A013918 -id:A013918 - cp's OEIS frontend

A189072 Semiprimes in A007504 (the sum of first n primes).

10, 58, 77, 129, 381, 501, 791, 1371, 1851, 2127, 2427, 2747, 3831, 4227, 4661, 6081, 6338, 7141, 7418, 9206, 9523, 11599, 12718, 15537, 20059, 20546, 21037, 26369, 27517, 29897, 34915, 36227, 45434, 47721, 48494, 49281, 50887, 51698, 52519, 54169, 57547

Offset: 1

Zak Seidov, Apr 16 2011

nonn

a(n) = A007504(k(n)), values of k(n) = 3, 7, 8, 10, 16, 18, 22, 28, 32, 34, 36, 38, 44, 46, 48, 54, 55, 58, 59, 65, 66, 72, 75, 82, 92, 93, 94, 104, 106, 110, 118, 120, 133, 136, 137, 138, 140, 141, 142, 144, 148, 150, 154, 156, 164, 168, 170, 174, 190, 194, 202, 210, 212, 218, 224, 226, 232, 234, 236, 244, 246, 249, 250, 256, 264, 272, 276, 277, 286, 294, 298, 300.

Intersection of A007504 and A001358. - Robert Israel, Jun 23 2017

			10 = 2*5 = A007504(3), 58 = 2*29 = A007504(7), 77 = 7*11 = A007504(8).

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A013918 (primes in A007504).

Cf. A007504, A001358.

PS:= ListTools:-PartialSums(select(isprime, [2,seq(i,i=3..10^4,2)])):
select(numtheory:-bigomega = 2, PS); # Robert Israel, Jun 23 2017

semiPrimeQ[n_Integer] := Total[FactorInteger[n]][[2]] == 2; Select[Accumulate[Prime[Range[100]]], semiPrimeQ] (* T. D. Noe, Apr 20 2011 *)
With[{nn=200},Select[Accumulate[Prime[Range[nn]]],PrimeOmega[#]==2&]] (* Harvey P. Dale, Dec 22 2018 *)

{a=0;s=[];forprime(p=2,10^4,2==bigomega(a=a+p)&s=concat(s,a));s}

A053790 Composite numbers arising as sum of first k primes.

Original entry on oeis.org

10, 28, 58, 77, 100, 129, 160, 238, 328, 381, 440, 501, 568, 639, 712, 791, 874, 963, 1060, 1161, 1264, 1371, 1480, 1593, 1720, 1851, 1988, 2127, 2276, 2427, 2584, 2747, 2914, 3087, 3266, 3447, 3638, 3831, 4028, 4227, 4438, 4661, 4888, 5117, 5350, 5589, 5830

Offset: 1

Enoch Haga, Mar 27 2000

Generate sum of first k primes; accept if not prime.

			a(4) = 77 because 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 = 77 = A007504(8) is composite, and 77 is the 4th such composite sum of k primes.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A053789, A013918, A066527.

Intersection of A002808 and A007504.

N:= 10^4: # to use primes <= N
P:= select(isprime,[2,seq(i,i=3..N,2)]):
remove(isprime,ListTools:-PartialSums(P)); # Robert Israel, Sep 22 2016

Cases[Accumulate[Prime[Range[100]]],Except[?PrimeQ]] (*_Fred Patrick Doty Aug 22 2017*)

first(n)=my(v=vector(n),s,i); forprime(p=2,, if(isprime(s+=p), next); v[i++]=s; if(i==n, break)); v \\ Charles R Greathouse IV, Aug 23 2017

from sympy import isprime, nextprime
def aupto(limit):
    alst, s, p = [], 2, 2
    while s < limit:
        if not isprime(s): alst.append(s)
        p = nextprime(p)
        s += p
    return alst
print(aupto(6000)) # Michael S. Branicky, Oct 28 2021

{k: A007504(k) in A002808}. - Michael S. Branicky, Oct 28 2021

A071150 Primes p such that the sum of all odd primes <= p is also a prime.

