A029707 -id:A029707 - cp's OEIS frontend

A117507 Numerators of partial sums of the Brun series divided by 4.

2, 23, 3919, 1400972, 1332221503, 2440266733544, 9013120937567806, 47710925260763230958, 503649376979113850651329, 5954610779280903922363948937, 114594038963707117577230115067496

Offset: 1

Wolfdieter Lang, Apr 13 2006

The Brun series is the sum over reciprocals of the (odd) twin primes (see the mathworld link).
The denominators divided by 5 are given in A117508.
A001359 gives the lesser of the twin primes (offset 1).
A006512 gives the greater of the twin primes (offset 1).
A029707=[2,3,5,7,10,..] gives the indices for the lesser of the (odd) twin primes (offset 0).
The proof that the partial sums of the Brun series have numerators divisible by 4 and denominators divisible by 5 can be given by induction.

			Rationals 4*A117507(n)/5*A117508(n): 8/15, 92/105, 15676/15015,
5603888/4849845, 5328886012/4360010655,...

W. Lang: Rationals and more.
Eric Weisstein's World of Mathematics, Brun's Constant

a(n)=numerator(r(n))/4, with r(n):=sum(1/ltp(k) + 1/(ltp(k)+2),k=1..n), n>=1, with ltp(k):=A001359(k) (lesser twin primes).

A163981 a(n) is the smallest prime of the form prime(n+1)*k - prime(n), k >= 1, where prime(n) is the n-th prime.

Original entry on oeis.org

7, 2, 2, 37, 2, 89, 2, 73, 151, 2, 43, 127, 2, 239, 59, 419, 2, 73, 359, 2, 401, 419, 1163, 881, 307, 2, 967, 2, 569, 3697, 397, 691, 2, 457, 2, 163, 821, 839, 179, 1259, 2, 2111, 2, 1777, 2, 223, 3803, 3863, 2, 3499, 1201, 2, 2269, 263, 269, 1889, 2, 283, 1409, 2, 2647

Offset: 1

Leroy Quet, Aug 07 2009

nonn

a(n) = 2 if and only if n is in A029707. - Robert Israel, Jan 16 2019

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

Cf. A029707, A129919.

Contains A085704.

a := proc (n) local k: for k while isprime(ithprime(n+1)*k-ithprime(n)) = false do end do: ithprime(n+1)*k-ithprime(n) end proc: seq(a(n), n = 1 .. 65); # Emeric Deutsch, Aug 10 2009

a[n_] := Module[{p, q, r}, For[p = Prime[n]; q = Prime[n + 1]; k = 1, True, k++, If[PrimeQ[r = q k - p], Return[r]]]];
Array[a, 100] (* Jean-François Alcover, Aug 28 2020 *)

a(n) = my(k=1); while (!isprime(p=prime(n+1)*k - prime(n)), k++); p; \\ Michel Marcus, Jul 02 2021

from sympy import isprime, nextprime, prime
def a(n):
    pn = prime(n); pn1 = nextprime(pn); k = 1
    while not isprime(pn1*k - pn): k += 1
    return pn1*k - pn
print([a(n) for n in range(1, 62)]) # Michael S. Branicky, Jul 02 2021

Extended by Emeric Deutsch, Aug 10 2009

A320704 Indices of primes followed by a gap (distance to next larger prime) of 12.

Original entry on oeis.org

46, 47, 91, 97, 114, 121, 139, 168, 197, 203, 214, 232, 239, 240, 242, 267, 278, 280, 290, 312, 317, 342, 357, 363, 376, 381, 404, 423, 437, 439, 449, 452, 461, 470, 472, 489, 499, 511, 546, 550, 562, 565, 599, 600, 617, 633, 634, 647, 653, 657, 675, 680, 692, 698, 716, 728

Offset: 1

M. F. Hasler, Oct 19 2018

nonn

Indices of the primes given in A031930.

Vincenzo Librandi, Table of n, a(n) for n = 1..10150
Index entries for primes, gaps between.

Equals A000720 o A031930.
Row 6 of A174349.
Indices of 12's in A001223.
Cf. A029707, A029709, A320701, A320702, ..., A320720 (analog for gaps 2, 4, 6, 8, ..., 44), A116493 (gap 70), A116496 (gap 100), A116497 (gap 200), A116495 (gap 210).

