A006992 -id:A006992 - cp's OEIS frontend

A076994 a(1) = 2, a(n+1) is the largest squarefree number < 2*a(n).

2, 3, 5, 7, 13, 23, 43, 85, 167, 331, 661, 1321, 2641, 5281, 10561, 21121, 42241, 84481, 168961, 337921, 675841, 1351681, 2703361, 5406721, 10813441, 21626881, 43253761, 86507521, 173015041, 346030081, 692060161, 1384120321, 2768240641

Offset: 1

Amarnath Murthy, Oct 26 2002

nonn

Analogous to Bertrand's primes.

Hugo Pfoertner, Table of n, a(n) for n = 1..360

Cf. A006992, A076995.

with(numtheory):a[1] := 2:for n from 2 to 84 do q := 2*a[n-1]-1:while(not issqrfree(q)) do q := q-1:od:a[n] := q:od:seq(a[l],l=1..84);

lsfn[n_]:=Module[{s=2n-1},While[!SquareFreeQ[s],s--];s]; NestList[ lsfn,2,40] (* Harvey P. Dale, Nov 28 2018 *)

More terms from Sascha Kurz, Jan 26 2003

A118909 a(1) = 4; a(n) is least semiprime > a(n-1)^2.

Original entry on oeis.org

4, 21, 445, 198026, 39214296677, 1537761063871773242347, 2364709089560047865452947255794201194068433, 5591849078247910476736920566826713466552016538943524658263883555662554776622687075541

Offset: 1

Jonathan Vos Post, May 05 2006

Semiprime analog of A055496 a(1) = 2; a(n) is smallest prime > 2*a(n-1). See also A059785 a(n+1)=prevprime(a(n)^2), with a(1) = 2. With that, of course, there's always a prime between n and 2n, so a(n) < 2^n. The obverse of this is A118908 a(1) = 4; a(n) is greatest semiprime < a(n-1)^2.

			a(8) = a(7)^2 + 52 and there is no smaller k such that a(7)^2 + k is semiprime.

Cf. A001358, A055496, A076656, A006992, A005384, A005385, A118908-A118913.

nxt[n_]:=Module[{sp=n^2+1},While[PrimeOmega[sp]!=2,sp++];sp]; NestList[nxt,4,7] (* Harvey P. Dale, Oct 22 2012 *)

from itertools import accumulate
from sympy.ntheory.factor_ import primeomega
def nextsemiprime(n):
  while primeomega(n + 1) != 2: n += 1
  return n + 1
def f(anm1, _): return nextsemiprime(anm1**2)
print(list(accumulate([4]*6, f))) # Michael S. Branicky, Apr 21 2021

A185231 a(n) = largest prime <= 2a(n-1), with a(0)=1.

Original entry on oeis.org

1, 2, 3, 5, 7, 13, 23, 43, 83, 163, 317, 631, 1259, 2503, 5003, 9973, 19937, 39869, 79699, 159389, 318751, 637499, 1274989, 2549951, 5099893, 10199767, 20399531, 40799041, 81598067, 163196129, 326392249, 652784471, 1305568919, 2611137817

Offset: 0

N. J. A. Sloane, Jan 24 2012, following a suggestion from Frank M Jackson

nonn

Equals 1 followed by A006992.

This is a complete sequence (cf. A075058).

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A006992, A075058.

np[n_]:=Module[{p=NextPrime[2n]},If[p<=2n,p,NextPrime[p,-1]]]; NestList[ np,1,40] (* Harvey P. Dale, Sep 29 2019 *)

lista(nn) = {p = 1; for (n = 1, nn, print1(p, ", "); p = precprime(2*p););} \\ Michel Marcus, Aug 26 2013

A380277 A version of the array A229607 without duplicates, read by antidiagonals: each row starts with the least prime not in a previous row, and each number p in a row is followed by the greatest prime q in the interval p < q < 2*p not in a previous row (or 0 if no such q exists).

Original entry on oeis.org

2, 3, 11, 5, 19, 17, 7, 37, 31, 29, 13, 73, 61, 53, 41, 23, 139, 113, 103, 79, 47, 43, 277, 223, 199, 157, 89, 59, 83, 547, 443, 397, 313, 173, 109, 67, 163, 1093, 883, 787, 619, 337, 211, 131, 71, 317, 2179, 1759, 1571, 1237, 673, 421, 257, 137, 97

Offset: 1

Pontus von Brömssen, Jan 18 2025

It appears that the first column is A104272.

