A151799 -id:A151799 - cp's OEIS frontend

A129912 Numbers that are products of distinct primorial numbers (see A002110).

1, 2, 6, 12, 30, 60, 180, 210, 360, 420, 1260, 2310, 2520, 4620, 6300, 12600, 13860, 27720, 30030, 37800, 60060, 69300, 75600, 138600, 180180, 360360, 415800, 485100, 510510, 831600, 900900, 970200, 1021020, 1801800, 2910600, 3063060, 5405400

Offset: 1

Bill McEachen, Jun 05 2007, Jun 06 2007, Jul 06 2007, Aug 07 2007

Conjecture: every odd prime p is either adjacent to a term of A129912 or a prime distance q from some term of A129912, where q < p. - Bill McEachen, Jun 03 2010, edited for clarity in Feb 26 2019

The first 2^20 terms k > 2 of A283477 all satisfy also the condition that the differences k-A151799(k) and A151800(k)-k are always either 1 or prime, like is also conjectured to hold for A002182 (cf. also the conjecture given in A117825). However, for A025487, which is a supersequence of both sequences, this is not always true: 512 is a member of A025487, but A151800(512) = 521, with 521 - 512 = 9, which is a composite number. - Antti Karttunen, Feb 26 2019

			For s = 4 there are 8 (generally 2^(s-1)) such numbers: 210 = 2*3*5*7, 420 = 2^2*3*5*7 = (2*3*5*7)*2, 1260 = 2^2*3^2*5*7 = (2*3*5*7)*(2*3), 6300 = 2^2*3^2*5^2*7 = (2*3*5*7)*(2*3*5), 2520 = 2^3*3^2*5*7 = (2*3*5*7)*(2*3)*2, 12600 = 2^3*3^2*5^2*7 = (2*3*5*7)*(2*3*5)*2, 37800 = 2^3*3^3*5^2*7 = (2*3*5*7)*(2*3*5)*(2*3), 75600 = 2^4*3^3*5^2*7 = (2*3*5*7)*(2*3*5)*(2*3)*2.

CRC Standard Mathematical Tables, 28th Ed., CRC Press

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Bill McEachen, Normalized A129912.
Robert Potter, Primorial Conjecture.
John Sokol, Sokol's Prime Conjecture, 2002.
Wikipedia, Primorial.
Index entries for sequences related to primorial numbers.

Subsequence of A025487. Sequence A283477 sorted into ascending order.

Cf. A002110, A117825, A151799, A151800.

Clear[f]; f[m_] := f[m] = Union[Times @@@ Subsets[FoldList[Times, 1, Prime[Range[m]]]]][[1 ;; 100]]; f[10]; f[m = 11]; While[f[m] != f[m-1], m++]; f[m] (* Jean-François Alcover, Mar 03 2014 *) (* or *)
pr[n_] := Product[Prime[n + 1 - i]^i, {i, n}]; upto[mx_] := Block[{ric, j = 1}, ric[n_, ip_, ex_] := If[n < mx, Block[{p = Prime[ip + 1]}, If[ex == 1, Sow@ n]; ric[n p^ex, ip + 1, ex]; If[ex > 1, ric[n p^(ex - 1), ip + 1, ex - 1]]]]; Sort@ Reap[ Sow[1]; While[pr[j] < mx, ric[2^j, 1, j]; j++]][[2, 1]]];
upto[10^30] (* faster, Giovanni Resta, Apr 02 2017 *)

is(n)=my(o=valuation(n,2),t); if(o<1||n<2, return(n==1)); n>>=o; forprime(p=3,, t=valuation(n,p); n/=p^t; if(t>o || tCharles R Greathouse IV, Oct 22 2015

Apart from 1 and 2, numbers of the form 2^k(1)*3^k(2)*5^k(3)*...*p(s)^k(s), where p(s) is s-th prime, k(i)>0 for i=1..s, k(i)-k(i-1) = 0 or 1 for i=2..s and |{k(1),k(2),..,k(s)}|=k(1). - Vladeta Jovovic, Jun 14 2007

