A124936 -id:A124936 - cp's OEIS frontend

A157483 Numbers k such that k-1 and k+1 are divisible by exactly 3 primes, counted with multiplicity.

19, 29, 43, 51, 67, 69, 77, 115, 171, 173, 187, 189, 237, 243, 245, 267, 274, 283, 285, 291, 317, 344, 355, 386, 403, 405, 411, 424, 427, 429, 435, 437, 476, 507, 597, 603, 604, 605, 638, 653, 664, 669, 723, 763, 776, 787, 789, 846, 891, 893, 907, 926, 963

Offset: 1

Vladimir Joseph Stephan Orlovsky, Mar 01 2009

nonn

Omega(a(n) - 1) = Omega(a(n) + 1) = 3, where Omega(n)=A001222(n). In general twin k-almost prime pairs are defined by Omega(a(n) - 1) = Omega(a(n) + 1) = k. Twin 1-almost primes are twin prime pairs (A014574). - Redjan Shabani, Jul 20 2012

			19 is a term: 19-1 = 18 = 2*3*3 and 19+1 = 20 = 2*2*5.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A124936, A014612, A156028.

with(numtheory); a := proc (n) if bigomega(n-1) = 3 and bigomega(n+1) = 3 then n else end if end proc: seq(a(n), n = 2 .. 1100); # Emeric Deutsch, Mar 03 2009

q=3;lst={};Do[If[Plus@@Last/@FactorInteger[n-1]==q&&Plus@@Last/@FactorInteger[n+1]==q,AppendTo[lst,n]],{n,7!}];lst
Mean/@SequencePosition[PrimeOmega[Range[1000]],{3,,3}] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale, Mar 21 2020 *)

is(n)=bigomega(n-1)==3 && bigomega(n+1)==3 \\ Charles R Greathouse IV, Feb 05 2017

More terms from Emeric Deutsch, Mar 03 2009

A157484 Numbers k such that k+-1 are divisible by exactly 4 primes, counted with multiplicity.

Original entry on oeis.org

55, 89, 151, 197, 233, 249, 295, 307, 329, 341, 343, 349, 461, 489, 491, 569, 571, 665, 713, 739, 775, 851, 857, 859, 869, 871, 949, 1013, 1015, 1061, 1097, 1111, 1149, 1191, 1205, 1207, 1209, 1211, 1219, 1255, 1275, 1277, 1291, 1303, 1315, 1421, 1431, 1449, 1483

Offset: 1

Vladimir Joseph Stephan Orlovsky, Mar 01 2009

nonn

			55 is a term: 55-1 = 54 = 2*3*3*3 and 55+1 = 56 = 2*2*2*7.

Harvey P. Dale, Table of n, a(n) for n = 1..2000

Cf. A014612, A124936, A157483.

q=4;lst={};Do[If[Plus@@Last/@FactorInteger[n-1]==q&&Plus@@Last/@FactorInteger[n+1]==q,AppendTo[lst,n]],{n,7!}];lst
SequencePosition[PrimeOmega[Range[1200]],{4,,4}][[All,1]]+1 (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale, May 08 2019 *)

is(k) = bigomega(k-1)==4 && bigomega(k+1)==4; \\ Jinyuan Wang, Mar 22 2020

A256388 Numbers n such that a single prime is the arithmetic mean of 2 semiprimes whose difference is 2*n.

Original entry on oeis.org

1, 3, 9, 15, 27, 39, 57, 69, 99, 105, 135, 147, 177, 189, 195, 225, 237, 267, 279, 309, 345, 417, 429, 459, 519, 567, 597, 615, 639, 657, 807, 819, 825, 855, 879, 1017, 1029, 1047, 1059, 1089, 1149, 1227, 1275, 1287, 1299, 1317, 1425, 1449, 1479, 1485, 1605

Offset: 1

Michel Marcus, Mar 27 2015

nonn

That is, there is a single prime p, such that p+n and p-n are both semiprime.

