A140077 -id:A140077 - cp's OEIS frontend

A168628 Numbers n such that n and n+-1 have 4 distinct prime factors.

37961, 44485, 45695, 50141, 51429, 55131, 55385, 61335, 63365, 64295, 67165, 68265, 68475, 70005, 70091, 71709, 72709, 76153, 80445, 81549, 81719, 82041, 84435, 85491, 86241, 90364, 95381, 97383, 98021, 99085, 99385, 99429, 99789, 100165, 100491, 100595

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 01 2009

nonn

Subsequence of A140078.

Cf. A140077, A168626

f[n_]:=Length[FactorInteger[n]]; lst={};Do[If[f[n]>=4&&f[n-1]>=4&&f[n+1]>=4,AppendTo[lst,n]],{n,9!}];lst
SequencePosition[PrimeNu[Range[110000]],{4,4,4}][[All,1]]+1 (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Apr 27 2018 *)

is(n)=omega(n)==4 && omega(n+1)==4 && omega(n-1)==4 \\ Charles R Greathouse IV, Jan 25 2025

Corrected and extended by Harvey P. Dale, Apr 27 2018

A321502 Numbers m such that m and m+1 have at least 2, but m or m+1 has at least 3 prime divisors.

Original entry on oeis.org

65, 69, 77, 84, 90, 104, 105, 110, 114, 119, 129, 132, 140, 153, 154, 155, 164, 165, 170, 174, 182, 185, 186, 189, 194, 195, 203, 204, 209, 219, 220, 221, 230, 231, 234, 237, 245, 246, 252, 254, 258, 259, 260, 264, 265, 266, 272, 273, 275, 279, 284, 285, 286, 290, 294, 299, 300, 305

Offset: 1

M. F. Hasler, Nov 27 2018

nonn

Since m and m+1 cannot have a common factor, m(m+1) has at least 2+3 prime divisors (= distinct prime factors), whence m+1 > sqrt(primorial(5)) ~ 48. It turns out that a(1)*(a(1)+1) = 2*3*5*11*13, i.e., the prime factor 7 is not present.

Cf. A321493, A321494, A321495, A321496, A321497 (analog for k = 3, ..., 7 prime divisors).
Cf. A074851, A140077, A140078, A140079 (m and m+1 have exactly k = 2, 3, 4, 5 prime divisors).
Cf. A255346, A321503 .. A321506, A321489 (m and m+1 have at least 2, ..., 7 prime divisors).
Cf. A006049, A006549, A093548.

select( is_A321502(n)=vecmax(n=[omega(n), omega(n+1)])>2&&vecmin(n)>1, [1..500])

Equals A255346 \ A074851.

A168629 Numbers n such that n,n+1 and sum of this two numbers have at least 3 distinct prime factors.

Original entry on oeis.org

1105, 1130, 1462, 1644, 1742, 1767, 2014, 2222, 2232, 2260, 2337, 2365, 2397, 2464, 2541, 2667, 2684, 2697, 2702, 2755, 2821, 2914, 3074, 3115, 3195, 3289, 3332, 3477, 3484, 3514, 3552, 3619, 3657, 3685, 3782, 3783, 3842, 3965, 4014, 4088, 4122, 4147, 4277

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 01 2009

nonn

			1105 = 5*13*17, 1106 = 2*7*79, 1105 + 1106 = 2211 = 3*11*67.

Cf. A140077, A140078, A168626, A168628.

q:= n-> andmap(x-> nops(numtheory[factorset](x))>2, [n, n+1, 2*n+1]):
select(q, [$1..4600])[];  # Alois P. Heinz, Jun 29 2021

f[n_]:=Length[FactorInteger[n]]; lst={};Do[If[f[n]>=3&&f[n+1]>=3&&f[n+n+1]>=3,AppendTo[lst,n]],{n,8!}];lst

A168630 Numbers n such that n, n+1, and the sum of those two numbers each have 4 or more distinct prime factors.

