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Search: a321493 -id:a321493
Displaying 1-6 of 6 results found. page 1
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A140077 Numbers n such that n and n+1 have 3 distinct prime factors. +10
16
230, 285, 429, 434, 455, 494, 560, 594, 609, 615, 644, 645, 650, 665, 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 (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, Sep 14 2015
See A321503 for numbers n such that n & n+1 have at least 3 prime divisors, disjoint union of this and A321493, the terms of A321503 which are not in this sequence. A321493 has A140078 as a subsequence, which in turn is subsequence of A321504, and so on. Since n and n+1 can't share a prime factor, we have a(1) > sqrt(p(3+3)#) > A000196(A002110(3+3)). Note that A000196(A002110(3+4)) = A321493(1) exactly! - M. F. Hasler, Nov 13 2018
LINKS
Harvey P. Dale, Table of n, a(n) for n = 1..10000
D. A. Goldston, S. W. Graham, J. Pintz, 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 A033992 and k+1 in A033992}. - R. J. Mathar, Jul 19 2023
MATHEMATICA
a = {}; Do[If[Length[FactorInteger[n]] == 3 && Length[FactorInteger[n + 1]] == 3, AppendTo[a, n]], {n, 1, 100000}]; a (*Artur Jasinski*)
SequencePosition[PrimeNu[Range[1250]], {3, 3}][[All, 1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Feb 27 2017 *)
PROG
(PARI) is(n)=omega(n)==3&&omega(n+1)==3 \\ Charles R Greathouse IV, Sep 14 2015
CROSSREFS
Cf. A074851, A140078, A140079.
Equals A321503 \ A321493.
KEYWORD
nonn
AUTHOR
Artur Jasinski, May 07 2008
STATUS
approved
A321494 Numbers k such that k and k+1 have at least 4 but not both exactly 4 distinct prime factors. +10
7
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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.
LINKS
Table of n, a(n) for n=1..42.
FORMULA
A321504 \ A140078.
MATHEMATICA
aQ[n_]:=Module[{v={PrimeNu[n], PrimeNu[n+1]}}, Min[v]>3 && v!={4, 4}]; Select[Range[120000], aQ] (* Amiram Eldar, Nov 12 2018 *)
PROG
(PARI) is(n)=vecmin(n=[omega(n), omega(n+1)])>=4&&n!=[4, 4]
CROSSREFS
Cf. A140078, A321504; A321493, A321496 (analog for 3 & 5 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
A321489 Numbers m such that both m and m+1 have at least 7 distinct prime factors. +10
4
965009045, 1068044054, 1168008204, 1177173074, 1209907985, 1218115535, 1240268490, 1338753129, 1344185205, 1408520805, 1477640450, 1487720234, 1509981395, 1663654629, 1693460405, 1731986894, 1758259425, 1819458354, 1821278459, 1826445984, 1857332840 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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)
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 102 terms from M. F. Hasler)
FORMULA
a(n) ~ n. - Charles R Greathouse IV, Nov 29 2018
EXAMPLE
a(1) = 5 * 7 * 11 * 13 * 23 * 83 * 101, a(1)+1 = 2 * 3 * 17 * 29 * 41 * 73 * 109.
MATHEMATICA
Select[Range[36000000], PrimeNu[#] > 6 && PrimeNu[# + 1] > 6 &]
PROG
(PARI) is(n)=omega(n)>6&&omega(n+1)>6
A321489=List(); for(n=965*10^6, 1.8e9, is(n)&&listput(A321489, n))
CROSSREFS
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.
KEYWORD
nonn
AUTHOR
Amiram Eldar and M. F. Hasler, Nov 12 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 09:09 EDT 2024. Contains 375511 sequences. (Running on oeis4.)