A157483 -id:A157483 - cp's OEIS frontend

A124941 Numbers k such that k and k+4 are 4-almost primes.

36, 56, 84, 100, 132, 136, 152, 228, 340, 344, 372, 376, 472, 484, 488, 532, 546, 564, 568, 580, 621, 632, 686, 708, 770, 804, 808, 820, 846, 852, 856, 868, 872, 950, 1012, 1192, 1204, 1206, 1208, 1274, 1304, 1326, 1336, 1444, 1524, 1550, 1572, 1576, 1690

Offset: 1

Zak Seidov, Nov 13 2006

nonn

			36=2^2*3^2, 40=2^3*5; 56=2^3*7, 60=2^2*3*5.

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

Cf. A014613, A124940, A124942, A157483, A180117.

Select[Range[1690], PrimeOmega[#]==PrimeOmega[#+4]==4 &] (* James C. McMahon, Dec 07 2024 *)

isok(n) = (bigomega(n) == 4) && (bigomega(n+4) == 4); \\ Michel Marcus, Oct 11 2013

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

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

A173966 Sums of two consecutive semiprimes.

Original entry on oeis.org

19, 29, 43, 51, 67, 69, 77, 115, 171, 173, 187, 189, 237, 243, 245, 267, 283, 285, 291, 317, 355, 403, 405, 411, 427, 429, 435, 437, 507, 597, 603, 605, 653, 669, 723, 763, 787, 789, 891, 893, 907, 963, 1003, 1029, 1053, 1075, 1085, 1107, 1131, 1245, 1267

Offset: 1

Vladimir Joseph Stephan Orlovsky, Mar 03 2010

nonn

First 16 terms:19,29,43,51,67,69,77,115,171,173,187,189,237,243,245,267 are the same as in A157483.

These are sums of two consecutive integers which are both semiprimes, whereas A118717 are sums of two semiprimes which are adjacent (consecutive) in A001358. [From R. J. Mathar, Mar 18 2010]

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

f[n_]:=Last/@FactorInteger[n]=={1,1}||Last/@FactorInteger[n]=={2};lst={};Do[If[f[n],If[f[n+1],AppendTo[lst,2*n+1]]],{n,7!}];lst
Total/@Select[Partition[Select[Range[700],PrimeOmega[#]==2&],2,1],#[[2]]- #[[1]] == 1&] (* Harvey P. Dale, Jun 22 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 *)

A157487 Numbers k such that k-1 and k+1 are each the product of exactly 7 primes, counted with multiplicity.

Original entry on oeis.org

10529, 15391, 17983, 18751, 22049, 23489, 24751, 26081, 29249, 32561, 35153, 43471, 49951, 52975, 58049, 58481, 67229, 67231, 70687, 71873, 72415, 76049, 77921, 79001, 79649, 82783, 83249, 85751, 88289, 93799, 95551, 97471, 102545

Offset: 1

Vladimir Joseph Stephan Orlovsky, Mar 01 2009

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..3000

Cf. A124936, A014612, A157483, A157484, A157485, A157486.

with(numtheory): a := proc (n) if bigomega(n-1) = 7 and bigomega(n+1) = 7 then n else end if end proc: seq(a(n), n = 2 .. 120000); # Emeric Deutsch, Mar 07 2009

q=7;lst={};Do[If[Plus@@Last/@FactorInteger[n-1]==q&&Plus@@Last/@FactorInteger[n+1]==q,AppendTo[lst,n]],{n,9!}];lst
Select[Range[110000],PrimeOmega[#+{1,-1}]=={7,7}&] (* Harvey P. Dale, Apr 04 2015 *)
Mean/@SequencePosition[PrimeOmega[Range[105000]],{7,,7}] (* _Harvey P. Dale, Sep 10 2022 *)

More terms from Emeric Deutsch, Mar 07 2009

A157489 Numbers n such that n-+5 are divisible by exactly 5 primes, counted with multiplicity.

Original entry on oeis.org

275, 373, 445, 755, 985, 1165, 1245, 1475, 1535, 1643, 1645, 1705, 1715, 1745, 2219, 2305, 2317, 2389, 2445, 2455, 2543, 2579, 2845, 2855, 2893, 3229, 3299, 3325, 3371, 3565, 3613, 3659, 3695, 3757, 3829, 3875, 4255, 4285, 4295, 4345, 4355, 4477, 4745, 5003, 5065

Offset: 1

Vladimir Joseph Stephan Orlovsky, Mar 01 2009

nonn

Let a, b and 10 be pairwise coprime, with A001222(a) = A001222(b) = 4. There exists c such that c == 5 (mod a) and c == -5 (mod b). Dickson's conjecture implies that (c+k*a*b-5)/a and (c+k*a*b+5)/b are prime for infinitely many k; for such k, c+k*a*b is in the sequence. - Robert Israel, Mar 22 2020

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

Cf. A001222, A124936, A014612, A157483, A157484, A157485, A157486

N:= 10^4: # for terms <= N
T5:= select(t -> numtheory:-bigomega(t)=5, {$1..N+5}):
S:= T5 intersect map(`+`,T5,10):
sort(convert(map(`-`,S,5),list)); # Robert Israel, Mar 22 2020

q=5;lst={};Do[If[Plus@@Last/@FactorInteger[n-q]==q&&Plus@@Last/@FactorInteger[n+q]==q,AppendTo[lst,n]],{n,8!}];lst
SequencePosition[PrimeOmega[Range[5100]],{5,,,_,,,_,,,_,5}][[All,1]]+5 (* Harvey P. Dale, Sep 23 2021 *)

cp's OEIS Frontend

A124941 Numbers k such that k and k+4 are 4-almost primes.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A173966 Sums of two consecutive semiprimes.

Views

Author

Keywords

Comments

Links

Programs

Mathematica

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

A157487 Numbers k such that k-1 and k+1 are each the product of exactly 7 primes, counted with multiplicity.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

A157489 Numbers n such that n-+5 are divisible by exactly 5 primes, counted with multiplicity.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica