(n) + A302048(n) = A101040(n) - A010051(n).
a(n) = A353478(n) + A353480(n) = A353477(n) + A353478(n) + A353479(n).
a(n) = A353475(n) + A353476(n).
(End)
a(n) = [Omega(n) = 2], where [ ] is the Iverson bracket. - Wesley Ivan Hurt, Jul 22 2025

Edited by M. F. Hasler, Oct 18 2017

A131284 Numbers n such that difference between prime factors of n-th semiprime is n.

Original entry on oeis.org

5, 80, 86, 613668, 6384425704

Offset: 1

Zak Seidov, Sep 25 2007

The 6384425704th semiprime is 44690979977 = 7*6384425711. 6384425711 - 7 = 6384425704. - Donovan Johnson, Jul 11 2010

			sp(5) = 14 = 2*7 and 7 - 2 = 5, sp(80) = 249 = 3*83 and 83 - 3 = 80, sp(86) = 267 = 3*89 and 89 - 3 = 86; sp(n) = n-th semiprime.

Cf. A064910, A084126, A084127, A109313.

{ n=0; j=1; /* n=3068365-1; j=613668;*/
while( l=(j\10^4+1)*10^4, until( l < j++, until(bigomega(n+=1)==2,);
if(2!=#f=factor(n)[,1],next); if(j==f[2]-f[1],print("\n",[j,n,f])));
print1(j-1,":"n", "))} \\ M. F. Hasler, Sep 28 2007

a(4) = 613668 (p=5, q=613673) from M. F. Hasler, Sep 28 2007

a(5) from Donovan Johnson, Jul 11 2010

A049236 a(n) is the number of distinct prime factors of prime(n) + 2.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 2, 1, 2, 1, 2, 2, 3, 1, 2, 1, 2, 1, 3, 1, 2, 2, 2, 1, 3, 2, 1, 1, 2, 2, 2, 1, 3, 2, 1, 3, 2, 2, 1, 3, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 3, 2, 3, 2, 2, 1, 2, 1, 3, 2, 2, 2, 2, 1, 3, 2, 2, 2, 2, 2, 2, 2, 1, 3, 2, 2, 2, 2, 1, 2

Offset: 1

Labos Elemer

			prime(27) = 103, prime(27) + 2 = 105 = 3*5*7 has 3 prime factors, so a(27) = 3.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A000040, A001221, A001359, A052147, A064909, A064910, A064911.

Table[Length[FactorInteger[Prime[n] + 2]], {n, 1, 50}] (* G. C. Greubel, May 12 2017 *)

a(n) = omega(prime(n) + 2); \\ Amiram Eldar, Sep 16 2024

a(n) = A001221(A052147(n)). - Amiram Eldar, Sep 16 2024

A386256 Smallest semiprime p1*p2 such that p2 mod p1 = n and no prime is used more than once in the sequence.

Original entry on oeis.org

6, 35, 377, 407, 817, 391, 3649, 3131, 4841, 4331, 11461, 5293, 7729, 8051, 12031, 25217, 34417, 29503, 24931, 33389, 26051, 57479, 78227, 44377, 68557, 15707, 78119, 64829, 197401, 77059, 166633, 71371, 140579, 86147, 96427, 109237, 84907, 142523, 213341, 158801

Offset: 1

Tamas Sandor Nagy, Aug 14 2025

Will every prime be used? If so, then the prime factors p1 and p2 of the terms, listed term by term, is a permutation of the primes.

			a(4) = 407 = 11 * 37 because 37 mod 11 = 4, and neither of these primes were used before in the sequence as a(1) = 2 * 3, a(2) = 5 * 7, and a(3) = 13 * 29, and so 11 and 37 are the earliest possible primes to satisfy the condition.
a(5) = 817 = 19 * 43 because 43 mod 19 = 5. Smaller candidate primes such as 13 and 31 would have been suitable, but 13 was already used for a(3) = 377 = 13 * 29. Therefore 19 and 43 are the earliest possible primes to satisfy the condition.

Robert G. Wilson v, Table of n, a(n) for n = 1..400

Cf. A000040, A064910.

q[k_, n_, ps_] := Module[{f = FactorInteger[k], p1, p2}, If[f[[;; , 2]] != {1, 1}, {}, p1 = f[[1, 1]]; p2 = f[[2, 1]]; If[Mod[p2, p1] == n && ! MemberQ[ps, p1] && ! MemberQ[ps, p2], {p1, p2}, {}]]];
seq[nmax_] := Module[{ps = {}, s = {}, k, p}, Do[k = 6; While[(p = q[k, n, ps]) == {}, k++]; AppendTo[s, Times @@ p]; ps = Join[ps, p], {n, 1, nmax}]; s]; seq[40] (* Amiram Eldar, Aug 14 2025 *)
nsp[n_Integer] := nsp[n] = Block[{sp = n + 1}, While[ PrimeOmega[sp] != 2, sp++]; sp]; a[n_] := Block[{sp = 4}, While[fi = Flatten[ Table[ #[[1]], {#[[2]]}] & /@ FactorInteger[ sp]]; Mod[ fi[[2]], fi[[1]]] != n || MemberQ[p, fi[[1]]] ||  MemberQ[p, fi[[2]]], sp = nsp[sp]]; AppendTo[p, fi[[1]]]; AppendTo[p, fi[[2]]]; sp]; p = {}; Do[Print[{n, f[n]}], {n, 50}] (* Robert G. Wilson v, Aug 20 2025 *)

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A064911 If n is semiprime (or 2-almost prime) then 1 else 0.

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A131284 Numbers n such that difference between prime factors of n-th semiprime is n.

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A049236 a(n) is the number of distinct prime factors of prime(n) + 2.

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A386256 Smallest semiprime p1*p2 such that p2 mod p1 = n and no prime is used more than once in the sequence.

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