The OEIS is supported by the many generous donors to the OEIS Foundation.



Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 60, we have over 367,000 sequences, and we’ve crossed 11,000 citations (which often say “discovered thanks to the OEIS”).

Other ways to Give
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A240670 Numbers n for which all exponents in the prime power factorization of (2*n)! are odious (A000069). 13
1, 3, 4, 5, 15, 16 (list; graph; refs; listen; history; text; internal format)
The next term, if it exists, must be more than 45000. - Peter J. C. Moses, Apr 11 2014
The sequence is finite.
Proof. For sufficiently large n, we always have a prime in (n/4, n/3]. Such primes p divide n! and, at the same time, for them we have 3<=n/p<4. Thus floor(n/p)=3, and in case sqrt(n)<n/4, floor(n/p^2)=0. Therefore, they involve in n! with exponent 3. Since 3 is evil, we are done. Moreover, numerically, using estimate for (4/3)-Ramanujan number (see Shevelev, Greathouse IV, and Moses link, Proposition 8), it is sufficient to consider n>=93 in order for the above arguments to be true. So 16 is the last term of the sequence. - Vladimir Shevelev, Apr 11 2014
D. Berend, G. Kolesnik, Regularity of patterns in the factorization of n!, J. Number Theory, 124 (2007), no. 1, 181-192.
Vladimir Shevelev, Charles R. Greathouse IV, Peter J. C. Moses, On intervals (kn, (k+1)n) containing a prime for all n>1, Journal of Integer Sequences, Vol. 16 (2013), Article 13.7.3. arXiv:1212.2785
32! = 2^31*3^14*5^7*7^4*11^2*13^2*17*19*23*29*31, and all exponents: 31,14,7,4,2,2,1,1,1,1,1 are odious, so 16 is in the sequence.
odiousQ[n_] := OddQ[DigitCount[n, 2][[1]]];
For[n = 1, True, n++, If[AllTrue[FactorInteger[(2 n)!][[All, 2]], odiousQ], Print[n]]] (* Jean-François Alcover, Sep 20 2018 *)
(PARI) isok(n) = {f = factor((2*n)!); sum(i=1, #f~, hammingweight(f[i, 2]) % 2) == #f; } \\ Michel Marcus, Apr 11 2014
Sequence in context: A346599 A330138 A051530 * A048040 A347391 A051033
Vladimir Shevelev, Apr 10 2014

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 1 23:26 EST 2023. Contains 367503 sequences. (Running on oeis4.)