A241535 -id:A241535 - cp's OEIS frontend

A241539 Smallest k>=1 such that the n-th semiprime + or - k are both primes, or a(n)=0 if there is no such k.

1, 1, 2, 3, 3, 2, 2, 9, 6, 3, 4, 3, 6, 9, 2, 15, 12, 8, 12, 4, 15, 9, 6, 2, 15, 6, 15, 12, 3, 14, 12, 4, 15, 6, 3, 2, 12, 9, 12, 18, 9, 14, 2, 6, 3, 10, 15, 6, 6, 33, 18, 9, 8, 12, 15, 12, 4, 15, 10, 6, 6, 3, 10, 9, 24, 6, 27, 18, 14, 15, 18, 6, 21, 8, 30, 3

Offset: 1

Vladimir Shevelev, Apr 25 2014

nonn

If the sequence has no zeros at least for sufficiently large n, then one can believe that, for a pair of sufficiently large semiprimes of the same parity {r,s}, there is a number k=k(r,s) such that either {r-k, s+k} or {r+k, s-k} is a pair of primes. Then, if a representation 2*n = r+s with, say, min{r,s} > log(n) is considered, then, at least for sufficiently large n, it is reduced to the Goldbach representation 2*n = p+q with primes p,q. It is natural to think that a Goldbach-like conjecture that at least every sufficiently large even number is a sum of two semiprimes could be proved more easily than the classic Goldbach conjecture (cf. Chen's theorem).

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Wikipedia, Chen's theorem

Cf. A001358, A241531, A241533, A241535, A241536.

sk[s_]:=Module[{k=1},While[!PrimeQ[s+k]||!PrimeQ[s-k],k++];k]; sk/@Select[Range[300],PrimeOmega[#]==2&] (* Harvey P. Dale, Jun 07 2025 *)

list(lim) = my(v=List(), t); forprime(p=2, sqrt(lim), t=p; forprime(q=p, lim\t, listput(v, t*q))); vecsort(Vec(v)) \\ From A001358
sp = list(1000); vector(#sp, n, k=1; while(!isprime(sp[n]+k) || !isprime(sp[n]-k), k++); k) \\ Colin Barker, May 31 2014

More terms from Peter J. C. Moses, Apr 28 2014

A241536 Smallest k>=1 such that prime(n)+k and prime(n)-k are both semiprimes, or a(n)=0 if there is no such k.

Original entry on oeis.org

0, 0, 1, 3, 0, 9, 8, 15, 2, 4, 27, 2, 8, 8, 8, 2, 10, 4, 2, 6, 4, 14, 28, 2, 32, 10, 8, 12, 14, 2, 6, 2, 4, 6, 6, 8, 2, 20, 34, 4, 24, 4, 14, 8, 12, 14, 2, 14, 8, 8, 14, 20, 6, 2, 8, 4, 20, 18, 10, 14, 16, 2, 2, 8, 8, 12, 4, 2, 8, 22, 12, 18, 26, 8, 2, 12, 18

Offset: 1

Vladimir Shevelev, Apr 25 2014

nonn

If a(n)=2, then prime(n)+2 and prime(n)-2 are both semiprimes; that is, prime(n) belongs to A063643. - Michel Marcus, Mar 26 2015

Cf. A001358, A241531, A241533, A241535.

sks[n_]:=Module[{k=1,p=Prime[n]},While[PrimeOmega[p+k]!=2||PrimeOmega[p-k]!=2||p-k<4,If[p-k<3,Break[]];k++];If[p-k<4,0,k]]; Array[sks,80] (* Harvey P. Dale, Dec 09 2016 *)

a(n) = {p = prime(n); for (k=1, p-1, if ((bigomega(p-k)==2) && (bigomega(p+k) == 2), return (k));); return (0);} \\ Michel Marcus, Apr 25 2014

More terms from Michel Marcus, Apr 25 2014

Name edited by Michel Marcus, Mar 26 2015

A253138 Number of ways to represent the n-th prime as the arithmetic mean of two semiprimes.

Original entry on oeis.org

0, 0, 1, 1, 0, 1, 1, 1, 1, 2, 1, 3, 2, 4, 3, 4, 3, 3, 5, 7, 6, 5, 5, 8, 8, 7, 9, 7, 10, 10, 12, 11, 15, 12, 14, 14, 13, 11, 13, 15, 15, 14, 15, 20, 14, 15, 19, 20, 16, 17, 17, 17, 21, 24, 23, 24, 28, 23, 25, 24, 27, 25, 32, 29, 25, 21, 26, 31, 31, 29, 36, 32

Offset: 1

Michel Lagneau, Mar 23 2015

nonn

Conjecture: a(n)>0 for n>5.
Note that a(n) = A241535(n) = A241536(n) = 0 for n=1,2 and 5. - Michel Marcus, Mar 26 2015
Among the a(n) decompositions of prime(n) into two semiprimes (prime(n)+ k)/2 and (prime(n)-k)/2, there is one where k is minimum with k = A241536(n) and there is one where k is maximum with k = prime(n) - A241535(n).

			a(12)=3 as prime(12) = 37 = (9+65)/2 = (25+49)/2 =(35+39)/2 where 9, 25, 35, 39, 49 and 65 are semiprime.

Michel Lagneau, Table of n, a(n) for n = 1..1000

Cf. A001358, A071681, A241535, A241536.

