A115186
Smallest number m such that m and m+1 have exactly n prime factors (counted with multiplicity).
Original entry on oeis.org
2, 9, 27, 135, 944, 5264, 29888, 50624, 203391, 3290624, 6082047, 32535999, 326481920, 3274208000, 6929459199, 72523096064, 37694578688, 471672487935, 11557226700800, 54386217385983, 50624737509375, 275892612890624, 4870020829413375, 68091093855502335, 2280241934368767, 809386931759611904, 519017301463269375
Offset: 1
a(10) = 3290624 = 6427 * 2^9, 3290624+1 = 13 * 5^5 * 3^4:
A001222(3290624) = A001222(3290625) = 10.
- J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 135, p. 46, Ellipses, Paris 2008.
-
f:= proc(n) uses priqueue; local t,x,p,i;
initialize(pq);
insert([-3^n, 3$n], pq);
do
t:= extract(pq);
x:= -t[1];
if numtheory:-bigomega(x-1)=n then return x-1
elif numtheory:-bigomega(x+1)=n then return x
fi;
p:= nextprime(t[-1]);
for i from n+1 to 2 by -1 while t[i] = t[-1] do
insert([t[1]*(p/t[-1])^(n+2-i), op(t[2..i-1]), p$(n+2-i)], pq)
od;
od
end proc:
seq(f(i),i=1..27); # Robert Israel, Sep 30 2024
A356855
a(n) is the least number m such that u defined by u(i) = bigomega(m + 2i) satisfies u(i) = u(0) for 0 <= i < n and u(n) != u(0), or -1 if no such number exists.
Original entry on oeis.org
1, 4, 3, 215, 213, 1383, 3091, 8129, 151403, 151401, 2560187, 33396293, 33396291, 56735777, 1156217487, 2514196079
Offset: 1
Let u be defined by u(i) = bigomega(3 + 2i). u(i) = 1 for 0 <= i < 3 and u(3) = 2 != 1, and 3 is the smallest such number, hence a(3) = 3.
Let u be defined by u(i) = bigomega(4 + 2i). u(i) = 2 for 0 <= i < 2 and u(3) = 3 != 2 , and 4 is the smallest such number, hence a(2) = 4.
Let u be defined by u(i) = bigomega(151403 + 2i). u(i) = 3 for 0 <= i < 9 and u(9) = 2 != 3, and 151403 is the smallest such number, hence a(9) = 151403.
-
u(m,i)=bigomega(m+2*i)
card(m)=my(k=u(m,0),c=0);while(u(m,c)==k,c++);c
a(n)=my(c=0);for(m=1,+oo,c=card(m);if(c==n,return(m)))
A356953
Least nonzero starting number in the first run of exactly n consecutive numbers having the same number of prime factors counted with multiplicity, or -1 if no such number exists.
Original entry on oeis.org
1, 2, 33, 1083, 602, 2522, 211673, 6612470, 3405122, 49799889, 202536181, 3195380868, 5208143601, 85843948321, 97524222465, 361385490681003, 441826936079342
Offset: 1
2 and 3 are 2 consecutive numbers and have the same number of prime factors, and 2 is the smallest such number, hence a(2) = 2.
-
card(m)=my(c=0,k=bigomega(m));if(bigomega(m-1)!=k,while(bigomega(m)==k,c++;m++));c
a(n)=if(n==1,return(1));for(m=2,+oo,if(card(m)==n,return(m)))
A357038
Numbers m such that each of the four consecutive numbers starting at m is the product of 8 prime factors (counting with multiplicity).
Original entry on oeis.org
4109290623, 10440390750, 24239110623, 63390659373, 66169625247, 67492525373, 72177640623, 74735721872, 88651359872, 97510501023, 99039940623
Offset: 1
4109290623 = 3^6*79*71353,
4109290624 = 2^7*32103833,
4109290625 = 5^5*11*173*691,
4109290626 = 2*3*7^2*17*19*109*397.
A387505
Smallest number m such that m, m+1, m+2, m+3 and m+4 have exactly n prime factors (counted with multiplicity).
Original entry on oeis.org
602, 12122, 632148, 101905622, 15605704374, 1300091574624, 431094129471872
Offset: 3
a(4) = 12122 = 2*11*19*29, a(4)+1 = 12123 = 3^3*449, a(4)+2 = 12124 = 2^2*7*433, a(4)+3 = 12125 = 5^3*97, a(4)+4 = 12126 = 2*3*43*47 (all products of four prime factors).
Showing 1-5 of 5 results.
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