Subsequence of A256389.

			A256381 is the list of numbers n such that n-3 and n+3 are semiprimes, and it contains a single prime, hence 3 is in the sequence.

Cf. A124936, A256381.

Cf. A256387 (no prime), A256389 (one or more primes).

Conjecture: a(n) = A001359(n)-2. - Benedict W. J. Irwin, Apr 26 2016

A157485 Numbers k such that k-+1 are divisible by exactly 5 primes, counted with multiplicity.

Original entry on oeis.org

271, 593, 701, 751, 919, 1169, 1241, 1639, 1649, 1673, 1711, 1751, 2071, 2311, 2393, 2549, 2551, 2609, 2729, 2861, 2863, 2897, 2899, 3185, 3331, 3569, 3631, 3823, 3849, 3851, 3943, 3977, 4231, 4265, 4649, 4663, 5071, 5081, 5237, 5239, 5391, 5551, 5561, 5585, 5741

Offset: 1

Vladimir Joseph Stephan Orlovsky, Mar 01 2009

nonn

Jinyuan Wang, Table of n, a(n) for n = 1..5000

Cf. A014612, A124936, A157483, A157484.

q=5;lst={};Do[If[Plus@@Last/@FactorInteger[n-1]==q&&Plus@@Last/@FactorInteger[n+1]==q,AppendTo[lst,n]],{n,7!}];lst
Select[Range[6000],PrimeOmega[#+{1,-1}]=={5,5}&] (* Harvey P. Dale, Jun 05 2021 *)
Mean/@SequencePosition[PrimeOmega[Range[6000]],{5,,5}] (* _Harvey P. Dale, Oct 19 2023 *)

is(k) = bigomega(k-1)==5 && bigomega(k+1)==5; \\ Jinyuan Wang, Mar 22 2020

More terms from Jinyuan Wang, Mar 22 2020

A256381 Numbers n such that n-3 and n+3 are semiprimes.

Original entry on oeis.org

7, 12, 18, 36, 52, 54, 88, 90, 118, 126, 158, 180, 206, 212, 216, 218, 250, 256, 262, 292, 298, 302, 306, 324, 326, 332, 338, 358, 368, 374, 410, 414, 448, 450, 508, 514, 530, 532, 540, 548, 556, 562, 576, 586, 594, 626, 632, 652, 682, 684, 692, 700, 710, 720

Offset: 1

Michel Marcus, Mar 27 2015

nonn

All but the first term are even.

Cf. A001358 (semiprimes).
Cf. A124936 (n-1 and n+1), A105571 (n-2 and n+2).
Cf. A256382 (n-4 and n+4), A256383 (n-5 and n+5).

IsSemiprime:=func< n | &+[k[2]: k in Factorization(n)] eq 2 >; [ n: n in [4..750] | IsSemiprime(n+3) and IsSemiprime(n-3) ]; // Vincenzo Librandi, Mar 28 2015

Select[Range[750], PrimeOmega[# + 3] == PrimeOmega[# - 3] == 2 &] (* Vincenzo Librandi, Mar 28 2015 *)
SequencePosition[Table[If[PrimeOmega[n]==2,1,0],{n,800}],{1,,,_,,,1}][[All,1]]+3 (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, May 21 2017 *)

lista(nn,m=3) = {for (n=m+1, nn, if (bigomega(n-m)==2 && bigomega(n+m)==2, print1(n, ", ")););}

A256382 Numbers n such that n-4 and n+4 are semiprimes.

Original entry on oeis.org

10, 18, 29, 30, 42, 53, 61, 73, 78, 81, 89, 90, 91, 115, 119, 125, 137, 138, 162, 165, 173, 181, 198, 205, 209, 210, 213, 217, 222, 258, 263, 291, 295, 299, 305, 323, 325, 330, 331, 390, 399, 407, 411, 441, 449, 450, 462, 477, 485, 489, 493, 497, 501, 515, 523

Offset: 1

Michel Marcus, Mar 27 2015

nonn

A117328 is the subsequence of primes.

