A014284 Partial sums of primes, if 1 is regarded as a prime (as it was until quite recently, see A008578).

1, 3, 6, 11, 18, 29, 42, 59, 78, 101, 130, 161, 198, 239, 282, 329, 382, 441, 502, 569, 640, 713, 792, 875, 964, 1061, 1162, 1265, 1372, 1481, 1594, 1721, 1852, 1989, 2128, 2277, 2428, 2585, 2748, 2915, 3088, 3267, 3448, 3639, 3832, 4029

Offset

1,2

Comments

Lexicographically earliest sequence whose first differences are an increasing sequence of primes. Complement of A175969. - Jaroslav Krizek, Oct 31 2010

A175944(a(n)) = A018252(n). - Reinhard Zumkeller, Mar 18 2011

Partial sums of noncomposite numbers (A008578). - Omar E. Pol, Aug 09 2012

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Formula

a(n) = Sum_{k <= n} [(A158611(k + 1)) * (A000012(n - k + 1))] = Sum_{k <= n} [(A158611(k + 1)) * (A000012(k))] = Sum_{k <= n} [(A008578(k)) * (A000012(n - k + 1))] = Sum_{k <= n} [(A008578(k)) * (A000012(k))] for n, k >= 1. - Jaroslav Krizek, Aug 05 2009

a(n + 1) = A007504(n) + 1. a(n + 1) - a(n) = A000040(n) = n-th primes. - Jaroslav Krizek, Aug 19 2009

a(n) = a(n-1) + prime(n-1), with a(1)=1. - Vincenzo Librandi, Jul 27 2013

G.f: (x*(1+b(x)))/(1-x) = c(x)/(1-x), where b(x) and c(x) are respectively the g.f. of A000040 and A008578. - Mario C. Enriquez, Dec 10 2016

Example

a(7) = 42 because the first six primes (2, 3, 5, 7, 11, 13) add up to 41, and 1 + 41 = 42.

Maple

A014284 := proc(n)
    add(A008578(i), i=1..n) ;
end proc:
seq(A014284(n), n=1..60) ; # R. J. Mathar, Mar 05 2017

Mathematica

Join[{1}, Table[1+Sum[Prime[j], {j, 1, n}], {n, 1, 50}]] (* Vladimir Joseph Stephan Orlovsky, Sep 25 2009, modified by G. C. Greubel, Jun 18 2019 *)
Accumulate[Join[{1}, Prime[Range[45]]]] (* Alonso del Arte, Oct 09 2012 *)

Prog

(Haskell)
a014284 n = a014284_list !! n
a014284_list = scanl1 (+) a008578_list
-- Reinhard Zumkeller, Mar 26 2015
(PARI) concat([1], vector(50, n, 1 + sum(j=1, n, prime(j)) )) \\ G. C. Greubel, Jun 18 2019
(Magma) [1] cat [1 + (&+[NthPrime(j): j in [1..n]]): n in [1..50]]; // G. C. Greubel, Jun 18 2019
(Sage) [1]+[1 + sum(nth_prime(j) for j in (1..n)) for n in (1..50)] # G. C. Greubel, Jun 18 2019

Crossrefs

Cf. A007504.

Equals A036439(n) - 1.

Cf. A175965, A175966, A175967, A175968, A175969, A051349, A175970. - Jaroslav Krizek, Oct 31 2010

Cf. A008578.

Sequence in context: A173690 A178855 A095944 * A118482 A281689 A026905

Adjacent sequences: A014281 A014282 A014283 * A014285 A014286 A014287

Keyword

nonn,easy

Author

Deepan Majmudar (dmajmuda(AT)esq.com)

Extensions

Correction for Aug 2009 change of offset in A158611 and A008578 by Jaroslav Krizek, Jan 27 2010

Status

approved