[seqfan] Constants derived from Artin's constants
Richard Mathar
mathar at strw.leidenuniv.nl
Fri Feb 13 18:38:17 CET 2009
Variants of the r-th order Artin's constants generate new constants,
some of which computed to 60 digits below. This is inspired by Jonathan's
observation in A112407 that each product over primes of Hardy-Littlewood
format is embedded in the associated product over all natural numbers.
I'd suspect that the products of the form
Product_(n=2...infinity) {1-1/(n^r*(n-1)}
have some tighter (perhaps "closed") forms, which I am not aware of. The
formulas related to zeta-functions from which I have computed them are
given below.
Richard J. Mathar
%I A000001
%S A000001 2,9,6,6,7,5,1,3,4,7,4,3,5,9,1,0,3,4,5,7,0,1,5,5,0,2,0,2,1,9,1,
%T A000001 4,2,8,6,4,8,6,4,8,3,1,5,1,9,1,7,8,9,4,7,8,9,0,8,1,6
%N A000001 Product_{n=2...infinity} (1-1/(n*(n-1))).
%C A000001 Product of Artin's constant A005596 and the equivalent almost-prime products.
%F A000001 The logarithm is -sum_{s>=2} sum_{j=1..floor[s/(1+r)]} binomial(s-r*j-1,j-1)*(1-Zeta(s))/j at r=1.
%F A000001 s*sum_{j=1..floor[s/2]} binomial(s-j-1,j-1)/j = A001610(s-1).
%e A000001 0.2966751347435910345.. = (1-1/2)*(1-1/6)*(1-1/12)*(1-1/20)*..
%Y A000001 Cf. A005596.
%K A000001 nonn,cons
%O A000001 0,1
%A A000001 R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 13 2009
%I A000002
%S A000002 8,3,9,0,4,2,1,5,4,2,7,4,4,6,8,6,0,0,7,6,8,4,6,2,1,1,1,1,9,4,5,
%T A000002 4,1,2,5,4,9,2,8,3,0,7,1,6,6,7,6,0,8,8,2,7,3,3,0,0,0
%N A000002 Product_{q in A001358} (1-1/(q*(q-1))).
%C A000002 Semiprime analog of A005596.
%F A000002 The logarithm is -sum_{s>=2} sum_{j=1..floor[s/(1+r)]} binomial(s-r*j-1,j-1)*P_2(s)/j at r=1,
where P_k(s) are the k-almost prime zeta functions of arXiv:0803.0900.
%e A000002 0.839042154274468600768... = (1-1/12)*(1-1/30)*(1-1/72)*(1-1/90)*(1-1/182)*..
%K A000002 nonn,cons,less
%O A000002 0,1
%A A000002 R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 13 2009
%I A000003
%S A000003 9,5,8,7,5,2,1,1,6,4,3,5,7,3,0,9,2,7,7,1,4,7,4,0,2,5,6,5,7,8,9,
%T A000003 2,8,6,1,2,6,5,9,4,9,0,4,4,8,5,0,2,3,5,9,9,0,1,5,9,2
%N A000003 Product_{q in A014612} (1-1/(q*(q-1))).
%C A000003 3-almost prime analog of A005596.
%F A000003 The logarithm is -sum_{s>=2} sum_{j=1..floor[s/(1+r)]} binomial(s-r*j-1,j-1)*P_3(s)/j at r=1,
where P_k(s) are the k-almost prime zeta functions of arXiv:0803.0900.
%e A000003 0.9587521164357309277147402... = (1-1/56)*(1-1/132)*(1-1/306)*(1-1/380)*..
%K A000003 nonn,cons,less
%O A000003 0,1
%A A000003 R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 13 2009
%I A000004
%S A000004 9,8,9,6,2,8,8,6,7,1,6,6,4,2,7,6,6,5,5,0,4,3,2,2,8,3,7,4,5,7,9,
%T A000004 2,4,3,0,8,0,5,7,5,5,7,5,8,9,3,5,0,2,9,6,5,3,4,8,4,4
%N A000004 Product_{q in A014613} (1-1/(q*(q-1))).
%C A000004 4-almost prime analog of A005596.
%F A000004 The logarithm is -sum_{s>=2} sum_{j=1..floor[s/(1+r)]} binomial(s-r*j-1,j-1)*P_4(s)/j at r=1,
where P_k(s) are the k-almost prime zeta functions of arXiv:0803.0900.
%e A000004 0.989628867166427665504.. = (1-1/240)*(1-1/552)*...
%K A000004 nonn,cons,less
%O A000004 0,1
%A A000004 R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 13 2009
%I A000005
%S A000005 6,7,3,9,1,7,3,6,3,3,7,6,3,5,7,5,4,1,6,6,4,4,0,8,9,7,9,3,2,2,6,
%T A000005 3,4,4,3,8,5,6,4,7,5,9,8,1,2,3,1,2,6,7,1,7,3,6,7,9,2
%N A000005 Product_{n=2...infinity} (1-1/(n^2*(n-1))).
