A060264 -id:A060264 - cp's OEIS frontend

A164960 The minimum number of steps needed to generate prime(n) under the map x -> A060264(x) starting from any x taken from {2,3} or from A164333.

0, 0, 1, 1, 2, 0, 2, 0, 3, 1, 0, 3, 1, 0, 4, 0, 2, 0, 1, 0, 0, 4, 2, 1, 5, 0, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, 5, 3, 0, 2, 0, 0, 0, 6, 0, 1, 1, 1, 0, 0, 0, 0, 0, 2, 1, 0, 0, 3, 1, 0, 0, 1, 0, 0, 1, 6, 4, 1, 0, 0, 3, 1, 0, 0, 1, 1, 7, 1

Offset: 1

Vladimir Shevelev, Sep 02 2009

nonn

			a(3) = 1 because prime(3)=5 can be generated in 1 step starting from x=2.
a(4) = 1 because prime(4)=7 can be generated in 1 step starting from x=3.

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

Cf. A164333.

# include source from A164333 and A060264 here
A164333 := proc(n)
        if n = 1 then
                13;
        else
                for a from procname(n-1)+1 do
                        if isA164333(a) then
                                return a;
                        end if;
                end do;
        end if;
end proc:
A164960aux := proc(p,strt)
        local a,x;
        if strt > p then
                return 1000000000;
        end if;
        a := 0 ;
        x := strt ;
        while x < p do
                x := A060264(x) ;
                a := a+1 ;
        end do;
        if x = p then
                return a ;
        else
                return 1000000000;
        end if;
end proc:
A164960 := proc(n)
        local p,a,strt,i;
        p := ithprime(n) ;
        a := A164960aux(p,2) ;
        a := min(a,A164960aux(p,3)) ;
        for i from 1 do
                strt := A164333(i) ;
                if strt > p then
                        return a;
                else
                        a := min(a, A164960aux(p,strt)) ;
                end if;
        end do:
        return a;
end proc:
seq(A164960(n),n=1..90) ; # R. J. Mathar, Oct 29 2011

nmax = 100; kmax = nmax + 5;
A164333 = Select[Table[{(Prime[k - 1] + 1)/2, (Prime[k] - 1)/2}, {k, 3, kmax}], AllTrue[Range[#[[1]], #[[2]]], CompositeQ] &][[All, 2]]*2 + 1;
A164960aux[p_, strt_] := Module[{a, x}, If[strt > p, Return[10^9]]; a = 0; x = strt; While[x < p, x = NextPrime[2 x]; a++]; If[x == p, Return[a], Return[10^9]]];
A164960[n_] := Module[{p, a, strt, i}, p = Prime[n]; a = A164960aux[p, 2]; a = Min[a, A164960aux[p, 3]]; For[i = 1, i < 100, i++, strt = A164333[[i]]; If[strt > p, Return[a], a = Min[a, A164960aux[p, strt]]]]; Return[a]];
Table[A164960[n], {n, 1, nmax}] (* Jean-François Alcover, Dec 13 2017, after R. J. Mathar *)

One term corrected, sequence extended, examples added by R. J. Mathar, Oct 29 2011

A118753 First prime after 4n. Smallest prime >= 4*n. Bisection of A060264.

Original entry on oeis.org

2, 5, 11, 13, 17, 23, 29, 29, 37, 37, 41, 47, 53, 53, 59, 61, 67, 71, 73, 79, 83, 89, 89, 97, 97, 101, 107, 109, 113, 127, 127, 127, 131, 137, 137, 149, 149, 149, 157, 157, 163, 167, 173, 173, 179, 181, 191, 191, 193, 197, 211, 211, 211, 223, 223, 223, 227, 229, 233

Offset: 0

Jonathan Vos Post, Apr 29 2006

Analogous to A060264 First prime after 2n; A118751 First prime after 3n.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A000040, A008586, A060308, A060264, A118747-56.

seq(nextprime(4*k),k=0..100); # Robert Israel, Dec 25 2017

NextPrime/@(4Range[0,60]) (* Harvey P. Dale, Nov 14 2021 *)

a(n) = min{A008586(n)+k such that k>0 and A008586(n)+k in A000040}.

A055496 a(1) = 2; a(n) is smallest prime > 2*a(n-1).