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

Cf. A001358 (semiprimes).
Cf. A117328 (with primes rather than semiprimes).
Cf. A124936 (n-1 and n+1), A105571 (n-2 and n+2).
Cf. A256381 (n-3 and n+3), A256383 (n-5 and n+5).

IsSemiprime:=func< n | &+[k[2]: k in Factorization(n)] eq 2 >; [ n: n in [6..750] | IsSemiprime(n+4) and IsSemiprime(n-4) ]; // Vincenzo Librandi, Mar 29 2015

Select[Range[600], PrimeOmega[# + 4] == PrimeOmega[# - 4] == 2 &] (* Vincenzo Librandi, Mar 29 2015 *)
Flatten[Position[Partition[Table[If[PrimeOmega[n]==2,1,0],{n,600}],9,1],?(#[[1]]==#[[9]]==1&),{1},Heads->False]]+4 (* _Harvey P. Dale, Mar 29 2015 *)

lista(nn,m=4) = {for (n=m+1, nn, if (bigomega(n-m)==2 && bigomega(n+m)==2, print1(n, ", ")););}

A256383 Numbers n such that n-5 and n+5 are semiprimes.

Original entry on oeis.org

9, 20, 30, 44, 60, 82, 90, 116, 124, 128, 138, 150, 164, 182, 208, 210, 214, 242, 254, 294, 296, 300, 304, 314, 324, 334, 360, 366, 376, 386, 398, 408, 412, 422, 432, 442, 476, 506, 510, 522, 524, 532, 538, 540, 548, 578, 584, 586, 628, 674, 676, 684

Offset: 1

Michel Marcus, Mar 27 2015

nonn

It appears that there are no primes in this sequence.

If n is odd, one of n+5 and n-5 is divisible by 4, so unless n = 9 it can't be a semiprime. Thus all terms except 9 are even. - Robert Israel, Apr 13 2020

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

Cf. A001358 (semiprimes).
Cf. A124936 (n-1 and n+1), A105571 (n-2 and n+2).
Cf. A256381 (n-3 and n+3), A256382 (n-4 and n+4).

IsSemiprime:=func< n | &+[k[2]: k in Factorization(n)] eq 2 >; [ n: n in [6..700] | IsSemiprime(n+5) and IsSemiprime(n-5) ]; // Vincenzo Librandi, Mar 29 2015

N:= 1000: # for terms <= N-5
PP:= select(isprime, {seq(i,i=3..N/3,2)}):
P:= select(`<=`,PP,floor(sqrt(N))):
SP:= {}:
for p in P do
  PP:= select(`<=`,PP,N/p);
  SP:= SP union map(`*`,PP,p);
od:
R:= {9} union (map(`+`,SP,5) intersect map(`-`,SP,5)):
sort(convert(R,list)); # Robert Israel, Apr 13 2020

Select[Range[2, 700], PrimeOmega[# + 5] == PrimeOmega[# - 5] == 2 &] (* Vincenzo Librandi, Mar 29 2015 *)

lista(nn,m=5) = {for (n=m+1, nn, if (bigomega(n-m)==2 && bigomega(n+m)==2, print1(n, ", ")););}

issemi(n)=bigomega(n)==2
list(lim)=my(v=List([9])); forprime(p=5,(lim-5)\3, if(issemi(3*p+10), listput(v,3*p+5))); forprime(p=29,(lim+5)\3, if(issemi(3*p-10), listput(v,3*p-5))); forstep(n=30,lim\=1,6, if(issemi(n-5) && issemi(n+5), listput(v, n))); Set(v) \\ Charles R Greathouse IV, Apr 13 2020

A157486 Numbers n such that n-+1 are divisible by exactly 6 primes, counted with multiplicity.

