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Search: id:A076789
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| A076789 |
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Phisumprimes. Phi-digit-sum index primes of order z. For the above, z = 1. The z is referred to as the order of the sequence. |
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+0 1
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| 13, 17, 47, 47, 61, 73, 113, 163, 199, 241, 269, 317, 373, 431, 449, 499, 523, 587, 599, 599, 617, 647, 701, 743, 809, 823, 853, 863, 911, 947, 991, 1013, 1061, 1063, 1069, 1117, 1181, 1193, 1193, 1217, 1217, 1283, 1289, 1321, 1427, 1471, 1471, 1493, 1553
(list; graph; listen)
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OFFSET
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0,1
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COMMENT
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The sum of the reciprocals of this sequence diverges; it grows as log log n, just as the sum of the reciprocals of the primes does. Note that this is based on phi - 1, not phi. - Frank Adams-Watters (FrankTAW(AT)Netscape.net), Mar 30 2006
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FORMULA
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The digits of Phi = (sqrt(5)-1)/2 are added d_1+d_2..d_i and the prime whose index is the i-th sum is chosen. E.g. for Phi = .618033989... the first Phisumprime is prime(6) the second is prime(7), 3rd is prime(15) etc. Let d_1, d_2, ..d_i be the expansion of the decimal digits of Phi. Then Phisumprime(n)= prime(d_1), prime (d_1+d_2), ...prime(sum(d_i, i=1..n)). This can be generalized to phisumprime(n, z) where z is the nesting level of prime(x). for z=1 we have prime() for z=2 we have prime (prime(x)), z=3 prime (prime(prime(x))) etc. In maple prime(x) is ithprime(x).
a(n)=A000040(A093083(n+1)-1) - Frank Adams-Watters (FrankTAW(AT)Netscape.net), Mar 30 2006
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EXAMPLE
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Let p = .618033989 Then the following are some phisumprimes of order z=1,2,3 phisumprime(4,1)= prime(6), prime(7), prime(15), prime(15) = 13,17,47,47 phisumprime(4,2)= prime(13), prime(17), prime(47), prime(47) = 41,59,211,211 phisumprime(4,3)= prime(41), prime(59), prime(211), prime(211)= 179,277,1297,1297 We can speak of this last expression as the phisumprime sequence of length 4 order 3.
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PROGRAM
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(PARI) \ phi digit sum index primes \ phisump.gp Primes whose index is the sequential sum of digits of phi { phisump(n) = default(realprecision, 100000); p = (sqrt(5)-1)/2; default(realprecision, 28); sr=0; s=0; for(x=1, n, d = p*10; d1=floor(d); s+=d1; p = frac(d); d = p*10; p2=prime(s); sr+=1/p2+0.; print1(p2" "); ); print(" "); print(sr);
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CROSSREFS
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Cf. pisumprimes, esumprime which is the same algorithm for the digits of pi and e.
Sequence in context: A043893 A068497 A125524 this_sequence A089577 A060569 A108265
Adjacent sequences: A076786 A076787 A076788 this_sequence A076790 A076791 A076792
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KEYWORD
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easy,nonn,uned
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AUTHOR
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Cino Hilliard (hillcino368(AT)gmail.com), Nov 16 2002
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