A076789 Phisumprimes: prime(k), where k is the sum of the first n digits of phi-1 and phi is the golden ratio.

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

Offset: 1

Author

Cino Hilliard, Nov 16 2002

Keywords

• easy
• nonn
• base
• less

Comments

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

Crossrefs

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

Programs

Mathematica

Prime[#]&/@Accumulate[RealDigits[GoldenRatio-1,10,50][[1]]] (* Harvey P. Dale, Sep 30 2012 *)

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); }

Formula

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.

a(n) = A000040(A093083(n+1)-1). - Franklin T. Adams-Watters, Mar 30 2006

Extensions

Edited by T. D. Noe, Jun 24 2009