A321496 -id:A321496 - cp's OEIS frontend

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

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
```

A321493 Numbers m such that m and m+1 both have at least 3, but m or m+1 has at least 4 distinct prime factors.

Original entry on oeis.org

714, 1364, 1595, 1770, 1785, 1869, 2001, 2090, 2145, 2184, 2210, 2261, 2345, 2379, 2414, 2639, 2805, 2820, 2849, 2870, 2925, 3002, 3009, 3059, 3080, 3219, 3255, 3289, 3354, 3366, 3444, 3450, 3485, 3534, 3654, 3689, 3705

Offset: 1

M. F. Hasler, Nov 13 2018

nonn

A321503 lists numbers m such that m and m+1 both have at least 3 distinct prime factors, while A140077 lists numbers such that m and m+1 have exactly 3 distinct prime factors. This sequence is the complement of the latter in the former, it consists of terms with indices (15, 60, 82, 98, 99, 104, ...) of the former.

Since m and m+1 can't share a prime factor, we have a(n)*(a(n)+1) >= p(3+4)# = A002110(7). Remarkably enough, a(1) = A000196(A002110(3+4)) exactly!

M. F. Hasler, Table of n, a(n) for n = 1..5000

Cf. A140077, A321503.

Cf. A321494, A321495, A321496, A321497 (analog for 4, 5, 6, 7 factors).

aQ[n_]:=Module[{v={PrimeNu[n],PrimeNu[n+1]}},Min[v]>2 && v!={3,3}]; Select[Range[120000], aQ] (* Amiram Eldar, Nov 12 2018 *)
dpfQ[{a_,b_}]:=a>2&&b>2&&(a>3||b>3); Position[Partition[PrimeNu[Range[4000]],2,1],?dpfQ]//Flatten (* _Harvey P. Dale, Apr 21 2025 *)

select( is(n)=omega(n)>2&&omega(n+1)>2&&(omega(n)>3||omega(n+1)>3), [1..1300])

A321503 \ A140077.

A321494 Numbers k such that k and k+1 have at least 4 but not both exactly 4 distinct prime factors.

Original entry on oeis.org

38570, 40754, 51414, 51765, 58695, 60605, 62985, 66044, 68585, 70889, 71070, 73185, 73814, 74865, 77349, 82004, 83265, 83720, 83979, 85085, 87009, 90804, 90915, 91805, 91884, 92378, 94094, 94829, 96459, 97565, 98769, 98889, 100814, 101269, 101660, 104005, 104754, 105468, 107184, 108030, 108185, 108965

Offset: 1

M. F. Hasler, Nov 12 2018

nonn

A321504 lists numbers n such that k and k+1 both have at least 4 distinct prime factors, while A140078 lists numbers such that k and k+1 have exactly 4 distinct prime factors. This sequence is the complement of the latter in the former, it consists of terms with indices (124, 214, 219, 276, 321, 415, ...) of the former.

Cf. A140078, A321504; A321493, A321496 (analog for 3 & 5 factors).

aQ[n_]:=Module[{v={PrimeNu[n],PrimeNu[n+1]}},Min[v]>3 && v!={4,4}]; Select[Range[120000], aQ] (* Amiram Eldar, Nov 12 2018 *)

is(n)=vecmin(n=[omega(n),omega(n+1)])>=4&&n!=[4,4]

A321504 \ A140078.

A321495 Numbers k such that k and k+1 have at least 5 but not both exactly 5 distinct prime factors.

Original entry on oeis.org

728364, 1565564, 1774409, 1817529, 1923635, 2162094, 2187185, 2199834, 2225894, 2369850, 2557190, 2594514, 2659734, 2671305, 2794154, 2944689, 2964884, 3126045, 3139730, 3170244, 3244955, 3273809, 3279639, 3382379, 3387054, 3506810, 3555110, 3585945, 3686969, 3711630

Offset: 1

M. F. Hasler, Nov 12 2018

nonn

Complement of A140079 (k and k+1 have exactly 5 distinct prime factors) in A321505 (k and k+1 have at least 5 distinct prime factors).

Cf. A140079, A321505; A321494, A321496 (analog for 4 & 6 factors).

aQ[n_]:=Module[{v={PrimeNu[n],PrimeNu[n+1]}},Min[v]>4 && v!={5,5}]; Select[Range[120000], aQ] (* Amiram Eldar, Nov 12 2018 *)

is(n)=vecmin(n=[omega(n), omega(n+1)])>4&&n!=[5,5]

A321505 \ A140079.

A321497 Numbers k such that both k and k+1 have at least 7 distinct prime factors and at least one has more than 7 distinct prime factors.

Original entry on oeis.org

5163068910, 5327923964, 6564937379, 6880516929, 7122669554, 8567026545, 8814635115, 9533531370, 9611079114, 10245081314, 10246336814, 10697507414, 10783550414, 10796559410, 11260076190, 11458770609, 11992960265, 12043540145, 12172828590, 12745759740, 12850545785, 12946979220

Offset: 1

Amiram Eldar and M. F. Hasler, Nov 13 2018

nonn

Terms of A321489 (k and k+1 have at least 7 distinct prime factors) which don't satisfy the definition with "exactly 7".

Cf. A321489, A321503, A321504, A321505, A321506, A321493, A321494, A321495, A321496 (analog for 3 .. 6 factors).

aQ[n_]:=Module[{v={PrimeNu[n], PrimeNu[n+1]}}, Min[v]>6 && v!={7, 7}]; Select[Range[10^10], aQ]

is(n)=omega(n)>6&&omega(n+1)>6&&(omega(n)>7||omega(n+1)>7)

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.

cp's OEIS Frontend

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

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A321493 Numbers m such that m and m+1 both have at least 3, but m or m+1 has at least 4 distinct prime factors.

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A321494 Numbers k such that k and k+1 have at least 4 but not both exactly 4 distinct prime factors.

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A321495 Numbers k such that k and k+1 have at least 5 but not both exactly 5 distinct prime factors.

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A321497 Numbers k such that both k and k+1 have at least 7 distinct prime factors and at least one has more than 7 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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