A034707 -id:A034707 - cp's OEIS frontend

A065867 Primes which are the sum of a prime number of consecutive primes.

5, 23, 31, 41, 53, 59, 67, 71, 83, 97, 101, 109, 131, 139, 173, 181, 197, 199, 211, 223, 233, 251, 263, 269, 271, 281, 311, 331, 349, 353, 373, 401, 421, 431, 439, 443, 449, 457, 463, 487, 491, 499, 503, 523, 563, 587, 593, 607, 617, 631, 647, 659, 661, 677

Offset: 1

Henry Bottomley, Dec 07 2001

			5 = 2 + 3.
23 = 5 + 7 + 11.
31 = 7 + 11 + 13.
41 = 11 + 13 + 17.
53 = 5 + 7 + 11 + 13 + 17.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A001043, A034707, A034962, A034965, A138591, A348790, A348791.

Cf. A034962 (subsequence).

lst={};Do[s=Prime[m];k=1;Do[p=Prime[n];s+=p;k++;If[PrimeQ[s]&&PrimeQ[k],If[s<=10837,AppendTo[lst,s]]],{n,m+1,5*5!}],{m,5*5!}];lst=Take[Union@lst,500] (* Vladimir Joseph Stephan Orlovsky, Sep 13 2009 *)
Module[{nn=60,prs},prs=Prime[Range[nn]];Take[Select[Union[ Flatten[ Table[ Total/@ Partition[prs,n,1],{n,prs}]]],PrimeQ],nn]] (* Harvey P. Dale, Aug 12 2016 *)

A133576 Numbers which are sums of consecutive composites.

Original entry on oeis.org

4, 6, 8, 9, 10, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 74, 75, 76, 77, 78, 79, 80, 81

Offset: 1

Jonathan Vos Post, Dec 26 2007

This is to composites A002808 as A034707 is to primes A000040. The complement of this sequence, numbers which are not sums of consecutive composites, begins 1, 2, 3, 5, 7, ... (A140464).

			Every composite is in this sequence as one consecutive composite. We account for primes thus:
a(10) = 17 = 8 + 9.
a(12) = 19 = 9 + 10.
a(16) = 23 = 6 + 8 + 9.
a(22) = 29 = 14 + 15.
a(24) = 31 = 9 + 10 + 12.
a(30) = 37 = 4 + 6 + 8 + 9 + 10.
a(34) = 41 = 20 + 21 = 12 + 14 + 15.
a(36) = 43 = 21 + 22.
Not included = 47.
a(45) = 53 = 26 + 27 = 8 + 9 + 10 + 12 + 14.
a(51) = 59 = 18 + 20 + 21 = 6 + 8 + 9 + 10 + 12 + 14.
Not included = 61.
a(58) = 67 = 33 + 34 = 21 + 22 + 24 = 10 + 12 + 14 + 15 + 16.
a(62) = 71 = 35 + 36 = 22 + 24 + 25 = 4 + 6 + 8 + 9 + 10 + 12 + 14.
Not included = 73.
a(69) = 79 = 39 + 40.
a(73) = 83 = 14 + 15 + 16 + 18 + 20.
a(79) = 89 = 44 + 45.
a(87) = 97 = 48 + 49 = 22 + 24 + 25 + 26.
a(91) = 101 = 50 + 51.
a(93) = 103 = 51 + 52.

