A273879 -id:A273879 - cp's OEIS frontend

A074851 Numbers k such that k and k+1 both have exactly 2 distinct prime factors.

14, 20, 21, 33, 34, 35, 38, 39, 44, 45, 50, 51, 54, 55, 56, 57, 62, 68, 74, 75, 76, 85, 86, 87, 91, 92, 93, 94, 95, 98, 99, 111, 115, 116, 117, 118, 122, 123, 133, 134, 135, 141, 142, 143, 144, 145, 146, 147, 152, 158, 159, 160, 161, 171, 175, 176, 177, 183, 184

Offset: 1

Benoit Cloitre, Sep 10 2002

Subsequence of A006049. - Michel Marcus, May 06 2016

			20=2^2*5 21=3*7 hence 20 is in the sequence.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A006049, A006549, A001221.
Analogous sequences for m distinct prime factors: this sequence (m=2), A140077 (m=3), A140078 (m=4), A140079 (m=5), A273879 (m=6).
Cf. A093548.
Equals A255346 \ A321502.

Filtered([1..200],n->[Size(Set(Factors(n))),Size(Set(Factors(n+1)))]=[2,2]); # Muniru A Asiru, Dec 05 2018

[n: n in [2..200] | #PrimeDivisors(n) eq 2 and #PrimeDivisors(n+1) eq 2]; // Vincenzo Librandi, Dec 05 2018

Flatten[Position[Partition[Table[If[PrimeNu[n]==2,1,0],{n,200}],2,1],{1,1}]] (* Harvey P. Dale, Mar 12 2015 *)

isok(n) = (omega(n) == 2) && (omega(n+1) == 2); \\ Michel Marcus, May 06 2016

import sympy
from sympy.ntheory.factor_ import primenu
for n in range(1,200):
    if primenu(n)==2 and primenu(n+1)==2:
        print(n, end=', '); # Stefano Spezia, Dec 05 2018

a(n) seems to be asymptotic to c*n*log(n)^2 with c=0.13...

{k: A001221(k) = A001221(k+1) = 2}. - R. J. Mathar, Jul 18 2023

A321506 Numbers m such that m and m+1 each have at least 6 distinct prime factors.

Original entry on oeis.org

11243154, 13516580, 16473170, 16701684, 17348330, 19286805, 20333495, 21271964, 21849905, 22054515, 22527141, 22754589, 22875489, 24031370, 25348070, 25774329, 28098245, 28618394, 28625960, 30259229, 31846269, 32642805, 32734910, 33205029, 33631520, 33641894, 35023365

Offset: 1

M. F. Hasler, Nov 12 2018

nonn

Equals A273879 up to a(138) = 58524465, which is not in A273879: see A321496 for the complement.

M. F. Hasler, Table of n, a(n) for n = 1..150

Cf. A273879 (variant with "exactly 6"), A321496 (terms not in A273879).

Cf. A321505 (analog for k=5 prime factors).

Select[Range[36000000], PrimeNu[#] > 5 && PrimeNu[# + 1] > 5 &] (* Amiram Eldar, Nov 12 2018 *)
Position[Partition[PrimeNu[Range[3503*10^4]],2,1],?(#[[1]]>5&&#[[2]]> 5&), 1,Heads-> False]//Flatten (* _Harvey P. Dale, Jul 28 2020 *)

PARI
```
is(n)=omega(n)>=6&&omega(n+1)>=6
```

A364265 The first term in a chain of at least 3 consecutive numbers each with exactly 6 distinct prime factors (i.e., belonging to A074969).

Original entry on oeis.org

323567034, 431684330, 468780388, 481098980, 577922904, 639336984, 715008644, 720990620, 726167154, 735965384, 769385252, 808810638, 822981560, 831034918, 839075510, 847765554, 879549670, 895723268, 902976710, 903293468, 904796814, 918520420, 940737005, 944087484, 982059364

Offset: 1

R. J. Mathar, Jul 16 2023

nonn

To distinguish this from A259349: "Numbers n with exactly k distinct prime factors" means numbers with A001221(n) = omega(n) = k, which specifies that in the prime factorization n = Product_{i>=1} p_i^(e_i), e_i >= 1, the exponents are ignored, and only the size of the set of the (distinct) p_i is considered. In A259349, the numbers n are products of k distinct primes, which means in the prime factorization of n, all exponents e_i are equal to 1. (If all exponents e_i = 1, the n are squarefree, i.e., in A005117.) Rephrased: the n which are products of k distinct primes have A001221(n) = omega(n) = A001222(n) = bigomega(n) = k, whereas the n which have exactly k distinct prime factors are the superset of (weaker) requirement A001221(n) = omega(n) = k. - R. J. Mathar, Jul 18 2023

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A259349 (requires squarefree). Subsequence of A273879.
Cf. A364266 (5 distinct factors).
See also A001221, A001222, A005117.
Numbers divisible by d distinct primes: A246655 (d=1), A007774 (d=2), A033992 (d=3), A033993 (d=4), A051270 (d=5), A074969 (d=6), A176655 (d=7), A348072 (d=8), A348073 (d=9).

omega := proc(n)
    nops(numtheory[factorset](n)) ;
end proc:
for k from 1 do
    if omega(k) = 6 then
        if omega(k+1) = 6 then
            if omega(k+2) = 6 then
                print(k) ;
            end if;
        end if;
    end if;
end do:

upto(n) = {my(res = List(), streak = 0); forfactored(i = 2, n, if(#i[2]~ == 6, streak++; if(streak >= 3, listput(res, i[1] - 2)), streak = 0)); res} \\ David A. Corneth, Jul 18 2023

a(1) = A138206(3).

{k: A001221(k) = A001221(k+1) = A001221(k+2) = 6}.

More terms from David A. Corneth, Jul 18 2023

A321496 Numbers k such that k and k+1 have at least 6 but not both exactly 6 distinct prime factors.

Original entry on oeis.org

58524465, 64822394, 130578734, 133595384, 143489709, 153124685, 155197965, 156532089, 157955720, 159653409, 165706904, 169075829, 170118234, 175458920, 184867605, 188377475, 193042394, 194236679, 195719810, 199150665, 202429590, 208727805, 209801514, 211588454

Offset: 1

M. F. Hasler, Nov 12 2018

nonn

Complement of A273879 (k and k+1 have exactly 6 distinct prime factors) in A321506 (k and k+1 have at least 6 distinct prime factors).

Cf. A273879, A321506; A321495 (analog for 5 factors).

aQ[n_]:=Module[{v={PrimeNu[n], PrimeNu[n+1]}}, Min[v]>5 && v!={6, 6}]; Select[Range[10^9], aQ] (* Amiram Eldar, Nov 12 2018 *)

is(n)=omega(n)>5&&omega(n+1)>5&&(omega(n)>6||omega(n+1)>6)

A321506 \ A273879.

More terms from Amiram Eldar, Nov 12 2018

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A074851 Numbers k such that k and k+1 both have exactly 2 distinct prime factors.

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A321506 Numbers m such that m and m+1 each have at least 6 distinct prime factors.

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A364265 The first term in a chain of at least 3 consecutive numbers each with exactly 6 distinct prime factors (i.e., belonging to A074969).

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A321496 Numbers k such that k and k+1 have at least 6 but not both exactly 6 distinct prime factors.

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Mathematica

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