Original entry on oeis.org

3, 29, 53, 61, 251, 263, 293, 317, 359, 383, 503, 641, 647, 787, 821, 827, 911, 1097, 1163, 1249, 1583, 1759, 1783, 1861, 1907, 2017, 2287, 2297, 2593, 2819, 2837, 2861, 3041, 3079, 3181, 3461, 3541, 3557, 3643, 3779, 4259, 4409, 4457, 4597, 4691, 4729, 4789

Offset: 1

Labos Elemer, May 13 2002

			29 is a prime and 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 = 127 (also a prime), so 29 is a term. - _Jon E. Schoenfield_, Mar 29 2021

R. J. Mathar, Table of n, a(n) for n = 1..5908

Analogous to A013917.

Cf. A065091, A071148, A071149, A013916, A013917, A013918, A007504.

SoddP := proc(n)
    option remember;
    if n <= 2 then
        0;
    elif isprime(n) then
        procname(n-1)+n;
    else
        procname(n-1);
    fi ;
end proc:
isA071150 := proc(n)
    if isprime(n) and isprime(SoddP(n)) then
        true;
    else
        false;
    end if;
end proc:
n := 1 ;
for i from 3 by 2 do
    if isA071150(i) then
        printf("%d %d\n",n,i) ;
        n := n+1 ;
    end if;
end do: # R. J. Mathar, Feb 13 2015

Function[s, Select[Array[Take[s, #] &, Length@ s], PrimeQ@ Total@ # &][[All, -1]]]@ Prime@ Range[2, 640] (* Michael De Vlieger, Jul 18 2017 *)
Module[{nn=650,pr},pr=Prime[Range[2,nn]];Table[If[PrimeQ[Total[Take[ pr, n]]], pr[[n]],Nothing],{n,nn-1}]] (* Harvey P. Dale, May 12 2018 *)

from sympy import isprime, nextprime
def aupto(limit):
  p, s, alst = 3, 3, []
  while p <= limit:
    if isprime(s): alst.append(p)
    p = nextprime(p)
    s += p
  return alst
print(aupto(4789)) # Michael S. Branicky, Mar 29 2021

A071151 Primes which are the sum of the first k odd primes for some k.

Original entry on oeis.org

3, 127, 379, 499, 6079, 6599, 8273, 9521, 11597, 13099, 22037, 33623, 34913, 49279, 52517, 54167, 64613, 92951, 101999, 116531, 182107, 222269, 225829, 240379, 255443, 283079, 356387, 360977, 448867, 535669, 541339, 552751, 611953, 624209

Offset: 1

Labos Elemer, May 13 2002

nonn

T. D. Noe, Table of n, a(n) for n = 1..1000

Analogous to A013918.

Cf. A013916-A013918, A065091, A071148-A071150, A007504.

s=0;lst={};Do[p=Prime[n];s+=p;If[PrimeQ[s],AppendTo[lst,s]],{n,2,7!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 28 2009 *)
Select[Accumulate[Prime[Range[2,500]]],PrimeQ]  (* Harvey P. Dale, Mar 14 2011 *)

list(lim)=my(v=List(),s); forprime(p=3,, if((s+=p)>lim, return(Vec(v))); if(isprime(s), listput(v,s))) \\ Charles R Greathouse IV, May 22 2017

from sympy import isprime, nextprime; m = 0; p = 2
while p < 3100:
    p = nextprime(p); m += p
    if isprime(m): print(m, end = ', ') # Ya-Ping Lu, Dec 24 2024

Name simplified by Charles R Greathouse IV, May 22 2017

A053845 Primes of form prime(1) + ... + prime(k) + 1.

