A076789 - OEIS

%I #24 Aug 22 2024 03:18:22

%S 13,17,47,47,61,73,113,163,199,241,269,317,373,431,449,499,523,587,

%T 599,599,617,647,701,743,809,823,853,863,911,947,991,1013,1061,1063,

%U 1069,1117,1181,1193,1193,1217,1217,1283,1289,1321,1427,1471,1471,1493,1553

%N Phisumprimes: prime(k), where k is the sum of the first n digits of phi-1 and phi is the golden ratio.

%C 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. - _Franklin T. Adams-Watters_, Mar 30 2006

%F 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_{i=1..n} d_i). 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)); for z=3 prime (prime(prime(x))); etc.

%F a(n) = A000040(A093083(n+1)-1). - _Franklin T. Adams-Watters_, Mar 30 2006

%t Prime[#]&/@Accumulate[RealDigits[GoldenRatio-1,10,50][[1]]] (* _Harvey P. Dale_, Sep 30 2012 *)

%o (PARI) \\ phi digit sum index primes; phisump.gp Primes whose index is the sequential sum of digits of phi

%o { 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); }

%Y Cf. A076787, which is the same algorithm for the digits of Pi.

%K easy,nonn,base

%O 1,1

%A _Cino Hilliard_, Nov 16 2002

%E Edited by _T. D. Noe_, Jun 24 2009