[math-fun] And I missed this Bernoulli bizarreness
A090495 Numbers n such that numerator(Bernoulli(2*n)/(2*n)) is different from numerator(Bernoulli(2*n)/(2*n*(2*n-1))). 12 574, 1185, 1240, 1269, 1376, 1906, 1910, 2572, 2689, 2980, 3238, 3384, 3801, 3904, 4121, Michael Somos <http://oeis.org/wiki/User:Michael_Somos> (Feb 01 2004) discovered the remarkable fact that A001067 <http://oeis.org/A001067> is different from A046968 <http://oeis.org/A046968>, even though they agree for the first 573 terms. Hey, Bernoulli(n+1)/n/(n+1) agrees with Bernoulli(n+1)/(n+1) for 1147 terms! Do[If[Numerator[BernoulliB[n + 1]/n/(n + 1)] != Numerator[BernoulliB[n + 1]/(n + 1)], Print[n.GCD[n, Numerator[BernoulliB[n + 1]/(n + 1)]]]], {n, 3511}] // tim 1147.37 2369.103 2479.37 2537.59 2751.131 8.898235 secs (Don't laugh. You should usually think in terms of Bernoulli *polynomials* .) As http://mathworld.wolfram.com/StirlingsSeries.html points out, Bernoulli(2*n)/(2*n*(2*n-1))) are the coefficients of the (divergent) Stirling series I mentioned yesterday. --rwg Mathematica now has anagrams: "deeeiimnorsstt" -> {"endometritises", "densitometries"}, ... "adegiiinnorstt" -> {"disorientating", "disintegration"}, ... "aeegiinorrsttv" -> {"tergiversation", "interrogatives"}, ... "eiinoopprssstu" -> {"superpositions", "propitiousness"}, ... Except that endometritises should be endometritides, which it doesn't even know: In[903]:= DictionaryWordQ@"endometritides" Out[903]= False
I did not verify it for more terms, but for the values given in Bill's mail the following holds: 2*A090495 - 1 = <the output of Bill's Mathematica program> E.g. 2*574-1=1147 Christoph Originalnachricht Von: Bill Gosper Gesendet: Samstag, 18. Juni 2016 01:50 An: math-fun@mailman.xmission.com Antwort an: math-fun Betreff: [math-fun] And I missed this Bernoulli bizarreness A090495 Numbers n such that numerator(Bernoulli(2*n)/(2*n)) is different from numerator(Bernoulli(2*n)/(2*n*(2*n-1))). 12 574, 1185, 1240, 1269, 1376, 1906, 1910, 2572, 2689, 2980, 3238, 3384, 3801, 3904, 4121, Michael Somos <http://oeis.org/wiki/User:Michael_Somos> (Feb 01 2004) discovered the remarkable fact that A001067 <http://oeis.org/A001067> is different from A046968 <http://oeis.org/A046968>, even though they agree for the first 573 terms. Hey, Bernoulli(n+1)/n/(n+1) agrees with Bernoulli(n+1)/(n+1) for 1147 terms! Do[If[Numerator[BernoulliB[n + 1]/n/(n + 1)] != Numerator[BernoulliB[n + 1]/(n + 1)], Print[n.GCD[n, Numerator[BernoulliB[n + 1]/(n + 1)]]]], {n, 3511}] // tim 1147.37 2369.103 2479.37 2537.59 2751.131 8.898235 secs (Don't laugh. You should usually think in terms of Bernoulli *polynomials* .) As http://mathworld.wolfram.com/StirlingsSeries.html points out, Bernoulli(2*n)/(2*n*(2*n-1))) are the coefficients of the (divergent) Stirling series I mentioned yesterday. --rwg Mathematica now has anagrams: "deeeiimnorsstt" -> {"endometritises", "densitometries"}, ... "adegiiinnorstt" -> {"disorientating", "disintegration"}, ... "aeegiinorrsttv" -> {"tergiversation", "interrogatives"}, ... "eiinoopprssstu" -> {"superpositions", "propitiousness"}, ... Except that endometritises should be endometritides, which it doesn't even know: In[903]:= DictionaryWordQ@"endometritides" Out[903]= False _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
I'm puzzled by Christoph's comment (and also by Bill's). Isn't this simply the fact that Bill is looking at Bernoulli(n) while A090495 is thinking about Bernoulli(2n)? But just to be safe, I created a new entry, A274297, which gives 2*A090495(n)-1. Please feel free to add further comments... Best regards Neil Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com On Sat, Jun 18, 2016 at 4:04 AM, Pacher Christoph < Christoph.Pacher@ait.ac.at> wrote:
I did not verify it for more terms, but for the values given in Bill's mail the following holds:
2*A090495 - 1 = <the output of Bill's Mathematica program>
E.g. 2*574-1=1147
Christoph Originalnachricht Von: Bill Gosper Gesendet: Samstag, 18. Juni 2016 01:50 An: math-fun@mailman.xmission.com Antwort an: math-fun Betreff: [math-fun] And I missed this Bernoulli bizarreness
A090495 Numbers n such that numerator(Bernoulli(2*n)/(2*n)) is different from numerator(Bernoulli(2*n)/(2*n*(2*n-1))). 12 574, 1185, 1240, 1269, 1376, 1906, 1910, 2572, 2689, 2980, 3238, 3384, 3801, 3904, 4121, Michael Somos <http://oeis.org/wiki/User:Michael_Somos> (Feb 01 2004) discovered the remarkable fact that A001067 <http://oeis.org/A001067> is different from A046968 <http://oeis.org/A046968>, even though they agree for the first 573 terms.
