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Search: a321494 -id:a321494
Displaying 1-7 of 7 results found. page 1
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A140078 Numbers k such that k and k+1 have 4 distinct prime factors. +10
19
7314, 8294, 8645, 9009, 10659, 11570, 11780, 11934, 13299, 13629, 13845, 14420, 15105, 15554, 16554, 16835, 17204, 17390, 17654, 17765, 18095, 18290, 18444, 18920, 19005, 19019, 19095, 19227, 20349, 20405, 20769, 21164, 21489, 21735 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Goldston, Graham, Pintz, & Yildirim prove that this sequence is infinite. - Charles R Greathouse IV, Jun 02 2016
The subsequence of terms where k and k+1 are also squarefree is A318896. - R. J. Mathar, Jul 15 2023
REFERENCES
David Wells, The Penguin Dictionary of Curious and Interesting Numbers (Rev. ed. 1997), p. 161 (entry for 7314).
LINKS
Seiichi Manyama, Table of n, a(n) for n = 1..10000
D. A. Goldston, S. W. Graham, J. Pintz and C. Y. Yildirim, Small gaps between almost primes, the parity problem and some conjectures of Erdos on consecutive integers, arXiv:0803.2636 [math.NT], 2008.
FORMULA
{k: k in A033993 and k+1 in A033993}. - R. J. Mathar, Jul 19 2023
MATHEMATICA
a = {}; Do[If[Length[FactorInteger[n]] == 4 && Length[FactorInteger[n + 1]] == 4, AppendTo[a, n]], {n, 1, 100000}]; a (* Artur Jasinski, May 07 2008 *)
Transpose[Position[Partition[PrimeNu[Range[20000]], 2, 1], _?(#[[1]] == #[[2]] == 4&), {1}, Heads->False]][[1]] (* Harvey P. Dale, Jun 21 2013 *)
SequencePosition[PrimeNu[Range[22000]], {4, 4}][[;; , 1]] (* Harvey P. Dale, Jun 20 2024 *)
PROG
(PARI) isok(n) = (omega(n)==4) && (omega(n+1)==4); \\ Michel Marcus, Sep 04 2015
CROSSREFS
Similar sequences with k distinct prime factors: A074851 (k=2), A140077 (k=3), this sequence (k=4), A140079 (k=5).
Cf. A093548.
Equals A321504 \ A321494.
KEYWORD
nonn
AUTHOR
Artur Jasinski, May 07 2008
EXTENSIONS
Link provided by Harvey P. Dale, Jun 21 2013
STATUS
approved
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. +10
7
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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!
LINKS
M. F. Hasler, Table of n, a(n) for n = 1..5000
FORMULA
A321503 \ A140077.
MATHEMATICA
aQ[n_]:=Module[{v={PrimeNu[n], PrimeNu[n+1]}}, Min[v]>2 && v!={3, 3}]; Select[Range[120000], aQ] (* Amiram Eldar, Nov 12 2018 *)
PROG
(PARI) select( is(n)=omega(n)>2&&omega(n+1)>2&&(omega(n)>3||omega(n+1)>3), [1..1300])
CROSSREFS
Cf. A140077, A321503.
Cf. A321494, A321495, A321496, A321497 (analog for 4, 5, 6, 7 factors).
KEYWORD
nonn
AUTHOR
M. F. Hasler, Nov 13 2018
STATUS
approved
A321504 Numbers k such that k and k+1 each have at least 4 distinct prime factors. +10
7
7314, 8294, 8645, 9009, 10659, 11570, 11780, 11934, 13299, 13629, 13845, 14420, 15105, 15554, 16554, 16835, 17204, 17390, 17654, 17765, 18095, 18290, 18444, 18920, 19005, 19019, 19095, 19227, 20349, 20405, 20769, 21164, 21489, 21735, 22010, 22154, 22659, 23001, 23114, 23484, 23529, 23540, 23919, 24395 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Equals A140078 up to a(123) but a({124, 214, 219, 276, 321, 415, ...}) = { 38570, 51414, 51765, 58695, 62985, 71070, ...} are not in A140078, see A321494.
LINKS
M. F. Hasler, Table of n, a(n) for n = 1..10000
MATHEMATICA
SequencePosition[Table[If[PrimeNu[n]>3, 1, 0], {n, 25000}], {1, 1}][[All, 1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Apr 29 2019 *)
PROG
(PARI) is(n)=omega(n)>=4&&omega(n+1)>=4
CROSSREFS
Cf. A140078, A321494.
Cf. A321505, A321506 (variant for k=5 & k=6 prime factors).