Proof: (This proof assumes that all terms of the array are nonzero. It would be nice to see a proof of this.) Let n >= 2 and p = T(n,1). To prove that p = A104272(n) we need to prove that pi(x)-pi(x/2) >= n for x >= p and that pi(p-1)-pi((p-1)/2) < n. Let x >= p and let q be the smallest prime larger than x. In each of the rows 1..n there are consecutive terms r < r' with r < q <= r' < 2*r, so q/2 < r < q. Hence there are at least n primes between q/2 and q (not counting q itself), i.e., pi(q)-pi(q/2) >= n+1. It follows that pi(x)-pi(x/2) = pi(q)-1-pi(x/2) >= pi(q)-1-pi(q/2) >= n. Finally, if pi(p-1)-pi((p-1)/2) >= n there would exist two consecutive terms r and r' in one of the rows 1..(n-1) with (p-1)/2 < r < r' <= p-1. This is impossible, because then p (or some larger prime) would have been chosen instead of r' as the successor of r. Hence pi(p-1)-pi((p-1)/2) < n. This concludes the proof (with the caveat above).

			Array starts:
   2,   3,   5,   7,   13,   23,   43,    83,   163,   317, ...
  11,  19,  37,  73,  139,  277,  547,  1093,  2179,  4357, ...
  17,  31,  61, 113,  223,  443,  883,  1759,  3517,  7027, ...
  29,  53, 103, 199,  397,  787, 1571,  3137,  6271, 12541, ...
  41,  79, 157, 313,  619, 1237, 2473,  4943,  9883, 19763, ...
  47,  89, 173, 337,  673, 1327, 2647,  5281, 10559, 21107, ...
  59, 109, 211, 421,  839, 1669, 3331,  6661, 13313, 26597, ...
  67, 131, 257, 509, 1013, 2017, 4027,  8053, 16103, 32203, ...
  71, 137, 271, 541, 1069, 2137, 4273,  8543, 17077, 34147, ...
  97, 193, 383, 761, 1511, 3019, 6037, 12073, 24137, 48271, ...
  ...
The least prime not in any of the first 6 rows is T(7,1) = 59. The greatest prime less than 2*59 = 118 is 113, but that number appears in a previous row as T(3,4). The next smaller prime is 109, which does not appear in a previous row, so T(7,2) = 109.

Pontus von Brömssen, Table of n, a(n) for n = 1..5050 (first 100 antidiagonals)

Cf. A000720, A006992 (first row), A104272, A229607.

A076995 a(1) = 4, a(n+1) is the largest composite number < 2a(n).

Original entry on oeis.org

4, 6, 10, 18, 35, 69, 136, 270, 539, 1077, 2152, 4303, 8605, 17208, 34415, 68829, 137657, 275313, 550625, 1101249, 2202497, 4404993, 8809985, 17619969, 35239936, 70479871, 140959741, 281919481, 563838961, 1127677921, 2255355841

Offset: 1

Amarnath Murthy, Oct 26 2002

nonn

Cf. A006992, A076994.

a[1] := 4:for n from 2 to 84 do q := 2*a[n-1]-1:while(isprime(q)) do q := q-1:od: a[n] := q:od:seq(a[l],l=1..84);

Corrected and extended by Sascha Kurz, Jan 26 2003

A118912 a(1) = 2; a(n) is greatest prime < a(n-1)^4.

Original entry on oeis.org

2, 13, 28559, 665230244078823349, 195833931687186822327230545227550596864953022841534058316595001440791433

Offset: 1

Jonathan Vos Post, May 05 2006

Exponent-4 analog of A059785 a(n+1)=prevprime(a(n)^2), with exponent 3 being A118910 a(1) = 2; a(n) is greatest prime < a(n-1)^3.

			a(1) = 2, by definition.
a(2) = 13 = 2^4 - 3.
a(3) = 28559 = 13^4 - 2.
a(4) = 665230244078823349 = 28559^4 - 12.
a(5) = 195833931687186822327230545227550596864953022841534058316595001440791433 = 665230244078823349^4 - 168.
a(6) is too large to include.

Cf. A001358, A055496, A076656, A006992, A005384, A005385, A118908-A118913.

NestList[NextPrime[#^4,-1]&,2,5] (* Harvey P. Dale, Feb 18 2025 *)

a(1) = 2; a(n) is greatest prime < a(n-1)^4.

A164920 Primes which are obtained at least by two ways using the iterations of the Bertrand operator (see A164917) beginning with primes of A164368.

Original entry on oeis.org

113, 139, 199, 211, 223, 277, 293, 397, 421, 443

Offset: 1

Vladimir Shevelev, Aug 31 2009

The sequence is connected with our sieve selecting the primes of A164368 from all primes.

			113 is in the sequence since it is obtained either by iterations of Bertrand operator beginning from 17 (17=>31=>61=>113) or by such iterations beginning with 59 (59=>113), and both of primes 17 and 59 are in A164368.

