A033560 -id:A033560 - cp's OEIS frontend

A126720 Primes p such that p - q = 24, where q is the previous prime before p; or prime numbers preceded by precisely 23 composite numbers.

1693, 2203, 4201, 4547, 4783, 5261, 6197, 6421, 6761, 7103, 7393, 7817, 8147, 8353, 9091, 11027, 11657, 11863, 12097, 12143, 13033, 13291, 16057, 16217, 16477, 16787, 16811, 17077, 17707, 18013, 18617, 18661, 19207, 19531, 20507, 22433, 22901

Offset: 1

Artur Jasinski, Feb 13 2007

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

Cf. A000230, A033560, A031505, A031925, A031927, A098974, A098976, A061779.

a = {}; Do[If[Prime[x + 1] - Prime[x] == 24, AppendTo[a, Prime[x + 1]]], {x, 1, 10000}]; a

q=2; forprime(p=3,1e5, if(p-q==24, print1(p", ")); q=p) \\ Charles R Greathouse IV, Mar 13 2020

a(n) = A098974(n) + 24. - Amiram Eldar, Mar 13 2020

a(n) >> n log^2 n. - Charles R Greathouse IV, Mar 13 2020

A257638 Numbers n such that n-25, n-1, n+1 and n+25 are consecutive primes.

Original entry on oeis.org

232962, 311712, 431832, 435948, 473352, 501342, 525492, 596118, 635388, 665922, 699792, 754182, 842448, 1013502, 1017648, 1036002, 1156848, 1255452, 1284738, 1306692, 1479912, 1516128, 1551732, 1560708, 1595928, 1659348, 1690572, 1745112

Offset: 1

Karl V. Keller, Jr., Nov 04 2015

nonn

This is a subsequence of A014574 (average of twin prime pairs) and A256753.
The terms ending in 2 and 8 are congruent to 12 mod 30 and 18 mod 30 respectively.
The numbers n-25 and n+1 belong to A033560 (p and p+24 are primes) and A098974 (p where p+24 is the next prime).
The numbers n-25 and n-1 belong to A252089 (p and p+26 are primes).

			232962 is the average of the four consecutive primes 232937, 232961, 232963, 232987.
311712 is the average of the four consecutive primes 311687, 311711, 311713, 311737.

Karl V. Keller, Jr., Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Twin Primes

Cf. A014574, A077800 (twin primes), A249674, A256753.

from sympy import isprime,prevprime,nextprime
for i in range(0,1000001,6):
  if isprime(i-1) and isprime(i+1) and prevprime(i-1) == i-25 and nextprime(i+1) == i+25: print (i,end=', ')

A263298 Numbers n such that n-23, n-1, n+1 and n+23 are consecutive primes.

Original entry on oeis.org

19890, 43890, 157770, 400680, 436650, 609780, 681090, 797310, 924360, 978180, 1093200, 1116570, 1179150, 1185930, 1313700, 1573110, 1663350, 2001510, 2110290, 2163570, 2336310, 2372370, 2408280, 2415630, 2562690, 2877840, 2896740, 2961900

Offset: 1

Karl V. Keller, Jr., Oct 13 2015

nonn

This is a subsequence of A014574 (average of twin prime pairs), A256753 and A249674 (30n).
From Michel Marcus, Oct 15 2015: (Start)
n-23 and n+1 belong to A242476 (p and p+22 are primes).
n-23 and n-1 belong to A033560 (p and p+24 are primes).
(End)

			19890 is the average of the four consecutive primes 19867, 19889, 19891, 19913.
43890 is the average of the four consecutive primes 43867, 43889, 43891, 43913.

Karl V. Keller, Jr., Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Twin Primes

Cf. A014574, A077800 (twin primes), A249674, A256753.

{p, q, r, s} = {2, 3, 5, 7};lst={}; While[p<5000000, If[Differences[{p, q, r, s}]=={22, 2, 22}, AppendTo[lst, q + 1]]; {p, q, r, s}={q, r, s,NextPrime@s}]; lst (* Vincenzo Librandi, Oct 14 2015 *)

isok(n) = isprime(n-1) && isprime(n+1) && (precprime(n-2) == n-23) && (nextprime(n+2) == n+23); \\ Michel Marcus, Oct 14 2015

from sympy import isprime,prevprime,nextprime
for i in range(0,5000001,6):
  if  isprime(i-1) and isprime(i+1) and prevprime(i-1) == i-23 and nextprime(i+1) == i+23: print (i,end=', ')

A274381 Safe primes p such that p + 24 is also a safe prime.

Original entry on oeis.org

23, 59, 83, 359, 479, 563, 839, 863, 1283, 2039, 2879, 2999, 3779, 4259, 4679, 5483, 7703, 10079, 12203, 13103, 23603, 26903, 27803, 30323, 31583, 33623, 35339, 41519, 43403, 44519, 44939, 53759, 55079, 57119, 57899, 58043, 65123, 66359, 70139, 70199, 76379, 77723, 79943

Offset: 1

Debapriyay Mukhopadhyay, Jun 23 2016

nonn

			83 is a safe prime and so is 83 + 24 = 107.
4679 is a safe prime and so is 4679 + 24 = 4703.

John Cerkan, Table of n, a(n) for n = 1..10000

Cf. A005385, A033560, A059323.