[n: n in [1..1000] | NthPrime(n+1) - NthPrime(n) eq 12]; // Vincenzo Librandi, Mar 21 2019

Select[Range[1000], Prime[#] + 12 == Prime[# + 1] &] (* Vincenzo Librandi, Mar 21 2019 *)

A320704_vec(N=100,g=12,p=2,i=primepi(p)-1,L=List())={forprime(q=1+p,,i++; if(p+g==p=q, listput(L,i); N--||break));Vec(L)} \\ returns the list of first N terms of the sequence

a(n) = A000720(A031930(n)).

A320704 = { i > 0 | prime(i+1) = prime(i) + 12 }.

A320705 Indices of primes followed by a gap (distance to next larger prime) of 14.

Original entry on oeis.org

30, 62, 66, 137, 146, 150, 162, 223, 250, 283, 309, 350, 360, 382, 402, 410, 424, 434, 503, 514, 526, 532, 536, 570, 610, 649, 654, 666, 687, 704, 706, 747, 780, 790, 867, 906, 919, 929, 967, 978, 981, 992, 1011, 1023, 1038, 1042, 1057, 1072, 1133, 1154, 1160, 1177, 1184

Offset: 1

M. F. Hasler, Oct 19 2018

nonn

Indices of the primes listed in A031932.

Vincenzo Librandi, Table of n, a(n) for n = 1..8000
Index entries for primes, gaps between.

Equals A000720 o A031932.
Row 7 of A174349.
Indices of 14's in A001223.
Cf. A029707, A029709, A320701, A320702, ..., A320720 (analog for gaps 2, 4, 6, 8, ..., 44), A116493 (gap 70), A116496 (gap 100), A116497 (gap 200), A116495 (gap 210).

[n: n in [1..1500] | NthPrime(n+1) - NthPrime(n) eq 14]; // Vincenzo Librandi, Mar 19 2019

Select[Range[1500], Prime[#] + 14 == Prime[# + 1] &] (* Vincenzo Librandi, Mar 19 2019 *)
Position[Differences[Prime[Range[1200]]],14]//Flatten (* Harvey P. Dale, Nov 28 2024 *)

A(N=100,g=14,p=2,i=primepi(p)-1,L=List())={forprime(q=1+p,,i++; if(p+g==p=q, listput(L,i); N--||break));Vec(L)} \\ returns the list of first N terms of the sequence

a(n) = A000720(A031932(n)).

A320705 = { i > 0 | prime(i+1) = prime(i) + 14 }.

A320706 Indices of primes followed by a gap (distance to next larger prime) of 16.

Original entry on oeis.org

282, 295, 319, 331, 335, 378, 409, 445, 476, 478, 481, 510, 560, 566, 619, 624, 674, 701, 739, 775, 856, 871, 881, 886, 935, 941, 1007, 1069, 1077, 1121, 1146, 1193, 1222, 1261, 1286, 1322, 1331, 1356, 1372, 1388, 1405, 1460, 1487, 1500, 1587, 1603, 1608, 1612, 1699, 1719, 1734, 1740, 1811, 1876, 1924, 1956, 1969, 1977, 2002, 2022, 2034, 2042, 2071

Offset: 1

M. F. Hasler, Oct 19 2018

nonn

Indices of the primes listed in A031934.

Vincenzo Librandi, Table of n, a(n) for n = 1..9790
Index entries for primes, gaps between.

Equals A000720 o A031934.
Row 8 of A174349.
Indices of 16's in A001223.
Cf. A029707, A029709, A320701, A320702, ..., A320720 (analog for gaps 2, 4, 6, 8, ..., 44), A116493 (gap 70), A116496 (gap 100), A116497 (gap 200), A116495 (gap 210).

[n: n in [1..2100] | NthPrime(n+1) - NthPrime(n) eq 16]; // Vincenzo Librandi, Mar 19 2019

Select[Range[2500], Prime[#] + 16 == Prime[# + 1] &] (* Vincenzo Librandi, Mar 19 2019 *)
Position[Partition[Prime[Range[2100]],2,1],?(#[[2]]-#[[1]]== 16&),1,Heads-> False]//Flatten (* _Harvey P. Dale, Nov 01 2020 *)

A(N=100,g=16,p=2,i=primepi(p)-1,L=List())={forprime(q=1+p,,i++; if(p+g==p=q, listput(L,i); N--||break));Vec(L)} \\ returns the list of first N terms of the sequence

a(n) = A000720(A031934(n)).