V. Shevelev, On critical small intervals containing primes, arXiv:0908.2319 [math.NT], 2009.

Cf. A006992, A164917, A164918, A104272, A164368, A164288.

A164962 a(n) is the least prime from the union {2,3} and A164333, beginning with which the n-th prime p_n is obtained by some number of iterations of the S operator g(see A164960).

Original entry on oeis.org

2, 3, 2, 3, 2, 13, 3, 19, 2, 13, 31, 3, 19, 43, 2, 53, 13, 61, 31, 71, 73, 3, 19, 43, 2, 101, 103, 53, 109, 113, 13, 131, 31

Offset: 1

Vladimir Shevelev, Sep 02 2009

The sequence is connected with our sieve selecting the primes of the union {2,3} and A164333 from all primes. Note that a(n)=n iff p_n is in the considered union, which corresponds to 0's iterations of g.

V. Shevelev, On critical small intervals containing primes, arXiv:0908.2319 [math.NT], 2009.

Cf. A164333, A055496, A164960, A104272, A080359, A164918, A006992, A164917, A164368, A164288.

A217836 a(n) is largest semiprime < 2*a(n-1), with a(1) = 4.

Original entry on oeis.org

4, 6, 10, 15, 26, 51, 95, 187, 371, 737, 1473, 2942, 5878, 11755, 23507, 47013, 94021, 188041, 376069, 752135, 1504261, 3008503, 6017001, 12034001, 24068001, 48135995, 96271987, 192543973, 385087943, 770175883, 1540351763, 3080703523, 6161407045, 12322814089, 24645628171

Offset: 1

Jonathan Vos Post, Oct 19 2012

This is to Bertrand primes A006992 as semiprimes A001358 are to primes A000040.

			a(2) = 6 because that is the largest semiprime < 2*a(1) = 8, where a(1) is the first semiprime.
a(3) = 10, the largest semiprime < 2*6 = 12.

Robert G. Wilson v, Table of n, a(n) for n = 1...225

Cf. A001358, A006992.

PrevSemiPrime[n_, k_] := Block[{c = 0, sp = n - 1}, While[c < k, While[ PrimeOmega[sp] != 2, sp--]; sp--; c++]; sp + 1]; NestList[ PrevSemiPrime[ 2#, 1] &, 4, 34] (* Robert G. Wilson v, Oct 19 2012 *)

Terms greater than a(15) from Robert G. Wilson v, Oct 19 2012

A227770 Bertrand primes II: a(n) is the largest prime < 2*a(n-1)-2.

Original entry on oeis.org

5, 7, 11, 19, 31, 59, 113, 223, 443, 883, 1759, 3511, 7019, 14033, 28057, 56101, 112199, 224363, 448703, 897401, 1794787, 3589571, 7179127, 14358247, 28716487, 57432961, 114865903, 229731787, 459463553, 918927083, 1837854119, 3675708217, 7351416419, 14702832827, 29405665651, 58811331281, 117622662557, 235245325061, 470490650107, 940981300211, 1881962600417

Offset: 1

Jonathan Sondow, Jul 30 2013

nonn

A strong form of Bertrand's postulate (Chebyshev's theorem) says there exists a prime number p with n < p < 2*n - 2 if n > 3.
The first prime > 3 is 5, so the sequence begins a(1) = 5.
For references, links, and crossrefs, see A006992 (Bertrand primes I).

			The largest prime < 2*a(1)-2 = 2*5-2 = 8 is 7, so a(2) = 7 = A006992(4).
The largest prime < 2*a(2)-2 = 2*7-2 = 12 is 11, so a(3) = 11 < 13 = A006992(5).

Cf. A006992.

NestList[NextPrime[2 # - 2, -1] &, 5, 40]

cp's OEIS Frontend

A076994 a(1) = 2, a(n+1) is the largest squarefree number < 2*a(n).

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A118909 a(1) = 4; a(n) is least semiprime > a(n-1)^2.

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A185231 a(n) = largest prime <= 2a(n-1), with a(0)=1.

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A380277 A version of the array A229607 without duplicates, read by antidiagonals: each row starts with the least prime not in a previous row, and each number p in a row is followed by the greatest prime q in the interval p < q < 2*p not in a previous row (or 0 if no such q exists).

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A076995 a(1) = 4, a(n+1) is the largest composite number < 2a(n).

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Maple

Extensions

A118912 a(1) = 2; a(n) is greatest prime < a(n-1)^4.

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A164920 Primes which are obtained at least by two ways using the iterations of the Bertrand operator (see A164917) beginning with primes of A164368.

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A164962 a(n) is the least prime from the union {2,3} and A164333, beginning with which the n-th prime p_n is obtained by some number of iterations of the S operator g(see A164960).

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A217836 a(n) is largest semiprime < 2*a(n-1), with a(1) = 4.

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