A111327 -id:A111327 - cp's OEIS frontend

A103208 Numbers k such that 3 divides prime(1) + ... + prime(k).

10, 16, 18, 20, 24, 26, 28, 30, 32, 34, 36, 40, 42, 44, 46, 52, 54, 57, 68, 70, 74, 76, 78, 80, 82, 84, 86, 88, 90, 97, 99, 103, 105, 107, 111, 113, 119, 121, 123, 125, 127, 129, 134, 136, 138, 161, 163, 166, 169, 175, 177, 179, 185, 187, 195, 197, 199, 203, 205, 207, 211, 213

Offset: 1

Robert G. Wilson v, Mar 19 2005

nonn

Also, numbers k such that 3 divides the concatenation of the first k primes (see A019518).

The first comment and the description are true whenever the number of primes congruent to 1 mod 6 exceeds the number of primes congruent to 5 mod 6 and the difference is congruent to 1 mod 3 or the number of primes congruent to 5 mod 6 exceeds the number of primes congruent to 1 mod 6 and the difference is congruent to 2 mod 3. - Roderick MacPhee, Oct 30 2015

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)
Hisanori Mishima, Smarandache consecutive prime sequences (n = 1 to 100).

Cf. A007504, A019518, A104644, A111287.

Cf. A111318, A111319, A111320, A111321, A111322, A111323, A111324, A111325, A111326, A111327.

s1:=[2]; M:=1000; for n from 2 to M do s1:=[op(s1),s1[n-1]+ithprime(n)]; od: s1;
f:=proc(k) global M,s1; local t1,n; t1:=[]; for n from 1 to M do if s1[n] mod k = 0 then t1:=[op(t1),n]; fi; od: t1; end; f(3);

f[n_] := FromDigits[ Flatten[ Table[ IntegerDigits[ Prime[i]], {i, n}]]]; Select[ Range[ 206], Mod[f[ # ], 3] == 0 &]
Flatten[Position[Accumulate[Prime[Range[250]]],?(Divisible[#,3]&)]] (* _Harvey P. Dale, Jan 14 2016 *)

a=0;b=0;for(x=3,1000,if(prime(x)%6==1,a+=1,b+=1);if((a-b)%3==1 || (b-a)%3==2,print1(x","))) \\ Roderick MacPhee, Oct 30 2015

lista(nn) = { s=0; for(k=1, nn, s += prime(k); if(s % 3 == 0, print1(k, ", ")););} \\ Altug Alkan, Dec 04 2015

Entry revised by N. J. A. Sloane, Nov 09 2005

A111319 Numbers k such that 5 divides prime(1) + ... + prime(k).

Original entry on oeis.org

2, 3, 9, 11, 17, 25, 29, 31, 51, 53, 57, 62, 71, 77, 85, 89, 91, 101, 103, 105, 116, 118, 131, 147, 153, 156, 159, 167, 173, 180, 186, 188, 190, 195, 209, 226, 230, 239, 242, 245, 256, 259, 261, 266, 268, 283, 292, 298, 303, 314, 317, 324, 349, 352, 357, 364, 366, 368, 376

Offset: 1

N. J. A. Sloane, Nov 05 2005

nonn

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

Cf. A007504, A103208, A111318, A111320, A111321, A111322, A111323, A111324, A111325, A111326, A111327.

Position[Accumulate[Prime[Range[400]]], ?(Divisible[#, 5] &)] // Flatten (* _Amiram Eldar, May 14 2024 *)

lista(pmax) = {my(s = 0, k = 0); forprime(p = 2, pmax, k++; s += p; if(!(s % 5), print1(k, ", ")));} \\ Amiram Eldar, May 14 2024

A111320 Numbers k such that 6 divides prime(1) + ... + prime(k).

Original entry on oeis.org

57, 97, 99, 103, 105, 107, 111, 113, 119, 121, 123, 125, 127, 129, 161, 163, 169, 175, 177, 179, 185, 187, 195, 197, 199, 203, 205, 207, 211, 213, 217, 233, 235, 237, 273, 293, 295, 297, 301, 303, 305, 307, 309, 311, 327, 329, 331, 333, 335, 337, 339, 343, 345, 347, 349

Offset: 1

N. J. A. Sloane, Nov 05 2005

nonn

Odd terms in A103208. - Robert Israel, Jan 26 2019

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

Cf. A007504, A103208, A111318, A111319, A111321, A111322, A111323, A111324, A111325, A111326, A111327.