Sum_{n>=1} 1/a(n) = Product_{n>=1} (1 + 1/A002110(n)) = 1.8177952875... . - Amiram Eldar, Jun 03 2023

Edited by N. J. A. Sloane, Jun 09 2007, Aug 08 2007
I corrected the Potter link to reflect its relocation. - Bill McEachen, Sep 12 2009
I added link to Wikicommons image. - Bill McEachen, Sep 16 2009
I again corrected the Potter link for its relocation - Bill McEachen, May 30 2013

A030459 Prime p concatenated with next prime is also prime.

Original entry on oeis.org

2, 31, 83, 151, 157, 167, 199, 233, 251, 257, 263, 271, 331, 353, 373, 433, 467, 509, 523, 541, 601, 653, 661, 677, 727, 941, 971, 1013, 1033, 1181, 1187, 1201, 1223, 1259, 1367, 1453, 1459, 1657, 1669, 1709, 1741, 1861, 1973, 2069, 2161

Offset: 1

Patrick De Geest

Terms 157, 257, 263, 541, 1187, 1459, 2179 also belong to A030460. - Carmine Suriano, Jan 27 2011

All terms, except for the first one, must be either in A185934 or in A185935, i.e., have the same residue (mod 6) as the subsequent prime. - M. F. Hasler, Feb 06 2011

M. F. Hasler and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 4110 terms from Hasler)

See A030461 for the concatenated primes.

Cf. A000720, A030460, A045533, A151799.

Select[Prime[Range[500]],PrimeQ[FromDigits[Join[IntegerDigits[#], IntegerDigits[ NextPrime[#]]]]]&] (* Harvey P. Dale, Jun 20 2011 *)

o=2;forprime(p=3,1e4, isprime(eval(Str(o,o=p))) & print1(precprime(p-1)","))  \\ M. F. Hasler, Feb 06 2011

a(n) = A151799(A030460(n)).

A030461(n) = concat(a(n), A030460(n)) = A045533(A000720(a(n))).

A123556 Number of elements in longest possible arithmetic progression of primes with difference n.

Original entry on oeis.org

2, 3, 2, 3, 2, 5, 1, 3, 2, 3, 2, 5, 1, 3, 2, 2, 2, 4, 1, 3, 2, 2, 1, 4, 1, 2, 2, 3, 2, 6, 1, 2, 1, 3, 2, 4, 1, 3, 2, 3, 2, 5, 1, 2, 2, 2, 1, 5, 1, 3, 2, 2, 1, 4, 1, 2, 2, 2, 2, 6, 1, 2, 1, 3, 2, 4, 1, 3, 2, 2, 2, 4, 1, 2, 1, 2, 2, 4, 1, 3, 2, 2, 1, 4, 1, 2, 2, 2, 1, 6, 1, 2, 1, 3, 2, 5, 1, 3, 2, 2, 2, 4, 1, 3, 2

Offset: 1

David W. Wilson, Nov 15 2006, revised Nov 25 2006

Length of n-th row of A124064.
The corresponding smallest term of the first such longest possible arithmetic progression of primes with common difference n is A342309(n). - Bernard Schott, Oct 12 2021
From Bernard Schott, Feb 24 2023: (Start)
For every positive integer n, there exists a smallest prime p that does not divide n = A053669(n); then, an AP of k primes with common difference n cannot contain more terms than this value of p so k <= p, moreover, the longest possible APs of primes have p-1 or p elements.
Proof: consider the AP of p elements (q, q+n, q+2*n, q+3*n, ..., q+(p-1)*n) with common difference n, q prime and p is the smallest prime that does not divide n; the modular arithmetic modulo p gives this set of remainders with p elements: {0, 1, 2, ..., p-1}, so there is always a multiple of p in each such AP with p terms, hence length k of longest possible AP of primes is >= p-1 and <= p.
Moreover, when the longest possible AP contains k = p elements, then this unique longest AP must start with p (corresponding to remainder = 0) and the common difference n is a multiple of A151799(p)# and not of p#, where # = primorial = A002110.
Now, always with a common difference n, when the longest possible AP contains k = p-1 elements, these longest APs with p-1 primes can start with p or with another prime q != p, and there are infinitely many such longest APs with p-1 terms (see Properties in Wikipedia link) in this case. When this AP starts with p, the set of remainders is {0, 1, ..., p-2} and when this AP starts with q, then the set of remainders becomes {1, 2, ..., p-1}.
Terms are ordered without repetition in A173919. (End)