Cf. A002808, A034707, A037174, A140464 (complement).

isA133576 := proc(n)
    local i,j ;
    for i from 1 do
        if A002808(i) > n then
            return false;
        end if;
        for j from i do
            s := add( A002808(l),l=i..j) ;
            if s > n then
                break;
            elif s = n then
                return true;
            end if;
        end do:
    end do:
end proc:
A133576 := proc(n)
    local a;
    if n = 1 then
        return A002808(1) ;
    else
        for a from procname(n-1)+1 do
            if isA133576(a) then
                return a;
            end if;
        end do:
    end if ;
end proc:
seq(A133576(n),n=1..71) ; # R. J. Mathar, Feb 14 2015

okQ[n_] := If[CompositeQ[n], True, MemberQ[IntegerPartitions[n, All, Select[Range[n], CompositeQ]], p_List /; Length[p] == Length[Union[p]] && AllTrue[Complement[Range[p[[-1]], p[[1]]], p], PrimeQ]]];
Select[Range[150], okQ] (* Jean-François Alcover, Oct 27 2023 *)

A166709 Number of distinct integers expressible as sums of consecutive primes up to n-th prime.

Original entry on oeis.org

1, 3, 5, 9, 14, 20, 25, 32, 39, 49, 56, 68, 78, 92, 105, 115, 128, 144, 157, 171, 192, 211, 231, 253, 276, 297, 319, 339, 366, 396, 419, 442, 473, 500, 533, 561, 592, 628, 665, 691, 726, 759, 794, 832, 868, 900, 936, 979, 1028, 1070, 1114, 1159, 1208, 1248, 1298

Offset: 1

Zak Seidov, Oct 18 2009

nonn

a(n) <= n(n+1)/2 (= T(n), A000217 Triangular numbers) because some sums give the same value. E.g., a(4)=9, T(4)=10, a(4)=T(4)-1, because 5 is equal to two sums 2+3, and 5. For n=100 "deficit" is 700: a(100)=4350=T(100)-700=5050-700.

			n=4: 9 distinct integers = 2, 3, 5, 7, 8(=3+5), 10(=2+3+5), 12(=5+7), 15(=3+5+7), and 17(=2+3+5+7);
n=10: 49 distinct integers: 2, 3, 5, 7, 8, 10, 11, 12, 13, 15, 17, 18, 19, 23, 24, 26, 28, 29, 30, 31, 36, 39, 41, 42, 48, 49, 52, 53, 56, 58, 59, 60, 67, 71, 72, 75, 77, 83, 88, 90, 95, 98, 100, 101, 112, 119, 124, 127, 129.
From _Rick L. Shepherd_, Oct 18 2009: (Start)
The first 6 rows of actual sums are:
n=1: 2
n=2: 2,3,5
n=3: 2,3,5,8,10
n=4: 2,3,5,7,8,10,12,15,17
n=5: 2,3,5,7,8,10,11,12,15,17,18,23,26,28
n=6: 2,3,5,7,8,10,11,12,13,15,17,18,23,24,26,28,31,36,39,41 (End)

Zak Seidov, Table of n, a(n) for n=1..1000

Cf. A034707 (numbers which are sums of consecutive primes).

Cf. A152415, A152430.

Table[Length[Union[Total/@Flatten[Table[Partition[Prime[Range[m]],k,1],{k,m}],1]]],{m,100}]

A166709(n)=#Set(concat(vector(n,i,vector(i,j,sum(k=j,i,prime(k)))))) \\ M. F. Hasler, Oct 18 2009

A163246 Squares which can be represented as the sum of consecutive primes in more than one way.

Original entry on oeis.org

36, 100, 576, 841, 961, 1764, 1849, 2209, 2304, 7056, 22801, 24649, 25600, 30276, 31684, 32400, 36481, 39601, 40000, 47524, 48400, 48841, 57600, 58081, 66564, 69169, 69696, 76729, 77284, 80089, 85849, 93636, 94864, 96721, 112896, 119716, 128164, 134689, 138384, 140625, 142884, 147456

Offset: 1

Gaurav Kumar, Jul 23 2009

nonn

			6^2 = 36 = 5 + 7 + 11 + 13 = 17 + 19.

Alois P. Heinz, Table of n, a(n) for n = 1..300

Cf. A000290, A034707, A050936, A067372, A084143, A163244.

