A163464 Cumulative sum of a repeated shift-and-add operation on the base-7 representation of prime(n).

0, 0, 0, 1, 1, 1, 2, 2, 3, 4, 4, 5, 5, 6, 6, 8, 9, 9, 10, 11, 11, 12, 12, 13, 14, 16, 16, 17, 17, 18, 20, 20, 21, 21, 24, 24, 25, 26, 26, 27, 28, 28, 30, 30, 32, 32, 34, 35, 36, 36, 37, 38, 38, 40, 41, 42, 43, 43, 44, 45, 45, 46, 49, 50, 50, 51, 53, 54, 57, 57, 58, 59, 60, 61, 62

Offset: 1

Author

Vladimir Joseph Stephan Orlovsky, Jul 28 2009

Keywords

• nonn
• easy
• base

Comments

Starting from the base-7 representation of prime(n) = d_m*7^m + ... + d_3*7^3 + d_2*7^2 + d_1*7 + d_0, the least-significant digit is recursively removed (a shift-right operation in base 7), and the intermediate numbers are all added up:

a(n) = (d_m*7^(m-1) + ... + d_3*7^2 + d_2*7 + d_1)

+ (d_m*7^(m-2) + ... + d_4*7^2 + d_3*7 + d_2)

+ (d_m*7^(m-3) + ... + d_4*7 + d_3)

+ ... + d_m

= Sum_{j=1..m} d_j*(7^j - 1)/6.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..5000

Crossrefs

Cf. A080085, A080086, A080087.

Programs

Maple

shiftadd := proc(n,b) dgs := convert(n,base,b) ; add( op(i,dgs)*(b^(i-1)-1),i=2..nops(dgs))/(b-1) ; end:
A163464 := proc(n) shiftadd(ithprime(n),7) ; end:
seq(A163464(n),n=1..40) ; # R. J. Mathar, Aug 02 2009

Mathematica

lst={}; Do[p=Prime[n]; s=0; While[p>1,p=IntegerPart[p/7];s+=p;]; AppendTo[lst, s],{n,6!}]; lst

Extensions

Definition rewritten by R. J. Mathar, Aug 02 2009