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A115186
Smallest number m such that m and m+1 have exactly n prime factors (counted with multiplicity).
16
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
1,1
COMMENTS
A001222(a(n)) = A001222(a(n)+1) = n: subsequence of A045920.
a(16) > 4*10^10. - Martin Fuller, Jan 17 2006
a(n) <= A093548(n) <= A052215(n). - Zak Seidov, Jan 16 2015
Apparently, 4*a(n)+2 is the least number k such that k-2 and k+2 have exactly n+2 prime factors, counted with multiplicity. - Hugo Pfoertner, Apr 02 2024
REFERENCES
J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 135, p. 46, Ellipses, Paris 2008.
EXAMPLE
a(10) = 3290624 = 6427 * 2^9, 3290624+1 = 13 * 5^5 * 3^4:
A001222(3290624) = A001222(3290625) = 10.
MAPLE
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
CROSSREFS
Equivalent sequences for longer runs: A113752 (3), A356893 (4).
Sequence in context: A123904 A327612 A019065 * A243560 A090900 A357721
KEYWORD
nonn
AUTHOR
Reinhard Zumkeller, Jan 16 2006
EXTENSIONS
a(13)-a(15) from Martin Fuller, Jan 17 2006
a(16)-a(17) from Donovan Johnson, Apr 08 2008
a(18)-a(22) from Donovan Johnson, Jan 21 2009
a(23)-a(25) from Donovan Johnson, May 25 2013
a(26)-a(27) from Robert Israel, Sep 30 2024
STATUS
approved