			a(1) = 2 for the AP (arithmetic progression) (2, 3) with A342309(1) = 2.
a(2) = 3 for the AP (3, 5, 7) with A342309(2) = 3.
a(3) = 2 for the AP (2, 5) with A342309(3) = 2.
a(6) = 5 for the AP (5, 11, 17, 23, 29) with A342309(6) = 5.
a(7) = 1 for the AP (2) with A342309(7) = 2.
a(18) = 4 for the AP (5, 23, 41, 59) with A342309(18) = 5.
a(30) = 6 for the AP (7, 37, 67, 97, 127, 157) with A342309(30) = 7.
a(150) = 7 for the AP (7, 157, 307, 457, 607, 757, 907) with A342309(150) = 7.
From _Bernard Schott_, Feb 25 2023: (Start)
For n = 12, p = A053669(12) = 5 and the AP (5, 17, 29, 41, 53) has 5 elements that are primes (the next should be 65 = 5*13), so a(12) = 5. This AP is the unique longest possible AP of primes with a common difference n = 12.
For n = 30, p = A053669(30) = 7 and the AP (7, 37, 67, 97, 127, 157) has 7-1 = 6 elements that are primes (the next should be 187 = 11*17) so a(30) = 6. Also, there are infinitely many such longest APs with common difference 30 and 6 elements. These other longest APs start with primes q that are > p = 7. The first few next q are 107, 359, 541, 2221, 6673, 7457, ...
For n = 60, p = A053669(60) = 7 and the longest AP that starts with 7 is (7, 67, 127) has only 3 elements that are primes (the next should be 187 = 11*17) so a(60) = 6. Also, there are infinitely many such longest APs with common difference 60 and 6 elements. All these longest APs start with primes q that are > p = 7. The first few such q are 11, 53, 641, 5443, 10091, 12457, ... and the smallest such AP is (11, 71, 131, 191, 251, 311). (End)

Thomas Bloom, Are there arbitrarily long arithmetic progressions of primes?, Erdős Problems.
Diophante, A1880. NP (Nombres Premiers) en PA (Progression Arithmétique) (in French).
Terence Tao, Erdős problem database, see no. 219.
Wikipedia, Primes in arithmetic progression.
Index entries for sequences related to primes in arithmetic progressions.

Cf. A007921, A053669, A124064, A342309.
Cf. A002110, A151799, A173919.
Sequences such that a(n) = k iff ...: A007921 (a(n)=1), A359408 (a(n)=2), A206037 (a(n)=3), A359409 (a(n)=4), A206039 (a(n)=5), A359410 (a(n)=6), A206041 (a(n)=7), A360146 (a(n)=10), A206045 (a(n)=11).

A053669(n) = forprime(p=2, , if(n%p, return(p)));
a(n) = my(p=A053669(n)); for (i=1, p-1, if (!isprime(p+i*n), return(p-1))); p; \\ Michel Marcus, Feb 26 2023

Assume the k-tuples conjecture. Let p = A053669(n). If the arithmetic progression of p elements starting at p with difference n consists of primes, then a(n) = p, otherwise a(n) = p-1.

A034808 Concatenation of 'prevprime(k) and k' is a prime.

