A007504 -id:A007504

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A045345

Numbers k such that k divides sum of first k primes A007504(k).

+20
126

1, 23, 53, 853, 11869, 117267, 339615, 3600489, 96643287, 2664167025, 43435512311, 501169672991, 745288471601, 12255356398093, 153713440932055, 6361476515268337

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

a(10) and a(11) were found by Giovanni Resta (Nov 15 2004). He states that there are no other terms for primes p < 4011201392413. See link to Prime Puzzles, Puzzle 31 below. - Alexander Adamchuk, Aug 21 2006

a(13) > pi(2*10^13). - Donovan Johnson, Aug 23 2010

a(15) > 1.42*10^13. - Giovanni Resta, Jan 07 2020

a(16) > 1.55*10^14. - Bruce Garner, Mar 06 2021

a(17) > 6.5*10^15. - Paul W. Dyson, Sep 26 2022

Numbers k such that A090396(k) = 0. - Felix Fröhlich, May 05 2021

LINKS

Table of n, a(n) for n=1..16.

Javier Cilleruelo and Florian Luca, On the sum of the first n primes, Q. J. Math. 59:4 (2008), 14 pp.

Karl-Heinz Hofmann, Listening to the terms of A090396, YouTube video, 2021.

Kaisa Matomäki, A note on the sum of the first n primes, Quart. J. Math. 61 (2010), pp. 109-115.

Carlos Rivera, Puzzle 31.- The Average Prime number, APN(k) = S(Pk)/k, The Prime Puzzles & Problems Connection.

Eric Weisstein's World of Mathematics, Prime Sums

OEIS wiki, Sums of powers of primes divisibility sequences.

FORMULA

Matomäki proves that a(n) >> n^(24/19). - Charles R Greathouse IV, Jun 13 2012

EXAMPLE

23 is in the sequence because the sum of the first 23 primes is 874 and that's 23 * 38.

53 is in the sequence because the sum of the first 53 primes is 5830 and that's 53 * 110.

83 is not in the sequence because the sum of the first 83 primes is 15968, which leaves a remainder of 32 when divided by 83.

The sum of the first a(14) primes is equal to a(14)*196523412770096.

MATHEMATICA

s = 0; t = {}; Do[s = s + Prime[n]; If[ Mod[s, n] == 0, AppendTo[t, n]], {n, 1000000}]; t (* Alexander Adamchuk, Aug 21 2006 *)

nn = 4000000; With[{acpr = Accumulate[Prime[Range[nn]]]}, Select[Range[nn], Divisible[acpr[[#]], #] &]] (* Harvey P. Dale, Sep 14 2012 *)

Select[Range[100], Mod[Sum[Prime[i], {i, #}], #] == 0 &] (* Alonso del Arte, Mar 22 2014 based on Bill McEachen's Wolfram Alpha example *)

A007504 = Cases[Import["https://oeis.org/A007504/b007504.txt", "Table"], {_, _}][[All, 2]]; Select[Range[10^5], Divisible[A007504[[# + 1]], #] &] (* Robert Price, Mar 13 2020 *)

PROG

(PARI) s=0; n=0; forprime(p=2, 1e7, s+=p; if(s%n++==0, print1(n", "))) \\ Charles R Greathouse IV, Jul 15 2011

(PARI) isok(n) = (vecsum(primes(n)) % n) == 0; \\ Michel Marcus, Nov 26 2020

(Python)

from itertools import accumulate, count, islice

from sympy import prime

def A045345_gen(): return (i+1 for i, m in enumerate(accumulate(prime(n) for n in count(1))) if m % (i+1) == 0)

A045345_list = list(islice(A045345_gen(), 5)) # Chai Wah Wu, Feb 23 2022

CROSSREFS

Cf. A007504, A090396, A171399.

KEYWORD

nonn,nice,more

AUTHOR

Jud McCranie

EXTENSIONS

More terms from Alexander Adamchuk, Aug 21 2006

a(12) from Donovan Johnson, Aug 23 2010

a(13) from Robert Price, Mar 17 2013

a(14) from Giovanni Resta, Jan 07 2020

a(15) from Bruce Garner, Mar 06 2021

a(16) from Paul W. Dyson, Sep 26 2022

STATUS

approved

A134128

a(n) = A007504(A134126(n)).