A320706 = { i > 0 | prime(i+1) = prime(i) + 16 }.

A320707 Indices of primes followed by a gap (distance to next larger prime) of 18.

Original entry on oeis.org

99, 180, 205, 221, 274, 293, 326, 368, 416, 529, 539, 573, 597, 602, 607, 623, 635, 639, 677, 693, 725, 785, 811, 838, 844, 852, 855, 916, 937, 939, 942, 945, 968, 997, 1028, 1093, 1130, 1151, 1203, 1227, 1252, 1304, 1311, 1349, 1508, 1514, 1519, 1523, 1540, 1547, 1629, 1636, 1641, 1654, 1656

Offset: 1

M. F. Hasler, Oct 19 2018

nonn

Indices of the primes listed in A031936.

Vincenzo Librandi, Table of n, a(n) for n = 1..10300
Index entries for primes, gaps between.

Equals A000720 o A031936.
Row 9 of A174349.
Indices of 18's in A001223.
Cf. A029707, A029709, A320701, A320702, ..., A320720 (analog for gaps 2, 4, 6, 8, ..., 44), A116493 (gap 70), A116496 (gap 100), A116497 (gap 200), A116495 (gap 210).

[n: n in [1..1700] | NthPrime(n+1) - NthPrime(n) eq 18]; // Vincenzo Librandi, Mar 22 2019

Select[Range[1700], Prime[#] + 18 == Prime[# + 1] &] (* Vincenzo Librandi, Mar 22 2019 *)
Flatten[Position[Differences[Prime[Range[2000]]],18]] (* Harvey P. Dale, May 12 2022 *)

A(N=100,g=18,p=2,i=primepi(p)-1,L=List())={forprime(q=1+p,,i++; if(p+g==p=q, listput(L,i); N--||break));Vec(L)} \\ returns the list of first N terms of the sequence

a(n) = A000720(A031936(n)).

A320707 = { i > 0 | prime(i+1) = prime(i) + 18 } = A001223^(-1)({18}).

A320713 Indices of primes followed by a gap (distance to next larger prime) of 30.

Original entry on oeis.org

590, 650, 708, 757, 842, 890, 928, 985, 1006, 1051, 1108, 1556, 1570, 1648, 1650, 1675, 1754, 1900, 1919, 2027, 2125, 2149, 2321, 2391, 2397, 2429, 2631, 2637, 2699, 2781, 2866, 2918, 2989, 2993, 3010, 3085, 3153, 3207, 3315, 3340, 3350, 3373, 3420, 3511, 3551, 3580, 3637, 3751, 3777, 3948

Offset: 1

M. F. Hasler, Oct 19 2018

nonn

Indices of the primes listed in A124596.

Index entries for primes, gaps between.

Equals A000720 o A124596.
Indices of 30's in A001223.
Row 15 of A174349.
Subsequence of A107730 (prime(n+1) ends in same digit as prime(n)).
Cf. A029707, A029709, A320701, A320702, ..., A320720 (analog for gaps 2, 4, 6, 8, ..., 44), A116493 (gap 70), A116496 (gap 100), A116497 (gap 200), A116495 (gap 210).

A(N=100,g=30,p=2,i=primepi(p)-1,L=List())={forprime(q=1+p,,i++; if(p+g==p=q, listput(L,i); N--||break));Vec(L)} \\ returns the list of first N terms of the sequence

a(n) = A000720(A124596(n)).

A320713 = { i>0 | prime(i+1) = prime(i) + 30 } = A001223^(-1)({30}).

A320718 Indices of primes followed by a gap (distance to next larger prime) of 40.

Original entry on oeis.org

2191, 2344, 2524, 2788, 3562, 4058, 4677, 5030, 5349, 6076, 6145, 6256, 6320, 6442, 6454, 6902, 7232, 7488, 8119, 8152, 8245, 8366, 8553, 8567, 8591, 8746, 9260, 9361, 10536, 10735, 11095, 11407, 11534, 11781, 12227, 12312, 12663, 12815, 12940, 13015, 13333, 13676, 13873, 14065, 14123

Offset: 1

M. F. Hasler, Oct 19 2018

nonn

Indices of the primes listed in A126721.

Index entries for primes, gaps between.