Original entry on oeis.org

3, 9, 37, 39, 51, 63, 87, 89, 111, 117, 123, 153, 157, 163, 173, 177, 183, 207, 211, 213, 217, 219, 239, 249, 257, 263, 267, 269, 273, 277, 279, 289, 321, 323, 327, 333, 337, 339, 343, 359, 369, 379, 407, 423, 439, 441, 459, 471, 473, 477, 479, 489, 497, 513

Offset: 1

Patrick De Geest, Oct 15 1998

Since there are primes in the sequence, and concat(p,p) = p*(10^x+1) is always composite, it is clear that here the variant 2 (A151799(n) < n) of the prevprime function is used, rather than the variant 1 (A007917(n) <= n). - M. F. Hasler, Sep 09 2015

			n=333 -> previous prime is 331, thus '331333' is a prime.

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

Cf. A034809-A034821.

Cf. A151799, A007917.

coQ[n_]:=PrimeQ[FromDigits[Flatten[IntegerDigits[{NextPrime[n,-1],n}]]]]; Select[Range[3,513],coQ[#]&] (* Jayanta Basu, May 30 2013 *)
Select[Range[2,550],PrimeQ[NextPrime[#,-1]*10^IntegerLength[#]+#]&] (* Harvey P. Dale, Nov 22 2020 *)

isok(n)=n>2 && isprime(fromdigits(concat(digits(precprime(n-1)), digits(n)))) \\ Andrew Howroyd, Aug 13 2024

from sympy import isprime, prevprime
def aupto(m):
  return [k for k in range(3, m+1) if isprime(int(str(prevprime(k))+str(k)))]
print(aupto(513)) # Michael S. Branicky, Mar 09 2021

Offset changed by Andrew Howroyd, Aug 13 2024

A175851 a(n) = 1 for noncomposite n, a(n) = n - previousprime(n) + 1 for composite n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 3, 4, 1, 2, 1, 2, 3, 4, 1, 2, 1, 2, 3, 4, 1, 2, 3, 4, 5, 6, 1, 2, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 1, 2, 1, 2, 3, 4, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 1, 2, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 1, 2, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4

Offset: 1

Jaroslav Krizek, Sep 29 2010

Sequence is cardinal and not fractal. Cardinal sequence is sequence with infinitely many times occurring all natural numbers. Fractal sequence is sequence such that when the first instance of each number in the sequence is erased, the original sequence remains.

Ordinal transform of the nextprime function, A151800(1..) = 2, 3, 5, 5, 7, 7, 11, 11, 11, 11, ..., also ordinal transform of A304106. - Antti Karttunen, Jun 09 2018

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000720, A007917, A008578, A151799, A151800, A305300.
Cf. A065358 for another way of visualizing prime gaps.
Cf. A304106 (ordinal transform of this sequence).
Cf. A049711.

a[n_] := If[!CompositeQ[n], 1, n - NextPrime[n, -1] + 1];
Array[a, 100] (* Jean-François Alcover, Dec 19 2021 *)

A175851(n) = if(1==n,n,1 + n - precprime(n)); \\ Antti Karttunen, Mar 04 2018

a(1) = 1, a(n) = n - A007917(n) + 1 for n >= 2. a(1) = 1, a(2) = 1, a(n) = n - A151799(n+1) + 1 for n >= 3.
a(n) = Sum_{i=1..n} floor(pi(i)/pi(n)), for n>1 with pi(n) = A000720(n). - Ridouane Oudra, Jun 24 2024
a(n) = A049711(n+1), for n>1. - Ridouane Oudra, Jul 16 2024

A059788 a(n) = largest prime < 2*prime(n).