+20
6

2, 5, 17, 58, 129, 5117, 43201, 329401, 1459228, 111461983, 269553485, 316504138, 734845192, 6185946407, 10731178047, 22691403557, 148086969623, 474635764987, 6777574922490, 30458710811303, 215730284567463, 761593685331414, 5875984396617486, 10893968395261326

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

This is the prime partial sum just before the prime is added to achieve an integer mean as defined in A134125.

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..32

EXAMPLE

The sequence begins with a(1) = 2 (to which 3 is added which leads to a sum 5 associated with A134125(1)).

a(4) = 58 (to which the prime 19 is added, a sum of 77, associated with A134125(4)).

PROG

(UBASIC) 10 'primes using counters 20 N=3:C=1:R=5:print 2; 3, 5 30 A=3:S=sqrt(N) 40 B=N\A 50 if B*A=N then N=N+2:goto 30 60 A=A+2:O=A 70 if A<=sqrt(N) then 40 80 C=C+1 90 R=R+N:T=R/C:U=R-N 100 if T=int(T) then print C; U; N; R; T:stop 110 N=N+2:goto 30

(PARI) lista(pmax) = {my(k = 0, s = 2); forprime(p = 3, pmax, k++; s += p; if(!(s % k), print1(s-p, ", "))); } \\ Amiram Eldar, Apr 30 2024

CROSSREFS

Cf. A134125, A134126, A134127, A134129.

KEYWORD

less,nonn

AUTHOR

Enoch Haga, Oct 09 2007

EXTENSIONS

Edited by R. J. Mathar, Apr 17 2009

a(21)-a(24) from Amiram Eldar, Apr 30 2024

STATUS

approved

A134129

Prime partial sums A007504(k+1) such that A007504(k+1)/k is integer.

+20
6

5, 10, 28, 77, 160, 5350, 43940, 331608, 1464099, 111509916, 269629588, 316586861, 734973855, 6186337680, 10731699088, 22692172980, 148089006456, 474639489984, 6777589645423, 30458742769120, 215730372141680, 761593852850347, 5875984874989879, 10893969051902225

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

See A134125 and cross-references.

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..32

FORMULA

a(n) = A007504(A134126(n+1)) .

EXAMPLE

A007504(2)/1 = 5/1 = 5 is integer, so 5 is added to the sequence.

A007504(3)/2 = 10/2 = 5 is integer, so 10 is added to the sequence.

A007504(4)/3 = 17/3 is not integer, so 17 is not added to the sequence.

PROG

(PARI) lista(pmax) = {my(k = 0, s = 2); forprime(p = 3, pmax, k++; s += p; if(!(s % k), print1(s, ", "))); } \\ Amiram Eldar, Apr 30 2024

CROSSREFS

Cf. A007504, A134125, A134126 A134127, A134128.

KEYWORD

nonn

AUTHOR

Enoch Haga, Oct 09 2007

EXTENSIONS

Edited by R. J. Mathar, May 15 2009

a(21)-a(24) from Amiram Eldar, Apr 30 2024

STATUS

approved

A189072

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

+20
5

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

LINKS

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

EXAMPLE

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

MAPLE

PS:= ListTools:-PartialSums(select(isprime, [2, seq(i, i=3..10^4, 2)])):

select(numtheory:-bigomega = 2, PS); # Robert Israel, Jun 23 2017

MATHEMATICA

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 *)

PROG

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

CROSSREFS

Cf. A013918 (primes in A007504).

Cf. A007504, A001358.

KEYWORD

nonn

AUTHOR

Zak Seidov, Apr 16 2011

STATUS

approved

A122207

Primes of the form A109724[n] = A007504[n^2] or the sum of the first n^2 primes.

+20
4

2, 17, 8893, 24133, 768373, 1583293, 2180741, 3875933, 6426919, 173472547, 289093219, 741938801, 2738357903, 2895147163, 3058653607, 17056871941, 24109439269, 26602406177, 29286422621, 62073696299, 65420584637, 68902997299

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Numbers n such that A109724[n] = A007504[n^2] is a prime are listed in A122208[n] = {1,2,8,10,22,26,28,32,36,78,88,110,150,152,154,...}.

LINKS

Ray Chandler, Table of n, a(n) for n = 1..1000

FORMULA

a(n) = A109724[ A122208[n] ] = A007504[ A122208[n]^2 ].

EXAMPLE

Prime 17 = 2 + 3 + 5 + 7 = a(2) is equal to the sum of the first 4 = 2^2 primes.