A274381:=n->`if`(type((n-1)/2, prime) and isprime(n+24) and isprime(n) and type((n+23)/2, prime), n, NULL): seq(A274381(n), n=1..10^5); # Wesley Ivan Hurt, Jun 25 2016

Select[Prime@ Range[10^4], PrimeQ[(# - 1)/2] && PrimeQ[# + 24] && PrimeQ[(23 + #)/2] &] (* Giovanni Resta, Jun 23 2016 *)
Select[Prime[Range[8000]],AllTrue[{(#-1)/2,(#+23)/2,#+24},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Dec 31 2019 *)

lista(nn) = forprime(p=3, nn, if (isprime((p-1)/2) && isprime(q=p+24) && isprime((q-1)/2), print1(p, ", "))); \\ Michel Marcus, Jun 23 2016

{ x | both x and x + 24 are safe primes }.

A207973 Primes p such that the equation prime(p-k)+k! = prime(p) has at least one solution k>0.

Original entry on oeis.org

2, 31, 43, 67, 107, 151, 167, 179, 227, 233, 389, 509, 547, 661, 719, 769, 823, 829, 967, 1033, 1093, 1259, 1321, 1493, 1567, 1733, 1873, 2099, 2341, 2539, 2621, 2683, 2819, 2927, 3119, 3169, 3373

Offset: 1

Gerasimov Sergey, Mar 02 2012

nonn

Apart from the first entry, the majority of the entries stem from k=4, i.e., this is essentially a reverse lookup within A033560. - R. J. Mathar, Mar 15 2012

			2 is in the sequence because prime(2) = prime(2-1)+1 = 3,
31 is in the sequence because prime(31) = prime(31-4)+1*2*3*4 = 103+24 = 127,
43 is in the sequence because prime(43) = prime(43-4)+1*2*3*4 = 167+24 = 191.

Cf. A000040, A000142.

is_A207973(n)={local(k);k=1;while((kMichael B. Porter, Mar 22 2012

A309392 Square array read by downward antidiagonals: A(n, k) is the k-th prime p such that p + 2*n is also prime, or 0 if that prime does not exist.

Original entry on oeis.org

3, 5, 3, 11, 7, 5, 17, 13, 7, 3, 29, 19, 11, 5, 3, 41, 37, 13, 11, 7, 5, 59, 43, 17, 23, 13, 7, 3, 71, 67, 23, 29, 19, 11, 5, 3, 101, 79, 31, 53, 31, 17, 17, 7, 5, 107, 97, 37, 59, 37, 19, 23, 13, 11, 3, 137, 103, 41, 71, 43, 29, 29, 31, 13, 11, 7, 149, 109

Offset: 1

Felix Fröhlich, Jul 28 2019

The same as A231608 except that A231608 gives the upward antidiagonals of the array, while this sequence gives the downward antidiagonals.
Conjecture: All values are nonzero, i.e., for any even integer e there are infinitely many primes p such that p + e is also prime.
The conjecture is true if Polignac's conjecture is true.

			The array starts as follows:
3,  5, 11, 17, 29, 41, 59,  71, 101, 107, 137, 149, 179, 191
3,  7, 13, 19, 37, 43, 67,  79,  97, 103, 109, 127, 163, 193
5,  7, 11, 13, 17, 23, 31,  37,  41,  47,  53,  61,  67,  73
3,  5, 11, 23, 29, 53, 59,  71,  89, 101, 131, 149, 173, 191
3,  7, 13, 19, 31, 37, 43,  61,  73,  79,  97, 103, 127, 139
5,  7, 11, 17, 19, 29, 31,  41,  47,  59,  61,  67,  71,  89
3,  5, 17, 23, 29, 47, 53,  59,  83,  89, 113, 137, 149, 167
3,  7, 13, 31, 37, 43, 67,  73,  97, 151, 157, 163, 181, 211
5, 11, 13, 19, 23, 29, 41,  43,  53,  61,  71,  79,  83,  89
3, 11, 17, 23, 41, 47, 53,  59,  83,  89, 107, 131, 137, 173
7, 19, 31, 37, 61, 67, 79, 109, 127, 151, 157, 211, 229, 241
5,  7, 13, 17, 19, 23, 29,  37,  43,  47,  59,  73,  79,  83

Wikipedia, Polignac's conjecture

Cf. A231608.

Cf. A001359 (row 1), A023200 (row 2), A023201 (row 3), A023202 (row 4), A023203 (row 5), A046133 (row 6), A153417 (row 7), A049488 (row 8), A153418 (row 9), A153419 (row 10), A242476 (row 11), A033560 (row 12), A252089 (row 13), A252090 (row 14), A049481 (row 15), A049489 (row 16), A252091 (row 17), A156104 (row 18), A271347 (row 19), A271981 (row 20), A271982 (row 21), A272176 (row 22), A062284 (row 25), A049490 (row 32), A020483 (column 1).

row(n, terms) = my(i=0); forprime(p=1, , if(i>=terms, break); if(ispseudoprime(p+2*n), print1(p, ", "); i++))
array(rows, cols) = for(x=1, rows, row(x, cols); print(""))
array(12, 14) \\ Print initial 12 rows and 14 columns of the array

cp's OEIS Frontend

A126720 Primes p such that p - q = 24, where q is the previous prime before p; or prime numbers preceded by precisely 23 composite numbers.

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A257638 Numbers n such that n-25, n-1, n+1 and n+25 are consecutive primes.

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A263298 Numbers n such that n-23, n-1, n+1 and n+23 are consecutive primes.

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A274381 Safe primes p such that p + 24 is also a safe prime.

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A207973 Primes p such that the equation prime(p-k)+k! = prime(p) has at least one solution k>0.

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A309392 Square array read by downward antidiagonals: A(n, k) is the k-th prime p such that p + 2*n is also prime, or 0 if that prime does not exist.

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