A071089 -id:A071089 - cp's OEIS frontend

A024011 Numbers k such that the k-th prime divides the sum of the first k primes.

1, 3, 20, 31464, 22096548, 1483892396791177

Offset: 1

a(6) > pi(10^12) = 37607912018. - Jon E. Schoenfield, Sep 11 2008
a(6) > pi(10^14) = 3204941750802. - Giovanni Resta, Jan 09 2014
a(7) > 6.5*10^15. - Paul W. Dyson, Sep 27 2022

			The third prime, 5, divides 2 + 3 + 5 = 10, so 3 is in the sequence.
2 + 3 + 5 + 7 = 17, which is not divisible by the fourth prime, 7, so 4 is not in the sequence.

Cf. A007506, A028581, A028582, A071089.

s = 0; For[i = 1, i <= 5 * 10^7, i++, s = s + Prime[i]; If[Mod[s, Prime[i + 1]] == 0, Print[i + 1]]]
With[{prs = Prime[Range[221000000]]}, PrimePi /@ Transpose[Select[ Thread[ {Accumulate[prs], prs}], Divisible[#[[1]], #[[2]]] &]][[2]]] (* Harvey P. Dale, Jul 23 2013 *)
nMax = 50000; primeSums = Accumulate[Prime[Range[nMax]]]; Select[Range[nMax], Divisible[primeSums[[#]], Prime[#]] &] (* Alonso del Arte, Nov 11 2019 *)

s=0; t=0; for(w=2,1000000000,if(isprime(w),s=s+w; t=t+1; if(s%w,print(t)),))

a(5) from Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), May 14 2000

a(6) from Paul W. Dyson, Apr 16 2022

A082974 a(n) = (a(n-1) + p(n)) mod p(n+1).

Original entry on oeis.org

2, 0, 5, 1, 12, 8, 6, 2, 25, 23, 17, 13, 11, 7, 1, 54, 52, 46, 42, 40, 34, 30, 24, 16, 12, 10, 6, 4, 0, 113, 109, 103, 101, 91, 89, 83, 77, 73, 67, 61, 59, 49, 47, 43, 41, 29, 17, 13, 11, 7, 1, 240, 230, 224, 218, 212, 210, 204, 200, 198, 188, 174, 170, 168, 164, 150, 144, 134

Offset: 1

Jon Perry, May 28 2003

Differences when decreasing are essentially A001223, so increases occur when primes being used are roughly double those at previous increase; e.g. a(3352)=(12+31123)mod 31139=31135 and a(6257)=(1+62273)mod 62297=62274 - Henry Bottomley, Jul 13 2003

			a(2) = (a(1) + 3) mod 5 = 5 mod 5 = 0.
a(3) = (a(2) + 5) mod 7 = 5 mod 7 = 5.
a(4) = (a(3) + 7) mod 11 = 12 mod 11 = 1.

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

Cf. A000040, A001223, A071089.

nxt[{n_,a_}]:={n+1,Mod[a+Prime[n+1],Prime[n+2]]}; NestList[nxt,{1,2},70][[All,2]] (* Harvey P. Dale, Sep 13 2016 *)

ps=0; pc=1; while (pc<100,ps+=prime(pc); ps%=prime(pc++); print1(ps","))

Edited by Henry Bottomley, Jul 13 2003

Definition clarified by Harvey P. Dale, Sep 13 2016

A274995 a(n) is the smallest odd prime that divides (-n) + the sum of all smaller primes, or 0 if no such prime exists.

Original entry on oeis.org

5, 19, 3, 7, 82811, 3, 11, 17, 3, 191, 5, 3, 37, 29, 3, 5, 69431799799, 3, 1105589, 28463, 3, 431, 2947308589, 3, 7, 5, 3, 59, 11, 3, 5, 7, 3, 41

Offset: 0

Neil Fernandez, Nov 11 2016

From Robert G. Wilson v, Nov 15 2016: (Start)
If n == 2 (mod 3) then a(n) = 3;
If n == 0 (mod 5) then a(n) = 5;
If n == 3 (mod 7) then a(n) = 7;
If n == 6 (mod 11) then a(n) = 11;
If n == 2 (mod 13) then a(n) = 13;
If n == 7 (mod 17) then a(n) = 17;
If n == 1 (mod 19) then a(n) = 19;
If n == 8 (mod 23) then a(n) = 23;
in that order; i.e., from smaller to greater prime modulus, etc.
First occurrence of p>2: 2, 0, 3, 6, 54, 7, 1, 123, 13, 36, 12, 33, 453, 46, ..., .
(End)
The congruence classes in the above list, modulo the prime bases, namely 2, 0, 3, 6, 2, ..., are given by A071089, in which each term is the remainder when the sum of the first n primes is divided by the n-th prime. - Neil Fernandez, Nov 23 2016

			a(1) = 19 because 19 is the smallest odd prime that divides the sum of (-1) + (sum of all primes smaller than itself), that is, -1 + 58 = 57.
a(7) = 17 because -7 + 2 + 3 + 5 + 7 + 11 + 13 + 17 = 49 and 49/7 = 7.