Original entry on oeis.org

3, 5, 7, 13, 19, 23, 31, 37, 43, 53, 61, 73, 79, 83, 89, 103, 113, 113, 131, 139, 139, 157, 163, 173, 193, 199, 199, 211, 211, 223, 251, 257, 271, 277, 293, 293, 313, 317, 331, 337, 353, 359, 379, 383, 389, 397, 421, 443, 449, 457, 463, 467, 479, 499, 509, 523

Offset: 1

Labos Elemer, Feb 22 2001

nonn

Also, smallest member of the first pair of consecutive primes such that between them is a composite number divisible by the n-th prime. - Amarnath Murthy, Sep 25 2002

Except for its initial term, A006992 is a subsequence based on iteration of n -> A151799(2n). The range of this sequence is a subset of A065091. - M. F. Hasler, May 08 2016

			n=18: p(18)=61, so a(18) is the largest prime below 2*61=122, which is 113.

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A005382, A005383, A005384, A005385, A055496, A006992, A052248.

with(numtheory):
A059788 := proc(n)
    prevprime(2*ithprime(n)) ;
end proc:
seq(A059788(n),n=1..50) ; # R. J. Mathar, May 08 2016

a[n_] := Prime[PrimePi[2Prime[n]]]
NextPrime[2*Prime[Range[100]], -1] (* Zak Seidov, May 08 2016 *)

a(n) = precprime(2*prime(n)); \\ Michel Marcus, May 08 2016

a(n) = A007917(A100484(n)). - R. J. Mathar, May 08 2016

A256891 Smallest primes of 3 X 3 magic squares formed from consecutive primes.

Original entry on oeis.org

1480028129, 1850590057, 5196185947, 5601567187, 5757284497, 6048371029, 6151077269, 9517122259, 19052235847, 20477868319, 23813359613, 24026890159, 26748150199, 28519991387, 34821326119, 44420969909, 49285771679, 73827799009, 73974781889, 74220519319, 76483907837, 76560277009, 80143089599, 85892025227, 89132925737, 95515449037, 99977424653

Offset: 1

Arkadiusz Wesolowski, Apr 12 2015

nonn

Let a = a(n) for some n and {a, b, c, d, e, f, g, h, i} be the set of consecutive primes. Then it is:
   +---+---+---+              +---+---+---+
   | d | c | h |              | c | d | h |
   +---+---+---+              +---+---+---+
   | i | e | a | (type 1), or | i | e | a | (type 2). See Harvey D. Heinz.
   +---+---+---+              +---+---+---+
   | b | g | f |              | b | f | g |
   +---+---+---+              +---+---+---+
  The type is determined by the sign of A343195.
For a given magic sum S, it is easy to calculate the unique set of n^2 consecutive primes that sum up to n*S (see PROGRAM MagicPrimes() in A073519), and in particular the smallest of these (cf. PROGRAM), listed here for n = 3, in A260673 for n = 4, in A272386 for n = 5, and in A272387 for n = 6. - M. F. Hasler, Oct 28 2018

Allan W. Johnson, Jr., Consecutive-Prime Magic Squares, Journal of Recreational Mathematics, vol. 15, 1982-83, pp. 17-18.
H. L. Nelson, A Consecutive Prime 3 x 3 Magic Square, Journal of Recreational Mathematics, vol. 20:3, 1988, p. 214.

A.H.M. Smeets, Table of n, a(n) for n = 1..759
Harvey D. Heinz, Prime Numbers Magic Squares: Minimum consecutive primes - 3, 1999-2010.
Eric Weisstein's World of Mathematics, Prime Magic Square
Index entries for sequences related to magic squares

Cf. A073519, A151799, A166113, A260673, A272386, A272387, A343194, A343195.

Subsequence of A265139.