Cf. A064911, A000040.

a253138 n = sum $ map a064911 $
   takeWhile (> 0) $ map (2 * p -) $ dropWhile (< p) a001358_list
   where p = a000040 n
-- Reinhard Zumkeller, Mar 27 2015

with(numtheory):for n from 1 to 100 do:c:=0:p:=ithprime(n):for m from 1 to p-1 do:p1:=p-m:p2:=p+m:if bigomega(p1)=2 and bigomega(p2)=2 then c:=c+1:else fi:od:printf(`%d, `,c):od:

Reap[For[n=1, n <= 100, n++, c=0; p = Prime[n]; For[m=1, m <= p-1, m++, p1 = p-m; p2 = p+m; If[PrimeOmega[p1] == 2 && PrimeOmega[p2] == 2 , c = c+1]]; Print[c]; Sow[c]]][[2, 1]] (* Jean-François Alcover, Mar 23 2015, translated from Maple *)

A241656 Smallest semiprime, sp, such that 2n - sp is a semiprime, or a(n)=0 if there is no such sp.

Original entry on oeis.org

0, 0, 0, 4, 4, 6, 4, 6, 4, 6, 0, 9, 4, 6, 4, 6, 9, 10, 4, 6, 4, 6, 21, 9, 4, 6, 15, 10, 9, 9, 4, 6, 4, 6, 15, 10, 9, 14, 4, 6, 25, 10, 4, 6, 4, 6, 9, 9, 4, 6, 9, 9, 15, 14, 4, 6, 21, 10, 25, 9, 4, 6, 4, 6, 9, 9, 15, 14, 4, 6, 9, 10, 4, 6, 4, 6, 9, 10, 15, 14, 4, 6, 21, 9, 4, 6, 15, 10, 9, 14

Offset: 1

Vladimir Shevelev and Robert G. Wilson v, Apr 26 2014

nonn

Conjecture: every even number greater than 22 is a sum of two semiprimes. Only 2, 4, 6 & 22 cannot be so represented.
If n is prime, then a(n) must be either 4 or an odd semiprime. See A241535.
First occurrence of the k-th semiprime (A001358): 4, 6, 12, 18, 38, 27, 23, 124, 41, 326, 127, 1344, 147, 1278, 189, 3294, 757, 317, 1362, 1775, 3300, 2504, 2025, 7394, 84848, 13899, 56584, 11347, 156396, 7667, 7905, 15447, 404898, 20937, ..., .

			a(12) = 9 because 2*12 = 24 = 9 + 15, two semiprimes.

Cf. A001358, A127018, A241535, A241658.

NextSemiPrime[n_, k_: 1] := Block[{c = 0, sgn = Sign[k]}, sp = n + sgn; While[c < Abs[k], While[ PrimeOmega[sp] != 2, If[ sgn < 0, sp--, sp++]]; If[ sgn < 0, sp--, sp++]; c++]; sp + If[sgn < 0, 1, -1]]; f[n_] := Block[{en = 2 n, sp = 4}, While[ PrimeOmega[en - sp] != 2, sp = NextSemiPrime[sp]]; If[en > sp, sp, 0]]; Array[ f, 90]

A241658 Smallest semiprime, sp, such that n - sp is a semiprime, or a(n)=0 if there is no such sp.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 4, 0, 4, 0, 6, 4, 4, 6, 6, 0, 4, 4, 6, 6, 0, 9, 9, 4, 4, 6, 6, 4, 4, 6, 6, 0, 9, 9, 10, 4, 4, 4, 6, 6, 4, 4, 6, 6, 21, 9, 9, 10, 4, 25, 6, 4, 15, 4, 10, 6, 9, 4, 9, 4, 4, 6, 6, 10, 4, 9, 6, 4, 15, 6, 10, 4, 9, 6, 14, 15, 4, 10, 6, 4, 25, 6, 10, 34, 4, 10, 6, 4, 4

Offset: 1

Vladimir Shevelev and Robert G. Wilson v, Apr 26 2014

nonn

Conjecture: every number greater than 33 is a sum of two semiprimes. Only 1, 2, 3, 4, 5, 6, 7, 9, 11, 17, 22 & 33 cannot be so represented.
If n is prime, then a(2n) must be either 4 or an odd semiprime. See A241535.
First occurrence of the k-th semiprime (A001358): 8, 12, 23, 36, 76, 54, 46, 113, 51, 185, 254, 85, 294, 1881, 378, 1035, 1514, 634, 1509, 3550, 1621, 2713, 4050, 14788, 1485, 26839, 1497, 22694, 11965, 15334, 15810, 30894, 2721, 16849, ..., .

			a(23) = 9 because 23 = 9 + 14, two semiprimes.

Cf. A001358, A241535, A241656.

NextSemiPrime[n_, k_: 1] := Block[{c = 0, sgn = Sign[k]}, sp = n + sgn; While[c < Abs[k], While[ PrimeOmega[sp] != 2, If[ sgn < 0, sp--, sp++]]; If[ sgn < 0, sp--, sp++]; c++]; sp + If[sgn < 0, 1, -1]]; f[n_] := Block[{sp = 4}, While[ PrimeOmega[n - sp] != 2, sp = NextSemiPrime[sp]]; If[n > sp, sp, 0]]; Array[ f, 90]

a(n) = {for (k=4, n-4, if ((bigomega(k) ==2) && (bigomega(n-k) == 2), return (k));); return (0);} \\ Michel Marcus, Jun 12 2014

cp's OEIS Frontend

A241539 Smallest k>=1 such that the n-th semiprime + or - k are both primes, or a(n)=0 if there is no such k.

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A241536 Smallest k>=1 such that prime(n)+k and prime(n)-k are both semiprimes, or a(n)=0 if there is no such k.

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A253138 Number of ways to represent the n-th prime as the arithmetic mean of two semiprimes.

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A241656 Smallest semiprime, sp, such that 2n - sp is a semiprime, or a(n)=0 if there is no such sp.

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A241658 Smallest semiprime, sp, such that n - sp is a semiprime, or a(n)=0 if there is no such sp.

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