Robert G. Wilson v, n and a(n), or 0 if no such value is known, for n=0..10000

Cf. A007504, A007506, A071089, A274649.

f[n_] := Block[{p = 3, s = 2 - n}, While[ Mod[s, p] != 0, s = s + p; p = NextPrime@ p]; p]; Array[f, 16, 0] (* Robert G. Wilson v, Nov 15 2016 *)

sump(n) = s = 0; forprime(p=2, n-1, s+=p); s;
a(n) = {my(p=3); while ((sump(p)-n) % p, p = nextprime(p+1)); p;} \\ Michel Marcus, Nov 12 2016

a(n)=my(s=2); forprime(p=3,, if((s-n)%p==0, return(p)); s+=p) \\ Charles R Greathouse IV, Nov 15 2016

a(16)-a(33) from Charles R Greathouse IV, Nov 15 2016

A330578 a(n) is the remainder when the sum of the first n composite numbers is divided by the n-th composite number.

Original entry on oeis.org

0, 4, 2, 0, 7, 1, 7, 3, 14, 4, 12, 6, 21, 7, 24, 16, 7, 25, 5, 15, 4, 26, 14, 1, 11, 36, 22, 34, 4, 33, 17, 31, 14, 46, 28, 9, 23, 3, 38, 17, 53, 9, 25, 2, 42, 18, 59, 9, 52, 26, 44, 64, 37, 9, 57, 28, 48, 18, 69, 7, 60, 28, 82, 49, 71, 37, 2, 59, 23, 81, 44

Offset: 1

Rémy Sigrist, Dec 18 2019

nonn

			a(3) = (4 + 6 + 8) mod 8 = 2.

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A002808, A053767, A071089 (prime variant), A330579 (positions of zeros).

s=0; forcomposite (c=4, 96, s+=c; print1 (s%c", "))

a(n) = A053767(n) mod A002808(n).

A066913 (sum of primes < n that do not divide n) (mod n).

Original entry on oeis.org

0, 0, 2, 3, 0, 5, 3, 7, 5, 0, 6, 11, 2, 4, 3, 7, 7, 17, 1, 10, 4, 20, 8, 23, 20, 7, 16, 7, 13, 29, 5, 30, 14, 5, 8, 11, 12, 24, 25, 30, 33, 16, 23, 4, 3, 26, 46, 35, 27, 21, 2, 1, 10, 52, 35, 36, 17, 2, 27, 10, 13, 34, 50, 51, 28, 23, 32, 5, 59, 64, 0, 58, 55, 7, 29, 7, 1, 70, 1

Offset: 1

Leroy Quet, Jan 22 2002

nonn

			a(8) = (3 + 5 + 7) (mod 8) = 7 because 3, 5 and 7 are the primes < 8 that do not divide 8.

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

Cf. A066911, A071089.

Table[Mod[Total[Select[Prime[Range[PrimePi[n]]],Mod[n,#]!=0&]],n],{n,80}] (* Harvey P. Dale, Aug 06 2019 *)

a(n) = sum(i=1, n-1, if (isprime(i) && (n%i), i)) % n; \\ Michel Marcus, May 20 2014

a(n) = A066911(n) modulo n. - Michel Marcus, May 20 2014

a(prime(n)) = A071089(n). - Michel Marcus, May 20 2014

A259643 Numbers n such that sum of first n odd primes divides product of first n odd primes.

Original entry on oeis.org

1, 3, 5, 11, 25, 29, 41, 49, 51, 59, 69, 81, 99, 103, 113, 131, 133, 135, 147, 149, 153, 181, 187, 193, 197, 199, 205, 211, 213, 217, 219, 229, 235, 239, 243, 255, 271, 277, 281, 287, 289, 303, 309, 313, 323, 333, 335, 343, 347, 357, 359, 365, 367, 381, 383, 389

Offset: 1

Altug Alkan, Oct 02 2015

Obviously, a(n) is always an odd number.

			a(1) = 1 because prime(2) mod prime(2) = 3 mod 3 = 0.
a(2) = 3 because (prime(2) * prime(3) * prime(4)) mod (prime(2) + prime(3) + prime(4)) = 105 mod 15 = 0.
a(3) = 5 because (prime(2) * prime(3) * prime(4) * prime(5) * prime(6)) mod (prime(2) + prime(3) + prime(4) + prime(5) + prime(6)) = 15015 mod 39 = 0.