{ A000290 } intersect { A034707 }.
{ A000290 } intersect { A050936 }.
k^2 such that A084143(k^2) > 1. - Georg Fischer, Jul 08 2022

Offset corrected by Arkadiusz Wesolowski, Mar 28 2012

Duplicate 2304 removed and some missing terms inserted by Georg Fischer, Jul 08 2022

A272713 Prime powers (p^k, k>=2) that are the sum of consecutive prime numbers.

Original entry on oeis.org

8, 49, 121, 128, 169, 243, 625, 841, 961, 1331, 1369, 1681, 1849, 2209, 3125, 5329, 6241, 6859, 6889, 8192, 10201, 11449, 11881, 12167, 12769, 16384, 18769, 22801, 24649, 26569, 32768, 36481, 39601, 44521

Offset: 1

Altug Alkan, May 05 2016

nonn

In other words, prime powers (p^k, k>=2) that are the sum of two or more consecutive prime numbers.
Intersection of A025475 and A034707.
Terms of this sequence are 2^3, 7^2, 11^2, 2^7, 13^2, 3^5, 5^4, 29^2, ...

			8 is a term because 8 = 2^3 = 3 + 5.
49 is a term because 49 = 7^2 = 13 + 17 + 19.
121 is a term because 121 = 11^2 = 37 + 41 + 43.

Charles R Greathouse IV, Table of n, a(n) for n = 1..4469

Cf. A025475, A034707, A067377, A050936.

list(lim)=my(v=List(),n=1,p,q,t,s); while(1, t=primes(n++); p=2; q=t[n]; s=vecsum(t); if(s>lim, return(Set(v))); while(s<=lim, if(isprimepower(s)>1, listput(v,s)); q=nextprime(q+1); s+=q-p; p=nextprime(p+1))) \\ Charles R Greathouse IV, May 05 2016

a(9)-a(34) from Charles R Greathouse IV, May 05 2016

A287027 Least sum s of consecutive prime numbers starting with prime(n) such that s is a perfect square.

Original entry on oeis.org

100, 961, 36, 14017536, 484, 49, 36, 134689, 354025, 80089, 443556, 121, 47524, 7744, 100, 700569, 344956329, 48841, 5329, 144, 324, 39601, 22801, 8649, 239438955625, 12250000, 197136, 222784, 147456, 319225, 316969, 24649, 576, 2975625, 7396, 21316, 70036245333532859364

Offset: 1

Zak Seidov, May 18 2017

nonn

Squares that are the sum of 4 consecutive primes: 36, 324, 576, 1764, 2304, 4900, 20736, 63504, 66564, 128164, 142884, 150544, 156816, 183184, 236196, 256036, 260100, 311364, 369664, 414736.
Squares that are the sum of 5 consecutive primes: 961, 1089, 1681, 17689, 18769, 21025, 23409, 45369, 76729, 80089, 97969, 124609, 218089, 235225, 290521, 421201, 434281.
Squares that are the sum of 6 consecutive primes: 3600, 24336, 25600, 47524, 66564, 98596, 129600, 138384, 228484, 236196, 331776, 379456, 404496, 490000, 559504.
Squares that are the sum of 7 consecutive primes: 169, 625, 2209, 10201, 25921, 235225, 342225, 361201, 380689, 383161, 426409, 508369, 531441, 537289, 543169, 564001, 603729.
Note that A007504(m) - A007504(n) ~ m^2 log(m)/2 as m -> infinity.  Heuristically this has probability ~ 1/(m sqrt(2 log(m))) of being a square.  Since the sum of these probabilities diverges, on the basis of the second Borel-Cantelli lemma we should expect a(n) to exist.  Of course, this is not a proof.  Moreover, since the sum diverges very slowly, we might expect some very large values of a(n). - Robert Israel, May 18 2017

			Sum of set {2,3,5,7,11,13,17,19,23} is 100 = 10^2, sum of set {3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83} is 961=31^2, sum of set {5,7,11,13}=36=6^2.