/* Brute-force search */ lst:=[]; n:=3; while n lt 10^11 do a:=NextPrime(n); q:=a; j:=a-n; if j mod 6 eq 0 then b:=NextPrime(a); if j eq b-a then c:=NextPrime(b); d:=c-b; if d mod 6 eq 0 then e:=NextPrime(c); k:=e-c; if k eq j then f:=NextPrime(e); if k eq f-e then g:=NextPrime(f); if g-f eq d then h:=NextPrime(g); m:=h-g; if m eq k then i:=NextPrime(h); if h-g eq i-h then Append(~lst, n); end if; end if; end if; end if; end if; end if; end if; end if; n:=q; end while; lst;

A256891(n)=MagicPrimes(A270305(n),3)[1] \\ See A073519 for MagicPrimes(). - M. F. Hasler, Oct 28 2018

a(n) = A151799(A151799(A151799(A151799(A166113(n))))). - Max Alekseyev, Nov 02 2015

Extended by Max Alekseyev, Nov 02 2015

A320873 List of 3 X 3 magic squares made of consecutive primes, in order of increasing magic sum. Only the lexicographically smallest variant of equivalent squares (modulo D4 symmetries) is listed, as a row containing the 3 rows of the square.

Original entry on oeis.org

1480028141, 1480028189, 1480028183, 1480028213, 1480028171, 1480028129, 1480028159, 1480028153, 1480028201, 1850590069, 1850590117, 1850590111, 1850590141, 1850590099, 1850590057, 1850590087, 1850590081, 1850590129, 5196185959, 5196186007, 5196186001, 5196186031, 5196185989, 5196185947, 5196185977, 5196185971, 5196186019

Offset: 1

M. F. Hasler, Oct 22 2018

The first row is the lexicographically first 3 X 3 magic square of consecutive primes with the smallest possible magic constant 4440084513 = A270305(1) = A073520(3).
The same 9 terms are also given in increasing order in sequence A073519. But this is equivalent of giving just the smallest of the terms (cf. A256891) or the central element (cf. A166113) or the magic constant itself (cf. A270305), which uniquely determines the sequence of primes since they have to be consecutive and their sum is equal to 3 times the magic constant.
In the case of 3 X 3 magic squares, however, the lexicographically smallest representative has its elements in a well-defined order, see comment in A320872. This allows the reconstruction of the square from the set of primes which can be computed from the central elements A166113 or magic constants A270305, cf. PROGRAM in A073519.

			The first row of 9 terms, (1480028141, 1480028189, 1480028183, 1480028213, 1480028171, 1480028129, 1480028159, 1480028153, 1480028201), corresponds to the following smallest 3 X 3 magic square of consecutive primes:
    [1480028141  1480028189  1480028183]
    [1480028213  1480028171  1480028129] .
    [1480028159  1480028153  1480028201]
The eleventh row yields the first example where the second term is smaller than the third one:
    [23813359643  23813359721  23813359727]
    [23813359781  23813359697  23813359613] .
    [23813359667  23813359673  23813359751]

Allan W. Johnson, Jr., Journal of Recreational Mathematics, vol. 23:3, 1991, pp. 190-191.
Clifford A. Pickover, The Zen of Magic Squares, Circles and Stars: An Exhibition of Surprising Structures across Dimensions, Princeton University Press, 2002.

Harvey Heinz, Prime Magic Squares
Index entries for sequences related to magic squares

Cf. A073519, A073520, A073521, A073522.
Cf. A073520 (smallest magic sum for a n X n magic square made from consecutive primes).
Cf. A104157 (smallest of n^2 consecutive primes forming a magic square).
Cf. A166113 (center element of 3 X 3 magic squares of consecutive primes).
Cf. A256891 (smallest entry of 3 X 3 magic squares of consecutive primes) = A151799^4(A166113).
Cf. A270305 (magic sums of 3 X 3 magic squares of consecutive primes) = 3*A166113.

