Re: Problem with a limit.

*To*: mathgroup at smc.vnet.net*Subject*: [mg66782] Re: Problem with a limit.*From*: Roger Bagula <rlbagulatftn at yahoo.com>*Date*: Tue, 30 May 2006 05:48:30 -0400 (EDT)*References*: <e5ei0n$7to$1@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

A little explaination might help. Sum[(PrimePi[k + 1] - PrimePi[k])/2^k, {k, 1, Infinity}] + Sum[(-1PrimePi[k + 1] + PrimePi[k])/2^k, {k, 1, Infinity}] =1 Dr. Ulam invented this for general sequences based on a van der Wearden theorem. This measure gives a binary measure of the primes and their co-set the composite numbers. Since the primes are more "dense" early their weight is larger, but if you reverse the direction of the sum and make it large to small, you get a more "representive" result in measure terms. That was what my effort was about. I have already put the two binaries in as digit sequences in OEIS as they seemed both important and interesting to me. > A119523 Decimal expansion of the van der Weaerden-Ulam binary measure of the primes as seen in the appearance PrimePi[n+1]-PrimePi[n] which is one when a prime number appears. +10 > 1 > 8, 2, 9, 3, 6, 5, 0, 1, 5, 0, 2, 9, 9, 0, 7, 2, 2, 6, 5, 6, 2, 5, 0, 0, 0 (list; graph) > OFFSET > > 0,1 > > COMMENT > > The Primes measure larger than the composites as they domiate the lower Integers. Sum[(PrimePi[k + 1] - PrimePi[k])/2^k, {k, 1, Infinity}]+Sum[(1 - PrimePi[k + 1] + PrimePi[k])/2^k, {k, 1, Infinity}]=1 > REFERENCES > > S. M. Ulam, Problems in Modern Mathematics,John Wiley and Sons, New York,1960, page 54 > FORMULA > > a(n) = RealDigits[Binary Sum of difference in PrimePi[m]] > MATHEMATICA > > a = Sum[(PrimePi[k + 1] - PrimePi[k])/2^k, {k, 1, Infinity}] Table[Floor[N[Mod[a*10^n, 10]]], {n, 1, 25}] RealDigits[N[a, 25]][[1]] > KEYWORD > > nonn,uned,new > AUTHOR > > Roger L. Bagula (rlbagultftn(AT)yahoo.com), May 27 2006 > A119524 Decimal expansion of the van der Weaerden-Ulam binary measure of the composites numbers as seen in the appearance 1-PrimePi[n+1]-PrimePi[n] which is one when a composite number appears. +10 > 1 > 1, 7, 0, 6, 3, 4, 9, 8, 4, 9, 7, 0, 0, 9, 2, 7, 7, 3, 4, 3, 7, 5, 0, 0, 0 (list; graph) > OFFSET > > 0,2 > > COMMENT > > The Primes measure larger than the composites as they dominate the lower Integers. Sum[(PrimePi[k + 1] - PrimePi[k])/2^k, {k, 1, Infinity}]+Sum[(1 - PrimePi[k + 1] + PrimePi[k])/2^k, {k, 1, Infinity}]=1 > REFERENCES > > S. M. Ulam, Problems in Modern Mathematics,John Wiley and Sons, New York,1960, page 54 > FORMULA > > a(n) = RealDigits[1-Binary Sum of difference in PrimePi[m]] > MATHEMATICA > > b = Sum[(1 - PrimePi[k + 1] + PrimePi[k])/2^k, {k, 1, Infinity}] Table[Floor[N[Mod[b*10^n, 10]]], {n, 1, 25}] RealDigits[N[b, 25]][[1]] > KEYWORD > > nonn,uned,new > AUTHOR > > Roger L. Bagula (rlbagultftn(AT)yahoo.com), May 27 2006