A101301 The sum of the first n primes, minus n.

1, 3, 7, 13, 23, 35, 51, 69, 91, 119, 149, 185, 225, 267, 313, 365, 423, 483, 549, 619, 691, 769, 851, 939, 1035, 1135, 1237, 1343, 1451, 1563, 1689, 1819, 1955, 2093, 2241, 2391, 2547, 2709, 2875, 3047, 3225, 3405, 3595, 3787, 3983, 4181, 4391, 4613, 4839

Offset: 1

Author

Jorge Coveiro, Dec 22 2004

Keywords

• nonn

Comments

Also: a(n) = sum_{k=1..n} phi(prime(k)).

Partial sums of A006093. - Omar E. Pol, Oct 31 2013

Difference minus n, between the constant term prime(n) for a polynomial P(x) built from the first n primes took as coefficients and the value that such term should have in order to make P(x) divisible by (x-1). See links. - R. J. Cano, Jan 14 2014

Sum of all deficiencies of the first n primes. - Omar E. Pol, Feb 21 2014

Links

• Enrique Pérez Herrero, Table of n, a(n) for n = 1..5000
• R. J. Cano, Additional information
• R. J. Cano, PARI/GP code: alternative sequencer

Crossrefs

Cf. A000010, A000040, A006093, A005867.

Programs

Haskell

a101301 n = a101301_list !! (n-1)
a101301_list = scanl1 (+) a006093_list
-- Reinhard Zumkeller, May 01 2013

Maple

seq((sum(phi(ithprime(x)),k=1..n)),n=1..100);

Mathematica

f[n_]:=Plus@@Prime[Range[n]]-n; Table[f[n],{n,1,50}] (* Enrique Pérez Herrero, Jun 10 2012 *)

PARI

a(n)=my(s);forprime(p=2,prime(n),s+=p); s-n \\ Charles R Greathouse IV, Oct 31 2013

PARI

\\ See links.

Formula

a(n)=sum_{k=1..n} (prime(k)-1)

a(n)=A007504(n)-n. - Juri-Stepan Gerasimov, Nov 23 2009

A027424(A000040(n)) < a(n). - Charles R Greathouse IV, Apr 07 2021

Extensions

Name simplified by Juri-Stepan Gerasimov, Nov 23 2009