Original entry on oeis.org

46189, 50634, 69597, 76797, 90117, 97954, 108205, 115804, 127347, 138957, 144627, 159340, 164020, 166022, 166497, 166705, 167205, 167485, 173194, 174454, 181670, 186294, 190014, 193154, 198789, 211029, 212134, 214225, 217217, 221815, 222547, 224146

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 01 2009

nonn

			FactorInteger[46189]=11*13*17*19, FactorInteger[46190]=2*5*31*149, FactorInteger[46189+46190]=3*7*53*83,..

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

Cf. A140077, A140078, A168626, A168628, A168629

g:= proc(n) option remember; nops(numtheory:-factorset(n))>=4 end proc:
filter:= n -> g(n) and g(n+1) and g(2*n+1):
select(filter, [$1..300000]); # Robert Israel, May 09 2018

f[n_]:=Length[FactorInteger[n]]; lst={};Do[If[f[n]>=4&&f[n+1]>=4&&f[n+n+1]>=4,AppendTo[lst,n]],{n,9!}];lst
Select[Range[225000],Min[Thread[PrimeNu[{#,#+1,2#+1}]]]>3&](* Harvey P. Dale, Nov 11 2017 *)

Definition modified and terms extended by Harvey P. Dale, Nov 11 2017

A247200 Odd numbers which are neither of the form p2^m + 1 nor of the form p2^m - 1 with p prime.

Original entry on oeis.org

71, 99, 101, 109, 131, 139, 155, 169, 181, 197, 199, 221, 229, 239, 241, 251, 259, 265, 281, 287, 289, 307, 309, 311, 323, 337, 339, 341, 349, 365, 371, 373, 379, 391, 401, 407, 409, 419, 431, 433, 439, 441, 443, 461, 469, 475, 485, 491, 493, 499, 505, 517, 519

Offset: 1

Arkadiusz Wesolowski, Nov 18 2014

nonn

For each n, the sequence has a set of n consecutive odd numbers.
For any n, the number 2*A140077(n) + 1 is in the sequence.
Every number of the form S*2^n + 1 or R*2^n - 1 with n > 0, where S is a composite Sierpiński number and R is a composite Riesel number, is in the sequence.
Odd numbers n such that (n-1)/A007814(n-1) and (n+1)/A007814(n+1) are composite. - Robert Israel, Nov 19 2014

Cf. A007814, A140077.

lst1:=[]; lst2:=[]; r:=519; t:=Floor(Log(2, r))-1; for m in [0..t] do e:=Floor(r/2^m); for p in [2..e] do if IsPrime(p) then a:=p*2^m-1; b:=p*2^m+1; if not a in lst1 then Append(~lst1, a); end if; if not b in lst1 then Append(~lst1, b); end if; end if; end for; end for; for n in [3..r by 2] do if not n in lst1 then Append(~lst2, n); end if; end for; lst2;

filter:= proc(n)
  local m1,m2;
  m1:= padic[ordp](n-1,2);
  if n-1 = 2^m1 then return false fi;
  m2:= padic[ordp](n+1,2);
  n+1 <> 2^m2 and not isprime((n-1)/2^m1) and not isprime((n+1)/2^m2);
end proc:
select(filter, [seq(2*i+1,i=0..1000)]); # Robert Israel, Nov 19 2014

b=0; forstep(n=1, 519, 2, c=2^floor(log(n)/log(2)); a=b; b=(n+1)/gcd(n+1, c); if(a>8&&!isprime(a)&&!isprime(b), print1(n, ", ")));

cp's OEIS Frontend

A168628 Numbers n such that n and n+-1 have 4 distinct prime factors.

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A321502 Numbers m such that m and m+1 have at least 2, but m or m+1 has at least 3 prime divisors.

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A168629 Numbers n such that n,n+1 and sum of this two numbers have at least 3 distinct prime factors.

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Maple

Mathematica

A168630 Numbers n such that n, n+1, and the sum of those two numbers each have 4 or more distinct prime factors.

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Author

Keywords

Examples

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Crossrefs

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Maple

Mathematica

Extensions

A247200 Odd numbers which are neither of the form p*2^m + 1 nor of the form p*2^m - 1 with p prime.

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Author

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Magma

Maple

PARI

A247200 Odd numbers which are neither of the form p2^m + 1 nor of the form p2^m - 1 with p prime.