Cf. A062703 (squares that are the sum of 2 consecutive primes), A080665 (squares that are the sum of 3 consecutive primes), A034707 (numbers that are sums of consecutive primes).

Cf. A007504.

f:= proc(n) local p, s;
  p:= ithprime(n); s:= p;
while not issqr(s) do p:= nextprime(p); s:= s+p od:
  s
end proc:
map(f, [$1..36]); # Robert Israel, May 18 2017

Table[Set[{k, s}, {n, 0}]; While[! IntegerQ@ Sqrt[AddTo[s, Prime@ k]], k++]; s, {n, 36}] (* Michael De Vlieger, May 20 2017 *)

a(n) = my(s=0); forprime(p=prime(n), , s=s+p; if(issquare(s), return(s))) \\ Felix Fröhlich, May 25 2017

Missing a(25) and a(37) from Giovanni Resta, May 18 2017

A309770 Numbers that are sums of one or more consecutive primes in more than one way.

Original entry on oeis.org

5, 17, 23, 31, 36, 41, 53, 59, 60, 67, 71, 72, 83, 90, 97, 100, 101, 109, 112, 119, 120, 127, 131, 138, 139, 143, 152, 173, 180, 181, 187, 197, 199, 204, 210, 211, 221, 223, 228, 233, 240, 251, 258, 263, 269, 271, 276, 281, 287, 300, 304, 311, 323, 330, 331, 340, 349

Offset: 1

Ilya Gutkovskiy, Aug 16 2019

nonn

Contains A067372 as a subsequence.

			5 is in the sequence because it can be written as either 5 or 2 + 3.
36 is the sequence because it can be written as either 5 + 7 + 11 + 13 or 17 + 19.

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime Sums

Cf. A000040, A034707, A050936, A054845, A054996, A067372, A067377.

N:= 1000: # for terms <= N
P:= select(isprime, [2, seq(i,i=3..N,2)]):
S:= [0,op(ListTools:-PartialSums(P))]:
V:= Vector(N):
for i from 1 to nops(S) do
  for j from i-1 to 1 by -1 do
    v:= S[i]-S[j];
    if v > N then break fi;
    V[v]:= V[v]+1;
od od:
select(t -> V[t]>1, [$1..N]); # Robert Israel, Aug 22 2019

A054845(a(n)) > 1.

A338446 Numbers that are sums of consecutive odd primes.

Original entry on oeis.org

3, 5, 7, 8, 11, 12, 13, 15, 17, 18, 19, 23, 24, 26, 29, 30, 31, 36, 37, 39, 41, 42, 43, 47, 48, 49, 52, 53, 56, 59, 60, 61, 67, 68, 71, 72, 73, 75, 78, 79, 83, 84, 88, 89, 90, 95, 97, 98, 100, 101, 102, 103, 107, 109, 112, 113, 119, 120, 121, 124, 127, 128, 131, 132, 137

Offset: 1

Ilya Gutkovskiy, Oct 28 2020

nonn

			67 is in the sequence because 67 = 7 + 11 + 13 + 17 + 19.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A007504, A034707, A050936, A053790, A065091, A071148.

q:= proc(n) local p, q, s; p, q, s:= prevprime(n+1)$3;
      do if p=2 then return false
       elif s=n then return true
       elif sAlois P. Heinz, Oct 31 2020

okQ[n_] := Module[{p, q, s}, {p, q, s} = Table[NextPrime[n + 1, -1], {3}]; While[True, Which[
     p == 2, Return@ False,
     s == n, Return@ True,
     s < n, p = NextPrime[p, -1]; s = s + p,
     True, s = s - q; q = NextPrime[q, -1]]]];
Select[Range[3, 150], okQ] (* Jean-François Alcover, Feb 21 2022, after Alois P. Heinz *)

from sympy import prevprime
def ok(n):
    if n < 2: return False
    p, q, s = [prevprime(n+1)] * 3
    while True:
        if p == 2: return False
        if s == n: return True
        elif s < n: p = prevprime(p); s += p
        else: s -= q; q = prevprime(q)
print([k for k in range(150) if ok(k)]) # Michael S. Branicky, Feb 21 2022 after Alois P. Heinz

A352423 Numbers that are the sum of some number of consecutive prime cubes.

