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
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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:
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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
A087977
a(n) is the first term in the first chain of at least n consecutive numbers each having exactly four distinct prime factors.
Original entry on oeis.org
210, 7314, 37960, 134043, 357642, 1217250, 1217250, 14273478, 44939642, 76067298, 163459742, 547163235, 2081479430, 2771263512, 11715712410, 17911205580, 56608713884, 118968284928, 118968284928, 585927201062, 585927201062, 585927201062, 585927201062
Offset: 1
a(6) = a(7) = 1217250 because the relevant 7 successive numbers have 4 distinct prime factors:
1217250 = 2 * 3^2 * 5^3 * 541;
1217251 = 7 * 17 * 53 * 193;
1217252 = 2^2 * 23 * 101 * 131;
1217253 = 3 * 47 * 89 * 97;
1217254 = 2 * 19 * 103 * 311;
1217255 = 5 * 13 * 61 * 307;
1217256 = 2^3 * 3 * 67 * 757.
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k=1; Do[While[Union[Table[Length[FactorInteger[i]], {i, k, k+n-1}]]!={4}, k++ ]; Print[k], {n, 1, 8}]
Module[{d4=Table[If[PrimeNu[n]==4,1,0],{n,143*10^5}]},Flatten[Table[ SequencePosition[d4,PadRight[{},n,1],1],{n,8}],1][[All,1]]] (* Requires Mathematica version 10 or later *) (* This generates the first 8 terms of the sequence *) (* Harvey P. Dale, Aug 25 2017 *)
A087978
a(n) is the first term in a chain of at least n consecutive numbers, each having exactly m = 5 distinct prime factors.
Original entry on oeis.org
2310, 254540, 1042404, 21871365, 129963314, 830692265, 4617927894, 18297409143, 41268813542, 287980277114, 1182325618032, 6455097761454, 14207465691240, 54049709480208, 90987640183352, 546525829796442, 546525829796442
Offset: 1
-
k=1; Do[While[Union[Table[Length[FactorInteger[i]], {i, k, k+n-1}]]!={5}, k++ ]; Print[k], {n, 1, 8}]
A273879
Numbers k such that k and k+1 have 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
Offset: 1
13516580 = 2^2 * 5 * 7 * 11 * 67 * 131 and 13516581 = 3 * 13 * 17 * 19 * 29 * 37 so 13516580 is in this sequence.
- Charles R Greathouse IV, 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 Erdős on consecutive integers, arXiv:0803.2636 [math.NT], 2008.
- D. A. Goldston, S. W. Graham, J. Pintz, C. Y. Yildirim, Small gaps between almost primes, the parity problem and some conjectures of Erdős on consecutive integers, International Mathematics Research Notices 7 (2011), pp. 1439-1450.
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SequencePosition[PrimeNu[Range[3265*10^4]],{6,6}][[All,1]] (* Harvey P. Dale, Nov 20 2021 *)
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is(n)=omega(n)==6 && omega(n+1)==6
A138207
a(n) is the first term in a chain of at least n consecutive numbers each with exactly 7 distinct prime factors.
Original entry on oeis.org
510510, 965009045, 30989984674, 1673602584618, 66298619667606
Offset: 1
A154573
a(n) is the first term in a chain of at least n consecutive numbers each with exactly 8 distinct prime factors.
Original entry on oeis.org
9699690, 65893166030, 10042712381260, 1126299564879684
Offset: 1
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