P:= [seq(ithprime(i),i=1..1000)]:
L:= ListTools:-PartialSums(P):
select(t -> L[t] mod 6 = 0, [seq(i,i=1..1000)]); # Robert Israel, Jan 26 2019

Position[Accumulate[Prime[Range[400]]], ?(Divisible[#, 6] &)] // Flatten (* _Amiram Eldar, May 14 2024 *)

lista(pmax) = {my(s = 0, k = 0); forprime(p = 2, pmax, k++; s += p; if(!(s % 6), print1(k, ", ")));} \\ Amiram Eldar, May 14 2024

A111324 Numbers k such that 10 divides prime(1) + ... + prime(k).

Original entry on oeis.org

3, 9, 11, 17, 25, 29, 31, 51, 53, 57, 71, 77, 85, 89, 91, 101, 103, 105, 131, 147, 153, 159, 167, 173, 195, 209, 239, 245, 259, 261, 283, 303, 317, 349, 357, 405, 453, 459, 475, 479, 491, 503, 505, 507, 511, 517, 531, 533, 545, 555, 565, 569, 583, 585, 591, 603, 617, 625

Offset: 1

N. J. A. Sloane, Nov 05 2005

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Cf. A007504, A103208, A111318, A111319, A111320, A111321, A111322, A111323, A111325, A111326, A111327.

Flatten[Position[Accumulate[Prime[Range[650]]],?(Divisible[#,10]&)]] (* _Harvey P. Dale, Sep 25 2011 *)

lista(pmax) = {my(s = 0, k = 0); forprime(p = 2, pmax, k++; s += p; if(!(s % 10), print1(k, ", ")));} \\ Amiram Eldar, May 14 2024

A111318 Numbers k such that 4 divides prime(1) + ... + prime(k).

Original entry on oeis.org

5, 9, 11, 15, 17, 19, 21, 25, 27, 29, 31, 33, 35, 37, 45, 49, 71, 79, 81, 83, 85, 87, 91, 95, 97, 99, 101, 103, 105, 107, 109, 111, 115, 117, 119, 121, 125, 129, 131, 135, 155, 159, 161, 163, 165, 169, 171, 173, 175, 177, 179, 181, 183, 185, 193, 195, 199, 201, 203, 205

Offset: 1

N. J. A. Sloane, Nov 05 2005

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Cf. A007504, A103208, A111319, A111320, A111321, A111322, A111323, A111324, A111325, A111326, A111327.

Module[{nn=300,prs},prs=Accumulate[Prime[Range[nn]]];Select[Range[ nn], Divisible[prs[[#]],4]&]] (* Harvey P. Dale, May 27 2012 *)

lista(pmax) = {my(s = 0, k = 0); forprime(p = 2, pmax, k++; s += p; if(!(s % 4), print1(k, ", ")));} \\ Amiram Eldar, May 14 2024

A111321 Numbers k such that 7 divides prime(1) + ... + prime(k).

Original entry on oeis.org

5, 8, 13, 22, 33, 40, 47, 50, 56, 63, 72, 84, 86, 104, 106, 110, 115, 126, 128, 139, 148, 150, 154, 157, 160, 180, 184, 186, 188, 200, 209, 220, 228, 230, 232, 236, 238, 240, 244, 253, 270, 274, 300, 302, 305, 322, 324, 331, 339, 354, 367, 371, 384, 415, 417, 420, 428, 433

Offset: 1

N. J. A. Sloane, Nov 05 2005

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Cf. A007504, A103208, A111318, A111319, A111320, A111322, A111323, A111324, A111325, A111326, A111327.

Position[Accumulate[Prime[Range[500]]],?(Divisible[#,7]&)]//Flatten (* _Harvey P. Dale, Oct 06 2017 *)

lista(pmax) = {my(s = 0, k = 0); forprime(p = 2, pmax, k++; s += p; if(!(s % 7), print1(k, ", ")));} \\ Amiram Eldar, May 14 2024

A111322 Numbers k such that 8 divides prime(1) + ... + prime(k).

Original entry on oeis.org

11, 15, 17, 19, 21, 27, 29, 31, 37, 49, 71, 79, 83, 85, 95, 99, 101, 103, 107, 109, 111, 115, 119, 121, 135, 155, 159, 161, 163, 169, 171, 177, 181, 183, 185, 201, 205, 209, 213, 235, 237, 239, 247, 255, 257, 259, 261, 263, 273, 275, 279, 283, 285, 287, 305, 309, 317, 319

Offset: 1

N. J. A. Sloane, Nov 05 2005

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A007504, A103208, A111318, A111319, A111320, A111321, A111323, A111324, A111325, A111326, A111327.

