0,1

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

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

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.

(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);

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

easy,nonn,uned

Cino Hilliard (hillcino368(AT)gmail.com), Nov 16 2002