%C A000005 Product of Artin's constant of rank 2 and the equivalent almost-prime products.
%F A000005 The logarithm is -sum_{s>=2} sum_{j=1..floor[s/(1+r)]} binomial(s-r*j-1,j-1)*(1-Zeta(s))/j at r=2.
%F A000005 s*sum_{j=1..floor[s/3]} binomial(s-2j-1,j-1)/j = A001609(s)-1.
%e A000005 0.6739173633763... = (1-1/4)*(1-1/18)*(1-1/48)*(1-1/100)*...
%Y A000005 Cf. A065414
%K A000005 nonn,cons
%O A000005 0,1
%A A000005 R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 13 2009
%I A000006
%S A000006 9,6,9,9,3,2,3,2,5,0,0,1,5,2,5,3,1,6,2,1,4,9,2,0,2,0,7,7,8,9,1,
%T A000006 2,9,5,7,5,9,6,1,1,4,5,7,9,4,7,9,6,6,9,6,0,8,8,0,0,6
%N A000006 Product_{q in A001358} (1-1/(q^2*(q-1))).
%C A000006 Semiprime analog of A065414.
%F A000006 The logarithm is -sum_{s>=2} sum_{j=1..floor[s/(1+r)]} binomial(s-r*j-1,j-1)*P_2(s)/j at r=2,
where P_k(s) are the k-almost prime zeta functions of arXiv:0803.0900.
%e A000006 0.969932325001525316214920... = (1-1/48)*(1-1/180)*(1-1/648)*(1-1/900)*..
%K A000006 nonn,cons,less
%O A000006 0,1
%A A000006 R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 13 2009
%I A000007
%S A000007 9,9,6,5,9,8,9,2,7,4,8,0,2,4,1,2,7,3,4,1,9,1,5,9,0,4,6,3,2,9,8,
%T A000007 9,4,6,9,2,2,9,1,0,1,0,3,9,1,0,1,1,7,8,3,8,2,0,6,5,8
%N A000007 Product_{q in A014612} (1-1/(q^2*(q-1))).
%C A000007 3-almost prime analog of A065414.
%F A000007 The logarithm is -sum_{s>=2} sum_{j=1..floor[s/(1+r)]} binomial(s-r*j-1,j-1)*P_3(s)/j at r=2,
where P_k(s) are the k-almost prime zeta functions of arXiv:0803.0900.
%e A000007 0.99659892748024127.. = (1-1/448)*(1-1/1584)*(1-1/5508)*(1-1/7600)*..
%K A000007 nonn,cons,less
%O A000007 0,1
%A A000007 R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 13 2009
%I A000008
%S A000008 9,9,9,5,9,5,2,7,8,5,8,6,5,3,5,5,3,5,6,3,7,4,5,2,4,9,3,2,4,8,3,
%T A000008 3,6,4,5,3,0,8,3,6,5,0,6,3,2,4,1,2,6,7,4,0,4,9,8,8,7
%N A000008 Product_{q in A014613} (1-1/(q^2*(q-1))).
%C A000008 4-almost prime analog of A065414.
%F A000008 The logarithm is -sum_{s>=2} sum_{j=1..floor[s/(1+r)]} binomial(s-r*j-1,j-1)*P_4(s)/j at r=2,
where P_k(s) are the k-almost prime zeta functions of arXiv:0803.0900.
%e A000008 0.99959527858653553563... = (1-1/3840)*(1-1/13248)*(1-1/45360)*(1-1/62400)*..
%K A000008 nonn,cons,less
%O A000008 0,1
%A A000008 R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 13 2009
%I A000009
%S A000009 8,5,0,6,7,0,6,3,0,7,9,1,1,0,4,3,5,3,7,5,0,3,0,9,5,2,1,2,5,0,0,
%T A000009 0,6,2,3,4,9,9,9,1,5,0,5,9,8,1,9,5,4,4,2,8,3,0,6,5,6
%N A000009 Product_{n=2...infinity} (1-1/(n^3*(n-1))).
%C A000009 Product of Artin's constant of rank 3 and the equivalent almost-prime products.
%F A000009 The logarithm is -sum_{s>=2} sum_{j=1..floor[s/(1+r)]} binomial(s-r*j-1,j-1)*(1-Zeta(s))/j at r=3.
%F A000009 s*sum_{j=1..floor[s/4]} binomial(s-3j-1,j-1)/j = A014097(s)-1.
%e A000009 0.85067063079110435... = (1-1/8)*(1-1/54)*(1-1/192)*(1-1/500)*(1-1/1080)*..
%Y A000009 Cf. A065415.
