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A339499 Decimal expansion of the generating constant for the composite numbers.

%I A339499 #18 Jun 17 2025 20:44:52
%S A339499 4,5,8,9,2,4,6,1,2,6,6,3,7,9,8,6,1,7,1,3,5,8,1,0,2,4,2,0,7,3,5,0,7,0,
%T A339499 7,3,6,9,2,7,4,1,4,8,3,3,8,6,1,6,7,4,8,3,0,6,5,0,1,9,9,9,5,7,4,4,4,9,
%U A339499 7,6,6,4,4,8,6,2,2,8,2,4,0,9,9,8,0,6,1,3,1,6,1,4,4,9,5,3,5,6,0,8
%N A339499 Decimal expansion of the generating constant for the composite numbers.
%C A339499 The integer parts of the sequence having this constant as starting value and thereafter a(n+1) = (frac(a(n))+1) * floor(a(n)), where floor and frac are integer and fractional part, are exactly the sequence of the composite numbers: see the Grime-Haran Numberphile video for details.
%H A339499 James Grime and Brady Haran, <a href="https://www.youtube.com/watch?v=_gCKX6VMvmU">2.920050977316</a>, Numberphile video, Nov 26 2020.
%F A339499 Sum_{k >= 1} (c(k) - 1)/(c(1) * c(2) * ... * c(k-1)), where c(k) is the k-th composite number.
%e A339499 4.5892461266379861713581024207350707369274148338616748...
%o A339499 (Python)
%o A339499 from mpmath import * #high precision computations
%o A339499                      #nsum function
%o A339499 from sympy import * # to generate prime numbers
%o A339499 mp.dps = 10000
%o A339499 #function that generates constant that encodes all composite numbers
%o A339499 #cnt - number of prime numbers
%o A339499 def composconst(cnt):
%o A339499     if cnt==1:
%o A339499         return 4-1
%o A339499     primlist=list()
%o A339499     i=0
%o A339499     while (i<cnt):
%o A339499         primlist.append(prime(i+1))
%o A339499         i=i+1
%o A339499     prims=set(primlist)
%o A339499     alllist=range(2,primlist[-1]+2) #all numbers [2..prime(cnt)+1]
%o A339499     alls=set(alllist)
%o A339499     comps=alls-prims #all composite numbers [4..prime(cnt)+1]
%o A339499     complist=list(comps)
%o A339499     cnt2 = len(complist)
%o A339499     return nsum(lambda k: (complist[int(k)]-1)/nprod(lambda l: complist[int(l)],[0,k-1]),[0,cnt2-1])
%o A339499 composconst(50)
%Y A339499 Cf. A002808, A249270, A339204.
%K A339499 nonn,cons
%O A339499 1,1
%A A339499 _Kamil Zabkiewicz_, Dec 07 2020