Revision History for A192455
(Underlined text is an addition;
strikethrough text is a deletion.)
Showing entries 1-10
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#12 by Russ Cox at Fri Mar 30 18:37:27 EDT 2012
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| AUTHOR
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_Paul D. Hanna (pauldhanna(AT)juno.com), _, Jul 14 2011
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Discussion
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| Fri Mar 30
| 18:37
| OEIS Server: https://oeis.org/edit/global/213
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#11 by Joerg Arndt at Fri Jul 15 03:28:34 EDT 2011
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#10 by Paul D. Hanna at Thu Jul 14 23:39:17 EDT 2011
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#9 by Paul D. Hanna at Thu Jul 14 23:39:07 EDT 2011
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| NAME
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G.f. A(x) satisfies: 1 = Sum_{n>=0} x^n*A(-x)^A001650(n+1), where A001650 is defined by "n appears n times (n odd).".
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| COMMENTS
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Compare the g.f. to a g.f. C(x) of the Catalan numbers: 1 = Sum_{n>=0} x^n*C(-x)^(2*n+1).
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| FORMULA
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G.f. satisfies: 1-x = Sum_{n>=1} x^(n^2)* () * (1-x^(2*n-1))* )) * A(-x)^(2*n-1).
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| EXAMPLE
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G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 27*x^4 + 112*x^5 + 492*x^6 +...
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| PROG
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((PARI) {a(n)=local(A=[1]); for(i=1, n, A=concat(A, 0); A[#A]=polcoeff(sum(m=1, #A, (-x)^m*Ser(A)^(1+2*sqrtint(m-1)) ), #A)); if(n<0, 0, A[n+1])}
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| CROSSREFS
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Cf. A193039, A193040, A193050, A001650.
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| STATUS
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proposed
editing
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#8 by Paul D. Hanna at Thu Jul 14 21:18:15 EDT 2011
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#7 by Paul D. Hanna at Thu Jul 14 21:17:29 EDT 2011
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| NAME
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allocated for Paul D. Hanna
G.f. A(x) satisfies: 1 = Sum_{n>=0} x^n*A(-x)^A001650(n+1), where A001650 is defined by "n appears n times (n odd).".
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| DATA
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1, 1, 2, 7, 27, 112, 492, 2249, 10580, 50885, 249067, 1236602, 6212563, 31523293, 161317863, 831615320, 4314659345, 22512421092, 118052038100, 621825506334, 3288597601727, 17455485596492, 92958082866815, 496535775228131, 2659574264906443
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| OFFSET
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0,3
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| COMMENTS
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Compare the g.f. to a g.f. C(x) of the Catalan numbers: 1 = Sum_{n>=0} x^n*C(-x)^(2*n+1).
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| FORMULA
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G.f. satisfies: 1-x = Sum_{n>=1} x^(n^2)* (1-x^(2*n-1))* A(-x)^(2*n-1).
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| EXAMPLE
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G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 27*x^4 + 112*x^5 + 492*x^6 +...
The g.f. satisfies:
1 = A(-x) + x*A(-x)^3 + x^2*A(-x)^3 + x^3*A(-x)^3 + x^4*A(-x)^5 + x^5*A(-x)^5 + x^6*A(-x)^5 + x^7*A(-x)^5 + x^8*A(-x)^5 + x^9*A(-x)^7 +...+ x^n*A(-x)^A001650(n+1) +...
where A001650 begins: [1, 3,3,3, 5,5,5,5,5, 7,7,7,7,7,7,7, 9,...].
The g.f. also satisfies:
1-x = (1-x)*A(-x) + x*(1-x^3)*A(-x)^3 + x^4*(1-x^5)*A(-x)^5 + x^9*(1-x^7)*A(-x)^7 + x^16*(1-x^9)*A(-x)^9 +...
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| PROG
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(PARI) {a(n)=local(A=[1]); for(i=1, n, A=concat(A, 0); A[#A]=polcoeff(sum(m=1, #A, (-x)^m*Ser(A)^(1+2*sqrtint(m-1)) ), #A)); if(n<0, 0, A[n+1])}
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| CROSSREFS
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Cf. A193039, A193040, A001650.
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| KEYWORD
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allocated
nonn
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| AUTHOR
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Paul D. Hanna (pauldhanna(AT)juno.com), Jul 14 2011
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| STATUS
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approved
editing
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#6 by Paul D. Hanna at Thu Jul 14 21:11:16 EDT 2011
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| NAME
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allocated for Paul D. Hanna
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| KEYWORD
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recycled
allocated
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#5 by T. D. Noe at Thu Jul 14 19:56:15 EDT 2011
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#4 by T. D. Noe at Thu Jul 14 19:56:11 EDT 2011
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| NAME
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a(n) is the smallest k such that prime(n) divides prime(k)+ prime(k+1).
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| DATA
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2, 3, 1, 8, 25, 9, 11, 21, 19, 69, 24, 29, 46, 23, 60, 115, 51, 111, 32, 82, 129, 185, 132, 71, 106, 155, 63, 116, 84, 203, 54, 77, 58, 145, 108, 87, 289, 93, 67, 443, 254, 460, 292, 76, 350, 300, 447, 86, 397, 124, 284, 808, 128, 335, 136, 547, 742, 361, 571, 475
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| OFFSET
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1,1
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| COMMENTS
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See the sequence A188815 for prime(k).
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| EXAMPLE
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a(4) = 8 because prime(4) = 7 divides prime(8) + prime(9) = 19 + 23 = 42.
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| MAPLE
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with(numtheory):for n from 1 to 60 do:p1:=ithprime(n):id:=0:for k from 1 to 1000 while(id=0) do: x:=ithprime(k)+ ithprime(k+1):if irem(x, p1)=0 then id:=1:printf(`%d, `, k):else fi:od:od:
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| CROSSREFS
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Cf.A188815.
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| KEYWORD
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nonn,changed
recycled
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| AUTHOR
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Michel Lagneau (mn.lagneau2(AT)orange.fr), Jul 01 2011
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| STATUS
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proposed
editing
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#3 by Michel Lagneau at Fri Jul 01 16:05:01 EDT 2011
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| COMMENTS
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See the sequence A188815 for prime(k).
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| EXAMPLE
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a(4) = 8 because prime(4) = 7 divides prime(8) + prime(9) = 19 + 23 = 42.
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| CROSSREFS
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Cf.A188815.
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Discussion
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| Wed Jul 13
| 14:05
| T. D. Noe: I do not see why we need this sequence if we already have A188815.
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