Original entry on oeis.org

3, 11, 29, 59, 101, 239, 569, 1061, 1481, 1721, 4889, 5351, 6871, 22549, 23593, 25801, 29297, 35569, 38239, 41023, 71209, 77137, 87517, 94057, 105541, 120349, 122921, 125509, 128113, 133387, 138869, 141677, 156109, 159073, 165041, 183707

Offset: 1

Enoch Haga, Mar 28 2000

			prime(1) + 1 = 2 + 1 = 3 (prime, thus a(1));
prime(1) + prime(2) + 1 = 2 + 3 + 1 = 6 (nonprime);
prime(1) + prime(2) + prime(3) + 1 = 2 + 3 + 5 + 1 = 11 (prime, thus a(2)); etc. - _Jon E. Schoenfield_, Jan 09 2015

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A007504, A013918, A089228.

p=1;lst={};Do[p+=Prime[n];If[PrimeQ[p],AppendTo[lst,p]],{n,7!}];lst (* Vladimir Joseph Stephan Orlovsky, Jun 14 2009 *)

lista(nn) = {s = 1; for (n=1, nn, s += prime(n); if (isprime(s), print1(s, ", ")););} \\ Michel Marcus, Jan 10 2015

10 x=x+1; 20 if x<>prmdiv(x) then 10; 30 y=x; 40 r=r+y; 50 if r=prmdiv(r) then print r;:p=p+1; 60 if p<100 then 10

a(n) = A007504(A089228(n)) + 1. - Amiram Eldar, Apr 29 2024

A070281 Smallest prime which is the sum of n consecutive primes, or 0 if no such prime exists.

Original entry on oeis.org

2, 5, 23, 17, 53, 41, 197, 0, 127, 0, 233, 197, 691, 281, 379, 0, 499, 0, 857, 0, 953, 0, 1151, 0, 1259, 0, 1583, 0, 2099, 0, 2399, 0, 2417, 0, 2579, 0, 2909, 0, 3803, 0, 3821, 0, 4217, 0, 4651, 0, 5107, 0, 5813, 0, 6829, 0, 6079, 0, 6599, 0, 14153, 0, 10091, 7699

Offset: 1

Amarnath Murthy, May 07 2002

nonn

Sequence A013918 lists the nonzero numbers occurring at even n.

			a(60) = 7699 because Sum_{k=1..60} prime(k) = 7699 and 7699 is the smallest possible prime formed by summing 60 consecutive primes. - _Sean A. Irvine_, Jun 07 2024

Cf. A070934.

f[n_] := Block[{k = 1, s},If[Mod[n, 2] == 0,s = Sum[Prime[i], {i, k, k + n - 1}];If[PrimeQ[s], s, 0],While[s = Sum[Prime[i], {i, k, k + n - 1}] ; ! PrimeQ[s], k++ ];s]];Table[f[n], {n, 65}] (* Ray Chandler, Sep 27 2006 *)

Corrected and extended by Jim Nastos, Jun 15 2002
Extended by Ray Chandler, Sep 27 2006
a(60) corrected by Giovanni Resta, May 31 2017
Original a(60) restored by Sean A. Irvine, Jun 07 2024

A154422 Continue with summing & priming the A153089 (level 3) list to level 4.

Original entry on oeis.org

2, 50575480511, 158413287841, 379787123171, 88082548147771, 3939163325960453, 4342203121792903, 41672041797268133, 92013021551247323, 145937058697288751, 157891295660264779, 270930872865589619

Offset: 1

Michael J. Crowe (michaelcrowe117(AT)btinternet.com), Jan 09 2009

nonn

See comments in A153089.

M. J. Crowe, Table of n, a(n) for n=1,...,10000

A000040(Level 1),A013918(Level 2),A153089(Level 3),A154423(Level 5),A154424(Level 6)

lst2={}; s2=0; Do[s2=s2+Prime[n]; If[PrimeQ[s2], AppendTo[lst2, s2]], {n, 1000000}]; lst3={}; s3=0; Do[s3=s3+lst2[[n]];If[PrimeQ[s3], AppendTo[lst3, s3]], {n,1,Length[lst2]}]; lst3; lst4={}; s4=0; Do[s4=s4+lst3[[n]];If[PrimeQ[s4], AppendTo[lst4, s4]], {n,1,Length[lst3]}]; lst4

A197421 Primes of the form Sum_{k=1..n} prime(k)*prime(k+1).