Original entry on oeis.org

1889, 3079, 4591, 5023, 6175, 6641, 7649, 9881, 10961, 11501, 11935, 12689, 13751, 14417, 15499, 15713, 16687, 18017, 18089, 18271, 19169, 19249, 19889, 19951

Offset: 1

Vladimir Joseph Stephan Orlovsky, Mar 01 2009

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..4000

Cf. A124936, A014612, A157483, A157484, A157485

q=6;lst={};Do[If[Plus@@Last/@FactorInteger[n-1]==q&&Plus@@Last/@FactorInteger[n+1]==q,AppendTo[lst,n]],{n,9!}];lst
Mean/@SequencePosition[PrimeOmega[Range[20000]],{6,,6}] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale, Sep 23 2019 *)

A167023 Fibonacci numbers where both neighbors are semiprimes.

Original entry on oeis.org

5, 34, 144, 46368

Offset: 1

Vladimir Joseph Stephan Orlovsky, Oct 27 2009

nonn

Next term (if it exists) is larger than 10^10000. I conjecture that this sequence is finite: if neighbors of Fibonacci numbers behave randomly, the expected number of remaining terms is about 0.0103 (or 0.00779 if their behavior mod 6 is taken into account). - Charles R Greathouse IV, Nov 09 2009

			5 is in the sequence because 4=2*2 and 6=2*3. 46368 is in the sequence because 46367 = 199 * 233 and 46369 = 89 * 521.

Cf. A000045, A001358.

u[n_]:=Plus@@Last/@FactorInteger[n]==2; lst={};Do[f=Fibonacci[n];If[u[f-1]&&u[f+1], Print[f];AppendTo[lst,f]],{n,3*5!}];lst
Select[Fibonacci[Range[200]],Union[PrimeOmega[#+{1,-1}]]=={2}&] (* Harvey P. Dale, Mar 16 2015 *)

for(n=5,99, f=fibonacci(n); if(bigomega(f-1)==2 && bigomega(f+1)==2, print1(f", "))) \\ Charles R Greathouse IV, Mar 21 2016

A124936 INTERSECT A000045. - R. J. Mathar, Nov 03 2009

Edited by R. J. Mathar, Nov 05 2009 and Charles R Greathouse IV, Nov 09 2009

A256389 Numbers n such that one or more primes can be the arithmetic mean of 2 semiprimes whose difference is 2*n.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 22, 24, 26, 27, 28, 30, 32, 34, 36, 38, 39, 40, 42, 44, 46, 48, 50, 52, 54, 56, 57, 58, 60, 62, 64, 66, 68, 69, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110

Offset: 1

Michel Marcus, Mar 27 2015

nonn

That is, there are several primes p, such that p+n and p-n are both semiprime.
Complement of A256387.
The terms of this sequence that do not belong to A256388 are even.

			A256381 is the list of numbers n such that n-3 and n+3 are semiprimes, and it contains a single prime, hence 3 is in the sequence.
A256382 is the list of numbers n such that n-4 and n+4 are semiprimes, and it contains several primes, hence 4 is in the sequence.

Cf. A124936, A105571, A256382.

Cf. A256387 (no prime), A256388 (a single prime).

cp's OEIS Frontend

A157483 Numbers k such that k-1 and k+1 are divisible by exactly 3 primes, counted with multiplicity.

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A157484 Numbers k such that k+-1 are divisible by exactly 4 primes, counted with multiplicity.

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A256388 Numbers n such that a single prime is the arithmetic mean of 2 semiprimes whose difference is 2*n.

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A157485 Numbers k such that k-+1 are divisible by exactly 5 primes, counted with multiplicity.

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A256381 Numbers n such that n-3 and n+3 are semiprimes.

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A256382 Numbers n such that n-4 and n+4 are semiprimes.

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A256383 Numbers n such that n-5 and n+5 are semiprimes.

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A157486 Numbers n such that n-+1 are divisible by exactly 6 primes, counted with multiplicity.

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