A099824 -id:A099824 - cp's OEIS frontend

A099825 Sum of the first 2^n primes.

2, 5, 17, 77, 381, 1851, 8893, 41741, 191755, 868151, 3875933, 17120309, 74950547, 325590115, 1405167561, 6029676711, 25750781177, 109495928099, 463852117169, 1958476902435, 8244703036797, 34615624751259, 144991244981985, 605994279458465, 2527803622205465

Offset: 0

Robert G. Wilson v, Oct 25 2004

nonn

Amiram Eldar, Table of n, a(n) for n = 0..65 (calculated using Kim Walisch's primesum program; terms 0..30 from Robert G. Wilson v)
Cino Hilliard, Sumprimes. [broken link]
Kim Walisch, Sum of the primes below x (primesum).

Cf. A000040, A000079, A006988, A007504, A033844, A099824, A099826, A121248.

NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; f[0] = 2; f[n_] := f[n] = Block[{k = 0, mx = 2^n/2, np = Prime[2^n/2], s = f[n - 1]}, While[k < mx, k++; np = NextPrim@np; s = s + np]; s]; Table[ f@n, {n, 0, 23}] (* Robert G. Wilson v, Aug 24 2006 *)
Module[{nn=22,ap},ap=Accumulate[Prime[Range[2^nn]]];Table[ap[[2^n]],{n,0,nn}]] (* Harvey P. Dale, Apr 12 2017 *)

a(n)=my(s); n=2^n; forprime(p=2,, s+=p; if(n--==0, return(s))) \\ Charles R Greathouse IV, Feb 16 2017 \\ corrected by David A. Corneth, Aug 05 2025

a(n) = A007504(A000079(n)). - Amiram Eldar, Jul 01 2024

A130739 Sum of primes < 2^n.

Original entry on oeis.org

0, 5, 17, 41, 160, 501, 1720, 6081, 22548, 80189, 289176, 1070091, 3908641, 14584641, 54056763, 202288087, 761593692, 2867816043, 10862883985, 41162256126, 156592635694, 596946687124, 2280311678414, 8729068693022

Offset: 1

Graeme McRae, Jul 06 2007

nonn

			a(3) is 17 because the sum of primes less than 2^3 is 2+3+5+7=17.

Kim Walisch, Table of n, a(n) for n = 1..80 (terms n=1..42 from Cino Hilliard)
Kim Walisch, primesum program.

Cf. A046731, A099824.

Table[Sum[Prime[i], {i, PrimePi[2^n-1]}], {n, 1, 10}]

a(n) = {s = 0; forprime(p=2, 2^n-1, s +=p); return (s);} \\ Michel Marcus, Jul 17 2013

a(n) = Sum_{i=2..2^n-1} A061397(i).

A099826 Sum of the first 3^n primes.

Original entry on oeis.org

2, 10, 100, 1264, 15116, 171148, 1864190, 19697700, 203534530, 2067129306, 20706364528, 205144046742, 2014349179358, 19632546354498, 190150622868298, 1831906588192414, 17567504017456404, 167794196312059488, 1597037992049539274

Offset: 0

Robert G. Wilson v, Oct 25 2004

nonn

Amiram Eldar, Table of n, a(n) for n = 0..36 (calculated using Kim Walisch's primesum program)
Cino Hilliard, Sum first 2^n Primes. [Link corrected by Alexey Beshenov, Jan 25 2009] [broken link]
Kim Walisch, Sum of the primes below x (primesum).

Cf. A000040, A000244, A006988, A007504, A038833, A099824, A099825.

NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; k = p = 1; s = 0; Do[ While[p = NextPrim[p]; s = s + p; k < 10^n, k++ ]; k++; Print[s], {n, 0, 16}]

a(n) = A007504(A000244(n)). - Amiram Eldar, Jul 01 2024

More terms from Cino Hilliard, Jan 14 2006

A134182 Difference between the sums of the first 10^n odd primes and the first 10^n odd positive integers > 1.

