[seqfan] Re: [math-fun] Numbers Aplenty

[Olivier Gerard]

Dear Seqfans,

On the math-fun discussion list, James Propp asked

What is the smallest value of n such that n+1 appears in more of
> the increasing sequences in the OEIS than n does?
> The reason I want to restrict attention to increasing sequences in the OEIS
> is that these correspond to interesting subsets of the positive integers. I
> suppose if anyone wants to answer my question with the word "increasing"
> omitted, I'd be interested in that too. Conjecture: The n that you get is
> the same for both versions of my question. Refined conjecture: in both
> cases, n is 11.


It was proposed to him that he joins seqfan and ask the question here.

I couldn't resist doing the computations myself.

Here are two graphics, based on the latest version of the database (January
4th 2014):

http://list.seqfan.eu/seqfans/olivier/OEIS_Frequencies_A.jpg

http://list.seqfan.eu/seqfans/olivier/OEIS_Frequencies_B.jpg

The horizontal scale is the integer under consideration
The vertical scale is logarithmic in base 10. It counts number of
apparitions in the OEIS sequences.
The upper curve is total apparitions in the stripped down version (the
first 3 lines for each sequence when available.)
The middle curve is the same for increasing sequences (after removing signs)
The lower curve is the same for strictly increasing sequences (after
removing signs)


Olivier

​Raw data (from 0 to 100, y-coordinates of each curve):

{{601621,39420,13687},{864198,117890,55500},{464087,50809,31725},{341000,42130,25098},{292704,36359,20578},{243521,33965,20092},{223598,30092,17380},{198621,28804,17858},{191759,25350,14853},{171932,23733,14187},{77475,20979,13224},{70794,22058,14699},{70521,19816,12692},{58798,19718,13050},{48073,15613,10211},{50902,16975,11024},{57752,17835,11269},{44147,17014,11693},{41275,14398,9725},{40192,16026,11314},{40556,14397,9964},{38703,14910,10197},{31788,12802,9084},{33746,14622,10787},{39539,14637,10189},{31437,13032,9357},{26774,11439,8286},{28370,12048,8735},{29996,11682,8661},{28940,13096,10100},{31545,12379,9124},{29626,13533,10443},{30311,11897,8589},{22845,10681,8225},{22619,10102,7641},{23732,10061,7835},{30560,11975,9217},{24972,12199,9825},{18603,8907,7088},{18721,9171,7316},{23604,10145,8057},{22989,11521,9434},{23154,9831,7836},{21326,10980,9127},{18837,8743,7162},{20340,9509,7681},{17241,8266,6799},{19696,10484,8727},{22422,9613,7842},{18847,9347,7692},{17662,8678,7324},{15732,8254,7039},{16529,8056,6771},{18441,9965,8542},{17221,8375,7069},{18464,8941,7527},{19625,8821,7431},{15242,8127,6906},{14527,7539,6436},{16641,9429,8261},{20172,9059,7666},{17459,9762,8567},{13673,7340,6362},{15871,8210,7083},{21238,9553,7732},{14275,7933,6876},{14530,7938,6959},{14968,9159,8135},{11905,7050,6246},{11644,7132,6343},{14028,7563,6678},{15146,9167,8168},{15900,8288,7204},{14636,9190,8283},{10121,6669,5977},{10997,7052,6280},{10732,6799,6121},{11025,7001,6273},{10898,6812,6123},{12531,8431,7655},{12694,7434,6596},{13401,7872,6868},{9218,6284,5687},{11686,8074,7409},{12596,7091,6263},{10180,6773,5981},{8514,5981,5435},{8658,6140,5552},{9798,6365,5705},{12748,8358,7330},{11717,6961,6182},{10454,6804,6022},{8269,5565,5069},{8219,5802,5259},{7812,5609,5075},{8076,5604,5149},{11367,6491,5757},{11145,7738,6990},{7880,5431,4948},{9100,6017,5413},{11002,6892,6133}}