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A045920
Numbers m such that the factorizations of m..m+1 have the same number of primes (including multiplicities).
40
2, 9, 14, 21, 25, 27, 33, 34, 38, 44, 57, 75, 85, 86, 93, 94, 98, 116, 118, 121, 122, 124, 133, 135, 141, 142, 145, 147, 153, 158, 164, 170, 171, 174, 177, 201, 202, 205, 213, 214, 217, 218, 230, 244, 245, 253, 284, 285, 296, 298, 301, 302, 326, 332, 334, 350, 356, 361
OFFSET
1,1
COMMENTS
A115186 is a subsequence: A001222(A115186(n)) = A001222(A115186(n)+1) = n. - Reinhard Zumkeller, Jan 16 2006
Indices k such that A076191(k) = 0. - Ray Chandler, Dec 10 2008
A045939 is a subsequence. - Zak Seidov, Jul 02 2020
This sequence is infinite (Heath-Brown, 1984). - Amiram Eldar, Jul 11 2020
REFERENCES
C. Clawson, Mathematical mysteries, Plenum Press 1996, p. 250.
LINKS
D. R. Heath-Brown, A parity problem from sieve theory, Mathematika, Vol. 29, No. 1 (1982), pp. 1-6.
D. R. Heath-Brown, The divisor function at consecutive integers, Mathematika, Vol. 31, No. 1 (1984), pp. 141-149.
Adolf Hildebrand, The divisor function at consecutive integers, Pacific journal of mathematics, Vol. 129, No. 2 (1987), pp. 307-319.
FORMULA
a(n) = A278291(n) - 1. - Zak Seidov, Nov 17 2018
MATHEMATICA
f[n_]:=Plus@@Last/@FactorInteger[n]; lst={}; Do[If[f[n]==f[n+1], AppendTo[lst, n]], {n, 0, 6!}]; lst (* Vladimir Joseph Stephan Orlovsky, May 12 2010 *)
Transpose[Transpose[Select[Partition[Table[{n, PrimeOmega[n]}, {n, 400}], 2, 1], #[[1, 2]]==#[[2, 2]]&]][[1]]][[1]] (* Harvey P. Dale, Feb 21 2012 *)
Position[Differences[PrimeOmega[Range[400]]], 0] // Flatten (* Zak Seidov, Oct 30 2012 *)
PROG
(Haskell)
import Data.List (elemIndices)
a045920 n = a045920_list !! (n-1)
a045920_list = map (+ 1) $ elemIndices 0 a076191_list
-- Reinhard Zumkeller, Mar 23 2012, Oct 11 2011
(PARI) is(n)=bigomega(n)==bigomega(n+1) \\ Charles R Greathouse IV, Sep 14 2015
CROSSREFS
Numbers m through m+k have the same number of prime divisors (with multiplicity): this sequence (k=1), A045939 (k=2), A045940 (k=3), A045941 (k=4), A045942 (k=5), A123103 (k=6), A123201 (k=7), A358017 (k=8), A358018 (k=9), A358019 (k=10).
Sequence in context: A288483 A353308 A304807 * A242466 A071344 A224855
KEYWORD
nice,nonn
AUTHOR
EXTENSIONS
More terms from David W. Wilson
STATUS
approved