Original entry on oeis.org

38, 14478, 2688838, 396250152, 52261798440, 6472980453364, 770530574266708, 89262852894258444, 10138479465982004008, 1134379338819040693132, 125436174619351016716668, 13738971133578180130155676, 1493061976858459711065006050, 161191473337955042966337346114

Offset: 1

Enoch Haga, Oct 13 2007

nonn

Original name: 10^n-th difference between cumulative prime and odd sums.

Beginning at 3, compute the sums of the prime and odd sequences at 10^n and take the difference.

			a(1) = (3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31) - (3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21) = 158 - 120 = 38.
a(2) = 14478 because at 10^2, 100 sums of primes and odds, the prime sum is 24678, the odd sum is 10200 and the difference is 14478.

Chai Wah Wu, Table of n, a(n) for n = 1..23 (using terms from A006988 and A099824)

Cf. A006988, A071148, A005563, A099824, A134181.

10 N=1: A=2
20 A=nxtprm(A): B=B+A
30 N=N+2: D=D+N
40 if C=9 then print A;N;B;D;B-D: stop
50 C=C+1: if C<10 then 20

a(n) = A134181(10^n).
a(n) = A099824(n) + prime(10^n+1) - (10^n*(10^n+2)) - 2. - Chai Wah Wu, Mar 30 2020
a(n) = A071148(10^n) - (10^n+1)^2 + 1, where A071148 are the partial sums of odd primes, and N^2 is the sum of the first N odd integers. - M. F. Hasler, Aug 08 2025

Edited by M. F. Hasler, Aug 08 2025

A113489 Composite numbers of the form "Sum of the first 10^k primes".

Original entry on oeis.org

129, 3682913, 496165411, 62260698721, 7472966967499, 99262851056183695, 1234379338586942892505, 135436174616790289414111, 14738971133550183905879827, 1593061976858155930556059673, 171191473337951767580578821529, 18305009217958957209544913346089

Offset: 1

Cino Hilliard, Jan 09 2006

nonn

			The first 10^10 primes add up to 1234379338586942892505, which is composite.

David Baugh, Table of n, a(n) for n = 1..18

Composite terms of A099824.

Select[{ (* list A099824 *) }, ! PrimeQ[#] &] (* Robert G. Wilson v, Jan 14 2006 *)

Terms a(8) and beyond from David Baugh, Sep 27 2016

A131558 Primes which are the sum of the first 10^k primes for some k.

Original entry on oeis.org

2, 24133, 870530414842019, 11138479445180240497, 1948856725874605823462196012533593

Offset: 1

Cino Hilliard, Aug 27 2007

Also numbers n such that A099824(n) is prime.

			The sum of the first 10^1 primes is 129 which is not prime and not in the sequence. The sum of the first 10^9 primes is 11138479445180240497 which is prime and in the sequence.

Cf. A099824.

sm=0;  ct=0;  p=1;  bb=1;
{ while ( 1,
    p = nextprime(p+1);  sm += p;  ct++;
    if ( (ct==bb), if ( isprime(sm),print1(sm,", ")); bb*=10; );
); }

a(5) from David Baugh, Sep 27 2016

A222558 Least prime p such that 2np is a sum of 10^6 subsequent primes.

Original entry on oeis.org

3736971300983, 1868582442157, 1245659681423, 934275734321, 747425233469, 622762733249, 534156162737, 467093343419, 415824854441, 373728877943, 339743670103, 311538175027, 287741107327, 266994001331, 249114901193, 233613943273, 219815919913, 208214150917

Offset: 1

Zak Seidov, Feb 25 2013

nonn

Indices of first primes are: 64, 89, 65, 81, 84, 13, 338, 35, 768, 105, 91, 256, 537, 186, 32, 174, 51, 1469, 519, 277, 2132, 232, 241, 310, 179, 744, 1835, 535, 787, 167, 664, 1538, 1253, 484, 620, 1450, 961, 649, 1472, 166, 480, 918, 107, 418, 173, 370, 871, 1967, 71, 534.