Hey, Bernoulli(n+1)/n/(n+1) agrees with Bernoulli(n+1)/(n+1) for 1147 terms! Do[If[Numerator[BernoulliB[n + 1]/n/(n + 1)] != Numerator[BernoulliB[n + 1]/(n + 1)], Print[n.GCD[n, Numerator[BernoulliB[n + 1]/(n + 1)]]]], {n, 3511}] // tim
1147.37
2369.103
2479.37
2537.59
2751.131
8.898235 secs
(Don't laugh. You should usually think in terms of Bernoulli *polynomials* .) As http://mathworld.wolfram.com/StirlingsSeries.html points out, Bernoulli(2*n)/(2*n*(2*n-1))) are the coefficients of the (divergent) Stirling series I mentioned yesterday. --rwg Mathematica now has anagrams: "deeeiimnorsstt" -> {"endometritises", "densitometries"}, ... "adegiiinnorstt" -> {"disorientating", "disintegration"}, ... "aeegiinorrsttv" -> {"tergiversation", "interrogatives"}, ... "eiinoopprssstu" -> {"superpositions", "propitiousness"}, ... Except that endometritises should be endometritides, which it doesn't even know: In[903]:= DictionaryWordQ@"endometritides"
Out[903]= False _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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Neil, since the B_n with odd n>1 are zero, you are completely right. Christoph Originalnachricht Von: Neil Sloane Gesendet: Samstag, 18. Juni 2016 15:42 An: math-fun Antwort an: math-fun Betreff: Re: [math-fun] And I missed this Bernoulli bizarreness I'm puzzled by Christoph's comment (and also by Bill's). Isn't this simply the fact that Bill is looking at Bernoulli(n) while A090495 is thinking about Bernoulli(2n)? But just to be safe, I created a new entry, A274297, which gives 2*A090495(n)-1. Please feel free to add further comments... Best regards Neil Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com On Sat, Jun 18, 2016 at 4:04 AM, Pacher Christoph < Christoph.Pacher@ait.ac.at> wrote:
I did not verify it for more terms, but for the values given in Bill's mail the following holds:
2*A090495 - 1 = <the output of Bill's Mathematica program>
E.g. 2*574-1=1147
Christoph Originalnachricht Von: Bill Gosper Gesendet: Samstag, 18. Juni 2016 01:50 An: math-fun@mailman.xmission.com Antwort an: math-fun Betreff: [math-fun] And I missed this Bernoulli bizarreness
A090495 Numbers n such that numerator(Bernoulli(2*n)/(2*n)) is different from numerator(Bernoulli(2*n)/(2*n*(2*n-1))). 12 574, 1185, 1240, 1269, 1376, 1906, 1910, 2572, 2689, 2980, 3238, 3384, 3801, 3904, 4121, Michael Somos <http://oeis.org/wiki/User:Michael_Somos> (Feb 01 2004) discovered the remarkable fact that A001067 <http://oeis.org/A001067> is different from A046968 <http://oeis.org/A046968>, even though they agree for the first 573 terms.
Hey, Bernoulli(n+1)/n/(n+1) agrees with Bernoulli(n+1)/(n+1) for 1147 terms! Do[If[Numerator[BernoulliB[n + 1]/n/(n + 1)] != Numerator[BernoulliB[n + 1]/(n + 1)], Print[n.GCD[n, Numerator[BernoulliB[n + 1]/(n + 1)]]]], {n, 3511}] // tim
1147.37
2369.103
2479.37
2537.59
2751.131
8.898235 secs
(Don't laugh. You should usually think in terms of Bernoulli *polynomials* .) As http://mathworld.wolfram.com/StirlingsSeries.html points out, Bernoulli(2*n)/(2*n*(2*n-1))) are the coefficients of the (divergent) Stirling series I mentioned yesterday. --rwg Mathematica now has anagrams: "deeeiimnorsstt" -> {"endometritises", "densitometries"}, ... "adegiiinnorstt" -> {"disorientating", "disintegration"}, ... "aeegiinorrsttv" -> {"tergiversation", "interrogatives"}, ... "eiinoopprssstu" -> {"superpositions", "propitiousness"}, ... Except that endometritises should be endometritides, which it doesn't even know: In[903]:= DictionaryWordQ@"endometritides"
Out[903]= False _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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Суббота, 18 июня 2016, 21:45 +03:00 от Dan Asimov <asimov@msri.org>:
This one is pretty well known:
On Jun 18, 2016, at 9:45 AM, Pacher Christoph < Christoph.Pacher@ait.ac.at > wrote:
"aeegiinorrsttv" -> {"tergiversation", "interrogatives"},
Btw, how does Mma interpret x/y/z ??? Is is (x/y)/z or something else?
—Dan 8/4/2 1 8/(4/2) 4 (8/4)/2 1
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Воскресенье, 19 июня 2016, 1:06 +03:00 от Zak Seidov via math-fun <math-fun@mailman.xmission.com>:
Суббота, 18 июня 2016, 21:45 +03:00 от Dan Asimov < asimov@msri.org >:
This one is pretty well known:
On Jun 18, 2016, at 9:45 AM, Pacher Christoph < Christoph.Pacher@ait.ac.at > wrote:
"aeegiinorrsttv" -> {"tergiversation", "interrogatives"},
Btw, how does Mma interpret x/y/z ??? Is is (x/y)/z or something else?
—Dan 8/4/2 1 8/(4/2) 4 (8/4)/2 1 and 8/4*2*3 12
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participants (5)
-
Bill Gosper -
Dan Asimov -
Neil Sloane -
Pacher Christoph -
Zak Seidov