Prime 24133 = a(4) is equal to sum of the first 100 = 10^2 primes.

MATHEMATICA

s=0; Do[s=Sum[Prime[k], {k, 1, n^2}]; If[PrimeQ[s], Print[{n, n^2, s}]], {n, 1, 154}]

Select[Table[Total[Prime[Range[n^2]]], {n, 500}], PrimeQ] (* Harvey P. Dale, Jul 20 2024 *)

CROSSREFS

Cf. A122208, A109724, A007504.

KEYWORD

nonn

AUTHOR

Alexander Adamchuk, Aug 25 2006

EXTENSIONS

More terms from Ray Chandler, Dec 02 2018

STATUS

approved

A122208

Numbers n such that the sum of the first n^2 primes A109724(n) = A007504(n^2) is a prime.

+20
4

1, 2, 8, 10, 22, 26, 28, 32, 36, 78, 88, 110, 150, 152, 154, 232, 252, 258, 264, 316, 320, 324, 368, 376, 426, 496, 516, 532, 608, 644, 666, 686, 764, 828, 832, 880, 932, 958, 1020, 1090, 1096, 1106, 1122, 1156, 1174, 1206, 1264, 1280, 1282, 1290, 1296, 1326

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Corresponding primes that are equal to the sum of the first a(n)^2 primes are listed in A122207(n) = {2, 17, 8893, 24133, 768373, 1583293, 2180741, 3875933, 6426919, 173472547, 289093219, 741938801, 2738357903, 2895147163, 3058653607, ...}. - Robert G. Wilson v, Sep 29 2006

LINKS

Ray Chandler, Table of n, a(n) for n = 1..1000

FORMULA

A122207(n) = A109724( a(n) ) = A007504( a(n)^2 ). - Robert G. Wilson v, Sep 29 2006

MATHEMATICA

s = 0; t = {}; Do[s = s + Sum[Prime@k, {k, (n - 1)^2 + 1, n^2}]; If[PrimeQ@s, AppendTo[t, n]], {n, 1341}]; t (* Robert G. Wilson v *)

CROSSREFS

Cf. A122207, A109724, A007504.

KEYWORD

nonn

AUTHOR

Alexander Adamchuk, Aug 25 2006

EXTENSIONS

More terms from Robert G. Wilson v, Sep 29 2006

STATUS

approved

A309036

a(n) = gcd(A007504(n), A014285(n)).

+20
4

2, 1, 1, 17, 2, 1, 1, 7, 2, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 3, 8, 1, 1, 1, 20, 43, 1, 3, 4, 1, 1, 1, 28, 1, 1, 3, 2, 1, 1, 1, 2, 3, 107, 1, 4, 1, 1, 1, 2, 7, 1, 1, 10, 3, 1, 1, 30, 1, 1, 1, 2, 5, 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 2, 1, 1, 1, 142, 1, 1, 3, 4, 1, 1, 11, 2, 1, 1, 1, 10

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

a(n) is even if n == 1 (mod 4).

LINKS

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

FORMULA

a(n) = A007504(n)/A307716(n) = A014285(n)/A306834(n).

EXAMPLE

a(4) = gcd(2+3+5+7, 1*2+2*3+3*5+4*7) = gcd(17,51) = 17.

MAPLE

S1:= 0: S2:= 0:

for n from 1 to 100 do

p:= ithprime(n);

S1:= S1 + p;

S2:= S2 + n*p;

A[n]:= igcd(S1, S2);

od:

seq(A[i], i=1..100);

MATHEMATICA

GCD @@ # & /@ Rest@ Nest[Append[#1, {#1[[-1, 1]] + #3, #1[[-1, -1]] + #2 #3}] & @@ {#1, #2, Prime@ #2} & @@ {#, Length@ #} &, {{0, 0}}, 89] (* Michael De Vlieger, Jul 08 2019 *)

PROG

(PARI) a(n) = gcd(sum(k=1, n, prime(k)), sum(k=1, n, k*prime(k))); \\ Michel Marcus, Jul 09 2019

(Magma) p:=PrimesUpTo(1000); [Gcd(&+[p[j]:j in [1..m]], &+[j*p[j]:j in [1..m]]): m in [1..90]]; // Marius A. Burtea, Jul 09 2019

CROSSREFS

Cf. A007504, A014285, A306834, A307414 (a(n)=1), A307716, A309055, A309056.

