A321502 -id:A321502 - 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

A321489 Numbers m such that both m and m+1 have at least 7 distinct prime factors.

Original entry on oeis.org

965009045, 1068044054, 1168008204, 1177173074, 1209907985, 1218115535, 1240268490, 1338753129, 1344185205, 1408520805, 1477640450, 1487720234, 1509981395, 1663654629, 1693460405, 1731986894, 1758259425, 1819458354, 1821278459, 1826445984, 1857332840

Offset: 1

Amiram Eldar and M. F. Hasler, Nov 12 2018

nonn

The first 300 terms of this sequence are such that m and m+1 both have exactly 7 prime divisors. See A321497 for the terms m such that m or m+1 has more than 7 prime factors: the smallest such term is 5163068910.
Numbers m and m+1 can never have a common prime factor (consider them mod p), therefore the terms are > sqrt(p(7+7)#) = A003059(A002110(7+7)). (Here we see that sqrt(p(7+8)#) is a more realistic estimate of a(1), but for smaller values of k we may have sqrt(p(2k+1)#) > m(k) > sqrt(p(2k)#), where m(k) is the smallest of two consecutive integers each having at least k prime divisors. For example, A321503(1) < sqrt(p(3+4)#) ~ A321493(1).)
From M. F. Hasler, Nov 28 2018: (Start)
The first 100 terms and beyond are all congruent to one of {14, 20, 35, 49, 50, 69, 84, 90, 104, 105, 110, 119, 125, 129, 134, 140, 144, 170, 174, 189, 195} mod 210. Here, 35, 195, 189, 14 140, 20 and 174 (in order of decreasing frequency) occur between 6 and 13 times, and {49, 50, 110, 129, 134, 144, 170} occur only once.
However, as observed by Charles R Greathouse IV, one can construct a term of this sequence congruent to any given m > 0, modulo any given n > 0.
The first terms of this sequence which are multiples of 210 are in A321497. An example of a term that is a multiple of 210 but not in A321497 is 29759526510, due to Charles R Greathouse IV. Such examples can be constructed by solving A*210 + 1 = B for A having 3 distinct prime factors not among {2, 3, 5, 7}, B having 7 distinct prime factors and gcd(B, 210*A) = 1. (End)

			a(1) = 5 * 7 * 11 * 13 * 23 * 83 * 101, a(1)+1 = 2 * 3 * 17 * 29 * 41 * 73 * 109.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 102 terms from M. F. Hasler)

Cf. A255346, A321503 .. A321506 (analog for k = 2, ..., 6 prime divisors).
Cf. A321502, A321493 .. A321497 (m and m+1 have at least but not both exactly k = 2, ..., 7 prime divisors).
Cf. A074851, A140077, A140078, A140079 (m and m+1 both have exactly k = 2, 3, 4, 5 prime divisors).
Cf. A006049, A006549, A093548.
Cf. A002110.

Select[Range[36000000], PrimeNu[#] > 6 && PrimeNu[# + 1] > 6 &]

is(n)=omega(n)>6&&omega(n+1)>6
A321489=List();for(n=965*10^6,1.8e9,is(n)&&listput(A321489,n))

a(n) ~ n. - Charles R Greathouse IV, Nov 29 2018

cp's OEIS Frontend

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

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