Original entry on oeis.org

44839, 60859, 130411, 204749, 303767, 902971, 1027969, 1471633, 2514257, 3658769, 6908719, 7415743, 21966317, 28168523, 32413109, 37049567, 44034163, 47856331, 373881787, 425445073, 443609813, 564963589, 732111109, 758871401, 857997893, 995046653, 2489902577

Offset: 1

Michel Lagneau, Oct 14 2011

nonn

The corresponding values of n are 22, 24, 30, 34, 38, 52, 54, 60, 70, 78, 94, 96, ....

			For n = 22, a(1) = 44839 = 2*3 + 3*5 + 5*7 + ....+ 79*83 where 79 = prime(22) and 83 = prime(23).

Charles R Greathouse IV, Table of n, a(n) for n = 1..2243

Primes in A074745.

Cf. A013918.

p:=0:for n from 1 to 600 do:p:=p+ithprime(n)*ithprime(n+1): if type(p,prime)=true then printf(`%d, `,p): else fi:od:

Select[Table[Sum[Prime[k] Prime[k + 1], {k, n}], {n, 400}], PrimeQ] (* Alonso del Arte, Oct 14 2011 *)

v=List();t=0;p=2;forprime(q=3,1e4,if(isprime(t+=p*q),listput(v,t));p=q);Vec(v) \\ Charles R Greathouse IV, Oct 14 2011

Incorrect a(2243) and beyond removed from b-file by Andrew Howroyd, Feb 27 2018

A364797 Prime powers that are equal to the sum of the first k prime powers (not including 1) for some k.

Original entry on oeis.org

2, 5, 9, 29, 49, 137, 281, 359, 449, 1579, 2029, 2281, 2677, 5519, 12527, 13229, 15451, 17047, 22409, 24389, 25931, 29191, 32687, 42937, 45757, 53239, 56443, 59743, 70201, 81677, 90863, 95087, 101627, 113111, 169343, 200407, 206911, 256049, 302977, 330133, 338707, 356263

Offset: 1

Ilya Gutkovskiy, Aug 08 2023

nonn

			49 is a term because 49 is a prime power and 49 = 2 + 3 + 4 + 5 + 7 + 8 + 9 + 11.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A013918, A013921, A013932, A246655, A364796.

Select[Accumulate[Select[Range[2250], PrimePowerQ]], PrimePowerQ]

list(lim) = {my(s = 0); for(p = 1, lim, if(isprimepower(p), s += p; if(isprimepower(s), print1(s, ", "))));} \\ Amiram Eldar, Jun 20 2025

A020641 a(n)-th prime is sum of first k primes for some k.

Original entry on oeis.org

1, 3, 7, 13, 45, 60, 977, 1108, 2470, 2687, 2784, 3126, 3470, 3977, 4100, 4511, 4644, 5668, 6148, 6627, 6963, 8407, 9767, 10379, 11007, 11220, 11449, 12111, 12332, 23080, 25001, 28009, 28357, 28709, 29060, 29404, 30824, 32271, 33397, 33764, 47735, 52278

Offset: 1

N. J. A. Sloane, Renaud Lifchitz (100637.64(AT)CompuServe.COM), David W. Wilson

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..2500

Cf. A013916, A013917, A013918.

PrimePi /@ Select[ FoldList[Plus, 0, Prime@ Range@450], PrimeQ@# &] (* Robert G. Wilson v Sep 28 2006 *)
PrimePi/@Select[Accumulate[Prime[Range[500]]],PrimeQ] (* Harvey P. Dale, Jun 05 2013 *)

cp's OEIS Frontend

A189072 Semiprimes in A007504 (the sum of first n primes).

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A053790 Composite numbers arising as sum of first k primes.

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A071150 Primes p such that the sum of all odd primes <= p is also a prime.

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Maple

Mathematica

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A071151 Primes which are the sum of the first k odd primes for some k.

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Mathematica

PARI

Python

Extensions

A053845 Primes of form prime(1) + ... + prime(k) + 1.

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UBASIC

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A070281 Smallest prime which is the sum of n consecutive primes, or 0 if no such prime exists.

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Mathematica

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A154422 Continue with summing & priming the A153089 (level 3) list to level 4.

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