KEYWORD

nonn

AUTHOR

Robert Israel, Jul 08 2019

STATUS

approved

A165659

Numerators of A007504 divided by A033955, starting from the second term of A033955.

+20
3

2, 5, 5, 17, 28, 41, 58, 77, 50, 129, 16, 197, 119, 281, 164, 127, 440, 501, 568, 71, 356, 791, 46, 321, 530, 1161, 1264, 457, 1480, 1593, 344, 1851, 284, 2127, 2276, 809, 2584, 2747, 1457, 441, 1633, 1149, 3638, 3831, 1007, 4227, 4438

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Conjecture: with the exception of the second term, 2 <= a(n)/A165660(n) < 3.

LINKS

Table of n, a(n) for n=1..47.

PROG

(PARI) a1(n)=sum(i=1, n, prime(i));

b1(n)=sum(i=1, n, prime(n+1)%prime(i));

a(n)=if(n<0, 0, numerator(a1(n)/b1(n)));

for(n=1, 50, print1(a(n) ", "))

CROSSREFS

Cf. A007504, A033995, A002110, A102647, A165657, A165658, A165660.

KEYWORD

easy,nonn

AUTHOR

Creighton Dement, Sep 24 2009

EXTENSIONS

Typo in definition corrected by Creighton Dement, Oct 09 2009

STATUS

approved

A165660

Denominators of A007504 divided by A033955, starting from the second term of A033955.

+20
3

1, 3, 2, 8, 13, 18, 27, 29, 23, 56, 7, 74, 44, 98, 67, 49, 171, 200, 217, 28, 137, 309, 17, 116, 209, 448, 471, 174, 571, 629, 137, 739, 111, 793, 853, 318, 997, 1002, 560, 164, 610, 446, 1419, 1466, 385, 1615, 1573, 1633, 1707, 1825, 946, 662, 2221, 781, 1198

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Conjecture: with the exception of the second term, 2 <= A165659(n)/a(n) < 3.

LINKS

Table of n, a(n) for n=1..55.

PROG

(PARI) a1(n)=sum(i=1, n, prime(i)); b1(n)=sum(i=1, n, prime(n+1)%prime(i)); a(n)=if(n<0, 0, denominator(a1(n)/b1(n))); for(n=1, 50, print1(a(n) ", "))

CROSSREFS

Cf. A007504, A033995, A002110, A102647, A165657, A165658, A165659

KEYWORD

easy,nonn

AUTHOR

Creighton Dement, Sep 24 2009

EXTENSIONS

Terms corrected by Creighton Dement, Oct 03 2009

Removed a conjecture - R. J. Mathar, Oct 09 2009

Typo in definition corrected by Creighton Dement, Oct 09 2009

STATUS

approved

A267084

a(n) = ceiling(A007504(n)/n) - floor(A007504(n)/n); a(n) is 0 if n divides the sum of first n primes, 1 otherwise.

+20
3

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

(list; graph; refs; listen; history; text; internal format)

OFFSET

COMMENTS

a(n) = 0 for n=1, 23, 53, 853, ... see A045345.

It is conjectured that there are infinitely many zeros, but that their density is zero.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..12000

Javier Cilleruelo and Florian Luca, On the sum of the first n primes, Q. J. Math. 59:4 (2008), 14 pp.

Index entries for characteristic functions

FORMULA

a(n) = A225804(n) - A060620(n).

MATHEMATICA

Table[Ceiling[(Plus@@Prime[Range[n]])/n]-Floor[(Plus@@Prime[Range[n]])/n], {n, 100}]

PROG

(PARI)

up_to = 105

v007504 = vector(up_to, i, prime(i));

for(i=2, up_to, v007504[i] = v007504[i-1]+v007504[i]); \\ Taking partial sums of primes here.

A007504(n) = v007504[n];

A267084(n) = if(!(A007504(n)%n), 0, 1); \\ Antti Karttunen, Sep 24 2017

(Scheme) (define (A267084 n) (if (zero? (modulo (A007504 n) n)) 0 1)) ;; Antti Karttunen, Sep 24 2017

CROSSREFS

Cf. A007504, A045345 (positions of zeros), A060620, A158682, A225804.

KEYWORD

nonn,easy

AUTHOR

Ctibor O. Zizka, Jan 10 2016

EXTENSIONS

More terms and the second description added to the name by Antti Karttunen, Sep 24 2017

STATUS

approved

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