Original entry on oeis.org

8, 27, 35, 125, 152, 160, 343, 468, 495, 503, 1331, 1674, 1799, 1826, 1834, 2197, 3528, 3871, 3996, 4023, 4031, 4913, 6859, 7110, 8441, 8784, 8909, 8936, 8944, 11772, 12167, 13969, 15300, 15643, 15768, 15795, 15803, 19026, 23939, 24389, 26136, 27467, 27810, 27935, 27962, 27970, 29791

Offset: 1

Michel Marcus, Apr 26 2022

nonn

Michael S. Branicky, Table of n, a(n) for n = 1..10000
Cathal O'Sullivan, Jonathan P. Sorenson, and Aryn Stahl, An Algorithm to Find Sums of Consecutive Powers of Primes, arXiv:2204.10930 [math.NT], 2022-2023. See S3 p. 10.

Cf. A030078, A034707, A340771.

lista(nn) = {my(list = List(), ip = primepi(nn), vp = primes(ip)); for(i=1, ip, my(s=vp[i]^3); listput(list, s); for (j=i+1, ip, s += vp[j]^3; if (s >vp[ip]^3, break); listput(list, s); ); ); Vec(vecsort(list, , 8)); }

import heapq
from sympy import prime
from itertools import islice
def agen(): # generator of terms
    p = prime(1)**3; h = [(p, 1, 1)]; nextcount = 2
    while True:
        (v, s, l) = heapq.heappop(h)
        yield v
        if v >= p:
            p += prime(nextcount)**3
            heapq.heappush(h, (p, 1, nextcount))
            nextcount += 1
        v -= prime(s)**3; s += 1; l += 1; v += prime(l)**3
        heapq.heappush(h, (v, s, l))
print(list(islice(agen(), 47))) # Michael S. Branicky, Apr 26 2022

A351125 Numbers that are sums of consecutive primorial numbers.

Original entry on oeis.org

1, 2, 3, 6, 8, 9, 30, 36, 38, 39, 210, 240, 246, 248, 249, 2310, 2520, 2550, 2556, 2558, 2559, 30030, 32340, 32550, 32580, 32586, 32588, 32589, 510510, 540540, 542850, 543060, 543090, 543096, 543098, 543099, 9699690, 10210200, 10240230, 10242540, 10242750

Offset: 1

Ilya Gutkovskiy, Feb 22 2022

nonn

			2550 is in the sequence because 2550 = 30 + 210 + 2310 = 2*3*5 + 2*3*5*7 + 2*3*5*7*11.

Eric Weisstein's World of Mathematics, Primorial
Index entries for sequences related to primorial numbers

Cf. A002110, A034707, A050936, A143293, A276156, A290249.

cp's OEIS Frontend

A065867 Primes which are the sum of a prime number of consecutive primes.

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Mathematica

A133576 Numbers which are sums of consecutive composites.

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Maple

Mathematica

A166709 Number of distinct integers expressible as sums of consecutive primes up to n-th prime.

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Mathematica

PARI

A163246 Squares which can be represented as the sum of consecutive primes in more than one way.

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A272713 Prime powers (p^k, k>=2) that are the sum of consecutive prime numbers.

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PARI

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A287027 Least sum s of consecutive prime numbers starting with prime(n) such that s is a perfect square.

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Maple

Mathematica

PARI

Extensions

A309770 Numbers that are sums of one or more consecutive primes in more than one way.

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Comments

Examples

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Programs

Maple

Formula

A338446 Numbers that are sums of consecutive odd primes.

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