Equals A000720 o A126721.
Row 20 of A174349.
Subsequence of A107730 (prime(n+1) ends in same digit as prime(n)).
Indices of 40's in A001223.
Cf. A029707, A029709, A320701, A320702, ..., A320720 (analog for gaps 2, 4, 6, 8, ..., 44), A116493 (gap 70), A116496 (gap 100), A116497 (gap 200), A116495 (gap 210).

A(N=100,g=40,p=2,i=primepi(p)-1,L=List())={forprime(q=1+p,,i++; if(p+g==p=q, listput(L,i); N--||break));Vec(L)} \\ returns the list of first N terms of the sequence

a(n) = A000720(A126721(n)).

A320718 = { i > 0 | prime(i+1) = prime(i) + 40 } = A001223^(-1)({40}).

A320719 Indices of primes followed by a gap (distance to next larger prime) of 42.

Original entry on oeis.org

1879, 3732, 4059, 4135, 4714, 5355, 5948, 6160, 6841, 7434, 7724, 7746, 7952, 7980, 8081, 8269, 8580, 9303, 9395, 9971, 10045, 10305, 10968, 11023, 11135, 11251, 11338, 11399, 11515, 11807, 11888, 11901, 12089, 12374, 12488, 13277, 13447, 14497, 14802, 15086, 15089, 15350, 15612, 15785

Offset: 1

M. F. Hasler, Oct 19 2018

nonn

Indices of the primes listed in A134120.

Harvey P. Dale, Table of n, a(n) for n = 1..3000
Index entries for primes, gaps between.

Cf. A029707, A029709 (analog for gaps 2 & 4), A320701, A320702, ... A320720 (analog for gaps 6, 8, ..., 44), A116493 (gap 70), A116496 (gap 100), A116497 (gap 200), A116495 (gap 210).
Equals A000720 o A134120.
Indices of 42's in A001223.
Row 21 of A174349.

Position[Differences[Prime[Range[16000]]],42]//Flatten (* Harvey P. Dale, Feb 22 2020 *)

A(N=100,g=42,p=2,i=primepi(p)-1,L=List())={forprime(q=1+p,,i++; if(p+g==p=q, listput(L,i); N--||break));Vec(L)} \\ returns the list of first N terms of the sequence

a(n) = A000720(A134120(n)).

A341284 a(n) is the least prime == -prime(n) (mod 2*prime(n+1)).

Original entry on oeis.org

7, 23, 37, 41, 89, 59, 73, 151, 157, 43, 127, 131, 239, 59, 419, 307, 73, 359, 367, 401, 419, 1163, 881, 307, 311, 967, 547, 569, 3697, 397, 691, 419, 457, 757, 163, 821, 839, 179, 1259, 907, 2111, 967, 1777, 599, 223, 3803, 3863, 2063, 3499, 1201, 3617, 2269, 263, 269, 1889, 2441, 283, 1409

Offset: 2

J. M. Bergot and Robert Israel, Feb 25 2021

a(k) is the least odd prime == -prime(k) (mod prime(k+1)).
a(k) = A163981(k) if and only if k is not in A029707.
a(k) = 2*prime(k+1)-prime(k) if and only if prime(k+1) is in A071680.

			a(3) = 23 is the least prime == -5 (mod 14), where prime(3) = 5 and prime(4) = 7.

Robert Israel, Table of n, a(n) for n = 2..10000

Cf. A029707, A071680, A163981, A342027.

f:= proc(n) local k;
  for k from 2*ithprime(n+1)-ithprime(n) by 2*ithprime(n+1)  do
    if isprime(k) then return k fi
  od;
end proc:
map(f, [$2..100]);

a(n) = forprime(p=2,, if (Mod(p, 2*prime(n+1)) == -prime(n), return (p))); \\ Michel Marcus, Feb 25 2021

(a(k) + prime(k)) mod (2*prime(k+1)) = 0.

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A117507 Numerators of partial sums of the Brun series divided by 4.

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A163981 a(n) is the smallest prime of the form prime(n+1)*k - prime(n), k >= 1, where prime(n) is the n-th prime.

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A320704 Indices of primes followed by a gap (distance to next larger prime) of 12.

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A320705 Indices of primes followed by a gap (distance to next larger prime) of 14.

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A320706 Indices of primes followed by a gap (distance to next larger prime) of 16.

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A320707 Indices of primes followed by a gap (distance to next larger prime) of 18.

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A320713 Indices of primes followed by a gap (distance to next larger prime) of 30.

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