KEYWORD
nonn
AUTHOR
M. F. Hasler, Nov 12 2018
STATUS
approved
A321495 Numbers k such that k and k+1 have at least 5 but not both exactly 5 distinct prime factors. +10
6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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).
LINKS
Table of n, a(n) for n=1..30.
FORMULA
A321505 \ A140079.
MATHEMATICA
aQ[n_]:=Module[{v={PrimeNu[n], PrimeNu[n+1]}}, Min[v]>4 && v!={5, 5}]; Select[Range[120000], aQ] (* Amiram Eldar, Nov 12 2018 *)
PROG
(PARI) is(n)=vecmin(n=[omega(n), omega(n+1)])>4&&n!=[5, 5]
CROSSREFS
Cf. A140079, A321505; A321494, A321496 (analog for 4 & 6 factors).
KEYWORD
nonn
AUTHOR
M. F. Hasler, Nov 12 2018
STATUS
approved
A321503 Numbers m such that m and m+1 both have at least 3 distinct prime factors. +10
6
230, 285, 429, 434, 455, 494, 560, 594, 609, 615, 644, 645, 650, 665, 714, 740, 741, 759, 804, 805, 819, 825, 854, 860, 884, 902, 935, 945, 969, 986, 987, 1001, 1014, 1022, 1034, 1035, 1044, 1064, 1065, 1070, 1085, 1104, 1105, 1130, 1196, 1209, 1220, 1221, 1235, 1239, 1245, 1265 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Disjoint union of A140077 (omega({m, m+1}} = {3}) and A321493 (not both have exactly 3 prime divisors). The latter contains terms with indices {15, 60, 82, 98, 99, 104, ...} of this sequence.
Numbers m and m+1 can never have a common prime factor (consider them mod p), therefore the terms are > sqrt(A002110(3+3)), A002110 = primorial.
LINKS
M. F. Hasler, Table of n, a(n) for n = 1..10000
MATHEMATICA
aQ[n_]:=Module[{v={PrimeNu[n], PrimeNu[n+1]}}, Min[v]>2]; Select[Range[1300], aQ] (* Amiram Eldar, Nov 12 2018 *)
PROG
(PARI) select( is(n)=omega(n)>2&&omega(n+1)>2, [1..1300])
CROSSREFS
Cf. A255346, A321504 .. A321506, A321489 (analog for k = 2, ..., 7 prime divisors).
Cf. A321493, A321494 .. A321497 (subsequences of the above: m or m+1 has more than k prime divisors).
Cf. A074851, A140077, A140078, A140079 (complementary subsequences: m and m+1 have exactly k = 2, 3, 4, 5 prime divisors).
KEYWORD
nonn
AUTHOR
M. F. Hasler, Nov 13 2018
STATUS
approved
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. +10
4
5163068910, 5327923964, 6564937379, 6880516929, 7122669554, 8567026545, 8814635115, 9533531370, 9611079114, 10245081314, 10246336814, 10697507414, 10783550414, 10796559410, 11260076190, 11458770609, 11992960265, 12043540145, 12172828590, 12745759740, 12850545785, 12946979220 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Terms of A321489 (k and k+1 have at least 7 distinct prime factors) which don't satisfy the definition with "exactly 7".
LINKS
Table of n, a(n) for n=1..22.
MATHEMATICA
aQ[n_]:=Module[{v={PrimeNu[n], PrimeNu[n+1]}}, Min[v]>6 && v!={7, 7}]; Select[Range[10^10], aQ]
PROG
(PARI) is(n)=omega(n)>6&&omega(n+1)>6&&(omega(n)>7||omega(n+1)>7)
CROSSREFS
Cf. A321489, A321503, A321504, A321505, A321506, A321493, A321494, A321495, A321496 (analog for 3 .. 6 factors).
KEYWORD
nonn
AUTHOR
Amiram Eldar and M. F. Hasler, Nov 13 2018
STATUS
approved
A321502 Numbers m such that m and m+1 have at least 2, but m or m+1 has at least 3 prime divisors. +10
2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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.
LINKS
Table of n, a(n) for n=1..58.
FORMULA
Equals A255346 \ A074851.
PROG
(PARI) select( is_A321502(n)=vecmax(n=[omega(n), omega(n+1)])>2&&vecmin(n)>1, [1..500])
CROSSREFS
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.
KEYWORD
nonn
AUTHOR
M. F. Hasler, Nov 27 2018
STATUS
approved
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Last modified August 29 06:09 EDT 2024. Contains 375510 sequences. (Running on oeis4.)