Position[Accumulate@ Prime@ Range@ 320, ?(Mod[#, 8] == 0 &)][[All, 1]] (* _Michael De Vlieger, Feb 05 2021 *)

isok(n) = !(vecsum(primes(n)) % 8); \\ Michel Marcus, Feb 05 2021

lista(pmax) = {my(s = 0, k = 0); forprime(p = 2, pmax, k++; s += p; if(!(s % 8), print1(k, ", ")));} \\ Amiram Eldar, May 14 2024

A111323 Numbers k such that 9 divides prime(1) + ... + prime(k).

Original entry on oeis.org

20, 24, 26, 30, 40, 42, 52, 74, 78, 80, 88, 113, 119, 127, 163, 177, 179, 187, 205, 207, 242, 248, 254, 258, 260, 262, 268, 270, 280, 282, 284, 288, 297, 311, 331, 357, 368, 372, 380, 394, 398, 400, 410, 412, 416, 428, 436, 443, 457, 466, 468, 470, 474, 490, 496, 505, 509

Offset: 1

N. J. A. Sloane, Nov 05 2005

nonn

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

Cf. A007504, A103208, A111318, A111319, A111320, A111321, A111322, A111324, A111325, A111326, A111327.

Flatten[Position[Accumulate[Prime[Range[509]]]/9,Integer]] (* _Jayanta Basu, May 18 2013 *)

lista(pmax) = {my(s = 0, k = 0); forprime(p = 2, pmax, k++; s += p; if(!(s % 9), print1(k, ", ")));} \\ Amiram Eldar, May 14 2024

A111325 Numbers k such that 11 divides prime(1) + ... + prime(k).

Original entry on oeis.org

8, 17, 53, 69, 76, 84, 87, 91, 167, 175, 179, 181, 188, 196, 201, 217, 219, 224, 240, 260, 275, 297, 312, 317, 319, 324, 340, 346, 376, 382, 386, 393, 417, 470, 503, 514, 526, 528, 542, 550, 562, 564, 584, 590, 607, 613, 615, 629, 637, 649, 691, 693, 732, 749, 752, 759

Offset: 1

N. J. A. Sloane, Nov 05 2005

nonn

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A007504, A103208, A111318, A111319, A111320, A111321, A111322, A111323, A111324, A111326, A111327.

Position[Accumulate[Prime[Range[400]]], ?(Divisible[#, 11] &)] // Flatten (* _Amiram Eldar, May 14 2024 *)

lista(pmax) = {my(s = 0, k = 0); forprime(p = 2, pmax, k++; s += p; if(!(s % 11), print1(k, ", ")));} \\ Amiram Eldar, May 14 2024

from sympy import nextprime
def aupto(limit):
    p, s, alst = 2, 2, []
    for k in range(1, limit+1):
        if s == 0: alst.append(k)
        p = nextprime(p); s = (s + p)%11
    return alst
print(aupto(759)) # Michael S. Branicky, Aug 17 2021

A111326 Numbers k such that 12 divides prime(1) + ... + prime(k).

Original entry on oeis.org

97, 99, 103, 105, 107, 111, 119, 121, 125, 129, 161, 163, 169, 175, 177, 179, 185, 195, 199, 203, 205, 207, 211, 213, 233, 235, 237, 273, 305, 307, 309, 311, 329, 335, 337, 343, 345, 347, 353, 357, 421, 423, 425, 439, 443, 445, 447, 449, 455, 463, 511, 513, 515, 539

Offset: 1

N. J. A. Sloane, Nov 05 2005

nonn

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

Cf. A007504, A103208, A111318, A111319, A111320, A111321, A111322, A111323, A111324, A111325, A111327.

Position[Accumulate[Prime[Range[400]]], ?(Divisible[#, 12] &)] // Flatten (* _Amiram Eldar, May 14 2024 *)

lista(pmax) = {my(s = 0, k = 0); forprime(p = 2, pmax, k++; s += p; if(!(s % 12), print1(k, ", ")));} \\ Amiram Eldar, May 14 2024

cp's OEIS Frontend

A103208 Numbers k such that 3 divides prime(1) + ... + prime(k).

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A111319 Numbers k such that 5 divides prime(1) + ... + prime(k).

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A111320 Numbers k such that 6 divides prime(1) + ... + prime(k).

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A111324 Numbers k such that 10 divides prime(1) + ... + prime(k).

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A111318 Numbers k such that 4 divides prime(1) + ... + prime(k).

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A111321 Numbers k such that 7 divides prime(1) + ... + prime(k).

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A111322 Numbers k such that 8 divides prime(1) + ... + prime(k).

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A111323 Numbers k such that 9 divides prime(1) + ... + prime(k).

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