First primes are: 311, 461, 313, 419, 433, 41, 2273, 149, 5849, 571, 467, 1619, 3877, 1109, 131, 1033, 233, 12281, 3719, 1787, 18671, 1459, 1523, 2053, 1063, 5653, 15737, 3853, 6037, 991, 4967, 12917, 10211, 3461, 4583, 12109, 7573, 4817, 12323, 983, 3413, 7187, 587, 2887, 1031, 2531, 6763, 17047, 353, 3851.

			a(1) = 3736971300983 = (p(64)+...+p(1000063))/2 = (311 + ... + 15486871)/2.
a(2) = 1868582442157 = (p(89)+...+p(1000088))/4 = (461 + ... + 15487253)/4.

Cf. A099824, A123086.

Do[s = 7472966967499 ; a = 2; b = 15485863; Do[s = s - a + (b = NextPrime[b]); a = NextPrime[a]; If[PrimeQ[s/m] , Print[{m, k, a, b, s/m}]; Break[]], {k, 2, 10^6}], {m, 2, 100, 2}]

A301599 Numbers k at which the ratio r(k) = (k-th prime) / (average of first k primes) reaches a record high.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 10, 12, 17, 25, 31, 35, 48

Offset: 1

Jon E. Schoenfield, Mar 24 2018

Equivalently, define the function f(k) = k*prime(k)/Sum_{j=1..k} prime(j); sequence lists numbers k such that f(k) > f(m) for all m < k.
a(14)=48 is the final term. Beyond k=48, r(k) decreases fairly smoothly (although nonmonotonically); see the Example section.
For m = 4..18, the first k > 48 at which r(k) < 2 - 1/m is 50, 53, 61, 775, 2678, 8973, 23483, 63535, 159863, 431988, 1091840, 2753459, 7186422, 18479367, 47260890, respectively. Does lim_{k->inf} r(k) equal 2? - Jon E. Schoenfield, Mar 27 2018

			The table below shows k, prime(k), the sum and average of the first k primes, and r(k), for each number k in the sequence, and also for k = 100, 1000, ..., 10^7.
.
   n|   a(n)=k  prime(k)             sum         avg    r(k)
  --+--------------------------------------------------------
   1|        1         2               2        2.000 1.00000
   2|        2         3               5        2.500 1.20000
   3|        3         5              10        3.333 1.50000
   4|        4         7              17        4.250 1.64706
   5|        5        11              28        5.600 1.96429
   6|        7        17              58        8.286 2.05172
   7|        9        23             100       11.111 2.07000
   8|       10        29             129       12.900 2.24806
   9|       12        37             197       16.417 2.25381
  10|       17        59             440       25.882 2.27955
  11|       25        97            1060       42.400 2.28774
  12|       31       127            1720       55.484 2.28895
  13|       35       149            2276       65.029 2.29130
  14|       48       223            4661       97.104 2.29650
           100       541           24133      241.330 2.24174
          1000      7919         3682913     3682.913 2.15020
         10000    104729       496165411    49616.541 2.11077
        100000   1299709     62260698721   622606.987 2.08753
       1000000  15485863   7472966967499  7472966.967 2.07225
      10000000 179424673 870530414842019 87053041.484 2.06110

Cf. A000040 (primes), A007504 (sum of first n primes), A006988 ((10^n)-th prime), A099824 (sum of first 10^n primes).

cp's OEIS Frontend

A099825 Sum of the first 2^n primes.

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A130739 Sum of primes < 2^n.

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A099826 Sum of the first 3^n primes.

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A134182 Difference between the sums of the first 10^n odd primes and the first 10^n odd positive integers > 1.

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UBASIC

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A113489 Composite numbers of the form "Sum of the first 10^k primes".

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A131558 Primes which are the sum of the first 10^k primes for some k.

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A222558 Least prime p such that 2*n*p is a sum of 10^6 subsequent primes.

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A301599 Numbers k at which the ratio r(k) = (k-th prime) / (average of first k primes) reaches a record high.

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A222558 Least prime p such that 2np is a sum of 10^6 subsequent primes.