Cf. A051838, A065091, A070826, A071089, A071148, A262807.

Module[{nn=400,op},op=Prime[Range[2,nn+1]];Select[Range[nn],Divisible[ Times@@ Take[op,#],Total[Take[op,#]]]&]] (* Harvey P. Dale, Nov 16 2022 *)

A262807 a(n) = (Product_{k=1..n} prime(k+1)) mod (Sum_{k=1..n} prime(k+1)) where prime(k) is the k-th prime number.

Original entry on oeis.org

0, 7, 0, 11, 0, 7, 45, 91, 24, 55, 0, 113, 93, 175, 308, 153, 414, 395, 273, 355, 609, 779, 558, 23, 0, 843, 962, 185, 0, 547, 1634, 21, 170, 1149, 1455, 2483, 1830, 2275, 2865, 1989, 0, 1515, 1211, 2013, 1105, 403, 2733, 819, 0, 4011, 0, 1457, 4278, 1155, 391, 1717, 2596, 2163, 0, 5985

Offset: 1

Altug Alkan, Oct 02 2015

Remainder when product of first n odd primes is divided by sum of first n odd primes.

Obviously a(2n) cannot be 0. Does 0 appear in the sequence infinitely often?

			a(1) = prime(2) mod prime(2) = 3 mod 3 = 0.
a(2) = (prime(2) * prime(3)) mod (prime(2) + prime(3)) = 15 mod 8 = 7.
a(3) = (prime(2) * prime(3) * prime(4)) mod (prime(2) + prime(3) + prime(4)) = 105 mod 15 = 0.
a(4) = (prime(2) * prime(3) * prime(4) * prime(5)) mod (prime(2) + prime(3) + prime(4) + prime(5)) = 1155 mod 26 = 11.

Cf. A065091, A070826, A071089, A071148, A259643.

Table[Mod[Product[Prime[k + 1], {k, n}], Sum[Prime[k + 1], {k, n}]], {n, 60}] (* Michael De Vlieger, Oct 02 2015 *)

a(n) = prod(k=1, n, prime(k+1)) % sum(k=1, n, prime(k+1));
vector(60, n, a(n))

a(n) = A070826(n+1) mod A071148(n).

A301852 Integers k such that the remainder of the sum of the first k primes divided by the k-th prime is equal to k.

Original entry on oeis.org

2, 7, 12, 83408, 5290146169416

Offset: 1

J. M. Bergot, Mar 27 2018

Integers k such that A071089(k) = k.
From Robert Israel, Mar 27 2018: (Start)
No more terms below 10^7.
Heuristically, the probability that k is a term is 1/prime(k) ~ 1/(k log k).
Since Sum_{k>=2} 1/(k log(k)) diverges, there should be infinitely many terms. However, the sum diverges very slowly, so terms may be very sparse: approximately log(log(k)) terms <= k. (End)
No more terms below 10^9. - Michel Marcus, Mar 28 2018
No more terms below 1.44*10^12. - Giovanni Resta, Apr 06 2018
No mroe terms below 10^13. - Lucas A. Brown, May 18 2023

			2 is a term because prime(1)+prime(2) = 5 = 2 mod prime(2).

Lucas A. Brown, Python program.

Cf. A000040, A071089.

res:= NULL: p:= 1: s:= 0:
for m from 1 to 10^6 do
  p:= nextprime(p);
  s:= s+p;
  if s mod p = m then res:= res, m fi
od:
res; # Robert Israel, Mar 27 2018

lista(nn)= my(p = 2, s = 2); for (n=1, nn, if ((s % p) == n, print1(n, ", ")); q = nextprime(p+1); s += q; p = q;); \\ Michel Marcus, Mar 27 2018

a(4) from Michel Marcus, Mar 27 2018

a(5) from Lucas A. Brown, May 18 2023

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A024011 Numbers k such that the k-th prime divides the sum of the first k primes.

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A082974 a(n) = (a(n-1) + p(n)) mod p(n+1).

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A274995 a(n) is the smallest odd prime that divides (-n) + the sum of all smaller primes, or 0 if no such prime exists.

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A330578 a(n) is the remainder when the sum of the first n composite numbers is divided by the n-th composite number.

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A066913 (sum of primes < n that do not divide n) (mod n).

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A259643 Numbers n such that sum of first n odd primes divides product of first n odd primes.

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A262807 a(n) = (Product_{k=1..n} prime(k+1)) mod (Sum_{k=1..n} prime(k+1)) where prime(k) is the k-th prime number.

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A301852 Integers k such that the remainder of the sum of the first k primes divided by the k-th prime is equal to k.

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