%K A000009 nonn,cons
%O A000009 0,1
%A A000009 R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 13 2009
%I A000010
%S A000010 9,9,3,5,2,1,5,8,9,7,1,0,5,0,5,4,6,0,6,7,5,4,0,9,2,6,9,2,4,1,4,
%T A000010 1,6,4,2,9,4,0,1,1,1,5,0,7,8,6,7,7,8,1,5,6,6,0,1,8,8
%N A000010 Product_{q in A001358} (1-1/(q^3*(q-1))).
%C A000010 Semiprime analog of A065415.
%F A000010 The logarithm is -sum_{s>=2} sum_{j=1..floor[s/(1+r)]} binomial(s-r*j-1,j-1)*P_2(s)/j at r=3,
where P_k(s) are the k-almost prime zeta functions of arXiv:0803.0900.
%e A000010 0.993521589710505460675409269.. = (1-1/192)*(1-1/1080)*(1-1/5832)*...
%K A000010 nonn,cons,less
%O A000010 0,1
%A A000010 R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 13 2009
%I A000011
%S A000011 9,9,9,6,4,5,2,3,8,3,3,2,6,1,3,3,6,7,7,3,0,2,0,6,6,3,9,1,2,0,7,
%T A000011 2,6,7,7,7,5,0,3,9,6,0,5,7,4,8,3,1,3,5,8,3,4,5,0,0,8
%N A000011 Product_{q in A014612} (1-1/(q^3*(q-1))).
%C A000011 3-almost prime analog of A065415.
%F A000011 The logarithm is -sum_{s>=2} sum_{j=1..floor[s/(1+r)]} binomial(s-r*j-1,j-1)*P_3(s)/j at r=3,
where P_k(s) are the k-almost prime zeta functions of arXiv:0803.0900.
%e A000011 0.999645238332613367730... = (1-1/3584)*(1-1/19008)*(1-1/99144)*..
%K A000011 nonn,cons,less
%O A000011 0,1
%A A000011 R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 13 2009
%I A000012
%S A000012 9,2,9,8,3,8,4,7,3,9,5,4,3,4,6,8,5,2,2,3,8,3,1,8,4,6,9,5,3,4,5,
%T A000012 5,3,5,4,8,9,4,4,9,0,8,3,0,5,4,8,2,2,5,3,6,3,5,2,3,6
%N A000012 Product_{n=2...infinity} (1-1/(n^4*(n-1))).
%C A000012 Product of Artin's constant of rank 4 and the equivalent almost-prime products.
%F A000012 The logarithm is -sum_{s>=2} sum_{j=1..floor[s/(1+r)]} binomial(s-r*j-1,j-1)*(1-Zeta(s))/j at r=4.
%F A000012 s*sum_{j=1..floor[s/5]} binomial(s-4j-1,j-1)/j = A058368(s)-1.
%e A000012 0.9298384739543468522383.. = (1-1/16)*(1-1/162)*(1-1/768)*(1-1/2500)*..
%Y A000012 Cf. A065416.
%K A000012 nonn,cons
%O A000012 0,1
%A A000012 R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 13 2009
%I A000013
%S A000013 9,9,8,5,0,9,5,0,0,6,0,7,5,7,3,7,5,4,5,8,7,1,5,0,2,1,3,5,7,8,1,
%T A000013 3,1,2,7,7,8,4,8,2,0,9,9,3,2,9,9,6,8,6,2,1,5,9,0,3,6
%N A000013 Product_{q in A001358} (1-1/(q^4*(q-1))).
%C A000013 Semiprime analog of A065416.
%F A000013 The logarithm is -sum_{s>=2} sum_{j=1..floor[s/(1+r)]} binomial(s-r*j-1,j-1)*P_2(s)/j at r=4,
where P_k(s) are the k-almost prime zeta functions of arXiv:0803.0900.
%e A000013 0.998509500607573754... = (1-1/768)*(1-1/6480)*(1-1/52488)*..
%K A000013 nonn,cons,less
%O A000013 0,1
%A A000013 R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 13 2009
%I A000014
%S A000014 9,9,9,9,5,9,6,4,3,4,7,6,3,9,2,5,0,7,1,6,7,5,0,5,9,4,1,2,1,8,7,
%T A000014 8,3,9,8,4,6,9,7,6,2,9,8,7,3,3,5,1,5,5,3,8,9,6,9,6,9
%N A000014 Product_{q in A014612} (1-1/(q^4*(q-1))).
%C A000014 3-almost prime analog of A065416.
%F A000014 The logarithm is -sum_{s>=2} sum_{j=1..floor[s/(1+r)]} binomial(s-r*j-1,j-1)*P_3(s)/j at r=4,
where P_k(s) are the k-almost prime zeta functions of arXiv:0803.0900.
%e A000014 0.99995964347639250716750... = (1-1/28672)*(1-1/228096)*(1-1/1784592)*...
%K A000014 nonn,cons,less
%O A000014 0,1
%A A000014 R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 13 2009
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