A320873_row(n)=vecextract(n=MagicPrimes(3*A166113[n],3),[2,6+n=n[2]*2==n[1]+n[3],7-n,9,5,1,3+n,4-n,8]) \\ For MagicPrimes() see A073519 (the set of primes of the first row).
/* the following allows the production of all 8 variants of a magic square that are equivalent modulo reflection on any of the 4 symmetry axes of the square */
REV(M)=matconcat(Vecrev(M)) \\ reverse the order of columns of M
FLIP(M)=matconcat(Colrev(M)) \\ reverse the order of rows of M
ALL(M,C(f,L)=concat(apply(f,L),L))=Set(C(REV,C(FLIP,[M,M~]))) \\ PARI orders the set according to the (first) columns of the matrices, so one must take the transpose to get them ordered according to elements of the first row.

a(9n-4) = A166113(n) = A270305(n)/3 for all n >= 1.

A013633 nextprime(n) - prevprime(n).

Original entry on oeis.org

3, 2, 4, 2, 6, 4, 4, 4, 6, 2, 6, 4, 4, 4, 6, 2, 6, 4, 4, 4, 10, 6, 6, 6, 6, 6, 8, 2, 8, 6, 6, 6, 6, 6, 10, 4, 4, 4, 6, 2, 6, 4, 4, 4, 10, 6, 6, 6, 6, 6, 12, 6, 6, 6, 6, 6, 8, 2, 8, 6, 6, 6, 6, 6, 10, 4, 4, 4, 6, 2, 8, 6, 6, 6, 6, 6, 10, 4, 4

Offset: 3

N. J. A. Sloane

nonn

			a(3) = nextprime(3) - prevprime(3) = 5 - 2 = 3: This shows that here the variants 2 (A151800 and A151799) of the nextprime and precprime functions are used, rather than the variants A007918 and A007917. - _M. F. Hasler_, Sep 09 2015

T. D. Noe, Table of n, a(n) for n=3..2000

Cf. A151800, A007918; A151799, A007917.

[ seq(nextprime(i)-prevprime(i),i=3..100) ];

A013633(n)=nextprime(n+1)-precprime(n-1) \\ M. F. Hasler, Sep 09 2015

a(n) = A151800(n) - A151799(n) = A007918(n+1) - A007917(n-1). - M. F. Hasler, Sep 09 2015

A305419 Largest k < n whose binary expansion encodes an irreducible (0,1)-polynomial over GF(2)[X], with a(1) = a(2) = 1.

Original entry on oeis.org

1, 1, 2, 3, 3, 3, 3, 7, 7, 7, 7, 11, 11, 13, 13, 13, 13, 13, 13, 19, 19, 19, 19, 19, 19, 25, 25, 25, 25, 25, 25, 31, 31, 31, 31, 31, 31, 37, 37, 37, 37, 41, 41, 41, 41, 41, 41, 47, 47, 47, 47, 47, 47, 47, 47, 55, 55, 55, 55, 59, 59, 61, 61, 61, 61, 61, 61, 67, 67, 67, 67, 67, 67, 73, 73, 73, 73, 73, 73, 73, 73, 73, 73, 73, 73, 73, 73, 87

Offset: 1

Antti Karttunen, Jun 07 2018

nonn

For n >= 3, a(n) is the largest term of A014580 less than n.

Antti Karttunen, Table of n, a(n) for n = 1..21845
Index entries for sequences related to polynomials in ring GF(2)[X]

Cf. A014580, A091225, A305420, A305422.

Cf. also A151799 (A136548), A305429.

A091225(n) = polisirreducible(Pol(binary(n))*Mod(1, 2));
A305419(n) = if(n<3,1, my(k=n-1); while(k>1 && !A091225(k),k--); (k));

cp's OEIS Frontend

A129912 Numbers that are products of distinct primorial numbers (see A002110).

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A030459 Prime p concatenated with next prime is also prime.

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A123556 Number of elements in longest possible arithmetic progression of primes with difference n.

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A034808 Concatenation of 'prevprime(k) and k' is a prime.

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A175851 a(n) = 1 for noncomposite n, a(n) = n - previousprime(n) + 1 for composite n.

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A059788 a(n) = largest prime < 2*prime(n).

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A256891 Smallest primes of 3 X 3 magic squares formed from consecutive primes.

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