cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A014148 a(n) = Sum_{m=1..n} Sum_{k=1..m} prime(k).

Original entry on oeis.org

2, 7, 17, 34, 62, 103, 161, 238, 338, 467, 627, 824, 1062, 1343, 1671, 2052, 2492, 2993, 3561, 4200, 4912, 5703, 6577, 7540, 8600, 9761, 11025, 12396, 13876, 15469, 17189, 19040, 21028, 23155, 25431, 27858, 30442, 33189, 36103, 39190, 42456, 45903
Offset: 1

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Author

N. J. A. Sloane

Keywords

Comments

Previous name was: Apply partial sum operator twice to sequence of primes.
Numbers n such that a(n) is prime are listed in A122381(n) = {1, 2, 3, 6, 10, 23, 31, 46, 55, 58, 66, 70, 82, 91, 118, 131, 151, 163, 182, 187, 198, 199, ...}. Corresponding primes a(n) = a( A122381(n) ) = A122382(n) = {2, 7, 17, 103, 467, 6577, 17189, 61627, 109919, 130531, 198109, 239579, 399557, 559313, ...}. - Alexander Adamchuk, Aug 30 2006
Row 2 in A254858. - Reinhard Zumkeller, Feb 08 2015
Partial sums of A007504, n>=1. - Omar E. Pol, Nov 23 2016

Links

Crossrefs

Cf. A000040, A007504, A014150, A122381, A122382, A178138, A254784, A254858.

Programs

  • Haskell
    a014148 n = a014148_list !! (n-1)
a014148_list = (iterate (scanl1 (+)) a000040_list) !! 2
-- Reinhard Zumkeller, Feb 08 2015
  • Maple
    b:= proc(n) option remember; `if`(n<1, [0$2],
      (p-> p+[ithprime(n), p[1]])(b(n-1)))
    end:
a:= n-> b(n+1)[2]:
seq(a(n), n=1..42);  # Alois P. Heinz, Oct 07 2021
  • Mathematica
    Table[Sum[Sum[Prime[k],{k,1,m}],{m,1,n}],{n,1,100}] (* Alexander Adamchuk, Aug 30 2006 *)
Accumulate[Accumulate[Prime[Range[50]]]] (* Harvey P. Dale, Dec 29 2011 *)

Formula

Convolution of the primes with the positive integers: Sum_{k=1..n} (n-k+1)*prime(k). - David Scambler, Oct 08 2006

Extensions

More terms from Alexander Adamchuk, Aug 30 2006
Name changed by Wesley Ivan Hurt, Oct 04 2021

A122381 Numbers n such that A014148[n] is a prime.

Original entry on oeis.org

1, 2, 3, 6, 10, 23, 31, 46, 55, 58, 66, 70, 82, 91, 118, 131, 151, 163, 182, 187, 198, 199, 203, 222, 275, 279, 334, 346, 351, 358, 402, 411, 462, 470, 515, 582, 591, 619, 639, 650, 667, 671, 679, 706, 739, 750, 767, 835, 851, 875, 882, 899, 919, 926, 962, 966
Offset: 1

Views

Author

Alexander Adamchuk, Aug 30 2006

Keywords

Comments

Corresponding primes are listed in A122382[n] = A014148[ a(n) ] = {2,7,17,103,467,6577,17189,61627,109919,130531,198109,239579,399557,559313,...}.

Examples

			A014148[n] begins {2,7,17,34,62,103,161,238,338,467,627,824,1062,1343,...}.
a(1) = 1 because A014148[1] = 2 is prime.
a(2) = 2 because A014148[2] = 7 is prime.

Crossrefs

Cf. A014148, A122382.

Programs

  • Mathematica
    p=0;s=0;f=0;Do[p=Prime[n];s=s+p;f=f+s;If[PrimeQ[f],Print[{n,f}]],{n,1,2000}]
Position[Nest[Accumulate,Prime[Range[1000]],2],?PrimeQ]//Flatten (* _Harvey P. Dale, Jul 15 2023 *)

Formula

A014148[n] = Sum[ Sum[ Prime[k], {k,1,m} ], {m,1,n} ]. A014148[ a(n) ] = A122382[n].

A157494 Primes in A014150.

Original entry on oeis.org

2, 1429, 32869, 3189059, 5157791, 62701339, 139181423, 296686879, 522304883, 5070516751, 6276844867, 7098350179, 8983996079, 9331926623, 21211375343, 31177858939, 34861039007, 38865340309, 39918757589, 62858815181
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Mar 01 2009

Keywords

Crossrefs

Cf. A001223, A014148, A014150, A061789, A069482, A122382, A129701, A157492, A157493

Programs

  • Mathematica
    s0=s1=s2=0;lst={};Do[p=Prime[n];s0+=p;s1+=s0;s2+=s1;If[PrimeQ[s2],AppendTo[lst,s2]],{n,7!}];lst
Select[Nest[Accumulate[#]&,Prime[Range[700]],3],PrimeQ] (* Harvey P. Dale, Jul 11 2025 *)
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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