Original entry on oeis.org

2, 5, 11, 23, 47, 97, 197, 397, 797, 1597, 3203, 6421, 12853, 25717, 51437, 102877, 205759, 411527, 823117, 1646237, 3292489, 6584983, 13169977, 26339969, 52679969, 105359939, 210719881, 421439783, 842879579, 1685759167, 3371518343

Offset: 1

N. J. A. Sloane, Jul 07 2000

nonn

It appears that lim_{n->infinity} a(n)/2^n exists and is approximately 1.569985585.... - Franklin T. Adams-Watters, Nov 11 2011
This is a B_2 sequence. - Thomas Ordowski, Sep 23 2014 See the link.
Conjecture: lim_{n->infinity} a(n)/A006992(n) = 5.1648264... - Thomas Ordowski, Apr 05 2015

Zak Seidov and Michael De Vlieger, Table of n, a(n) for n = 1..1000 (First 100 terms from Zak Seidov)
V. Shevelev, Ramanujan and Labos primes, their generalizations, and classifications of primes, J. Integer Seq. 15 (2012), #12.5.4.
Eric Weisstein's World of Mathematics, B_2-Sequence.

Cf. A006992, A060264, A163963.
Values of a(n)-2*a(n-1) in A163469. - Zak Seidov, Jul 28 2009
Cf. A065545 (with a(1)=3). - Zak Seidov, Feb 04 2016
Row 1 of A229608.

A055496 := proc(n) option remember; if n=1 then 2 else nextprime(2*A055496(n-1)); fi; end;

NextPrim[n_Integer] := Block[ {k = n + 1}, While[ !PrimeQ[k], k++ ]; Return[k]]; a[1] = 2; a[n_] := NextPrim[ 2*a[n - 1]]; Table[ a[n], {n, 1, 31} ]
a[1]=2;a[n_]:=a[n]=Prime[PrimePi[2*a[n-1]]+1];Table[a[n],{n,40}] (* Zak Seidov, Feb 16 2006 *)
NestList[ NextPrime[2*# ]&,2,100] (* Zak Seidov, Jul 28 2009 *)

print1(a=2);for(n=2,20,print1(", ",a=nextprime(a+a))) \\ Charles R Greathouse IV, Jul 19 2011

a(n+1) = A060264(a(n)). - Peter Munn, Oct 23 2017

Mathematica updated by Jean-François Alcover, Jun 19 2013

A020482 Greatest p with p, q both prime, p+q = 2n.

Original entry on oeis.org

2, 3, 5, 7, 7, 11, 13, 13, 17, 19, 19, 23, 23, 23, 29, 31, 31, 31, 37, 37, 41, 43, 43, 47, 47, 47, 53, 53, 53, 59, 61, 61, 61, 67, 67, 71, 73, 73, 73, 79, 79, 83, 83, 83, 89, 89, 89, 79, 97, 97, 101, 103, 103, 107, 109, 109, 113, 113, 113, 109, 113, 113, 109, 127, 127, 131, 131

Offset: 2

David W. Wilson

nonn

a(n) = A171637(n,A035026(n)). - Reinhard Zumkeller, Mar 03 2014

H. J. Smith, Table of n, a(n) for n = 2..20000

Cf. A060264, A020481.

a020482 = last . a171637_row  -- Reinhard Zumkeller, Mar 03 2014

a[n_] := {p, q} /. {ToRules @ Reduce[p+q == 2*n, {p, q}, Primes]} // Max; Table[a[n], {n, 2, 100}] (* Jean-François Alcover, Dec 19 2013 *)
Table[Max[Flatten[Select[IntegerPartitions[2n,{2}],AllTrue[#,PrimeQ]&]]],{n,2,70}] (* Harvey P. Dale, Sep 04 2024 *)

a(n)=forprime(q=2,n,if(isprime(2*n-q), return(2*n-q))) \\ Charles R Greathouse IV, Apr 28 2015

from sympy import primerange, isprime
def A020482(n): return next(m for p in primerange(2*n) if isprime(m:=(n<<1)-p)) # Chai Wah Wu, Nov 19 2024

A060308 Largest prime <= 2n.

Original entry on oeis.org

2, 3, 5, 7, 7, 11, 13, 13, 17, 19, 19, 23, 23, 23, 29, 31, 31, 31, 37, 37, 41, 43, 43, 47, 47, 47, 53, 53, 53, 59, 61, 61, 61, 67, 67, 71, 73, 73, 73, 79, 79, 83, 83, 83, 89, 89, 89, 89, 97, 97, 101, 103, 103, 107, 109, 109, 113, 113, 113, 113, 113, 113, 113, 127, 127, 131

Offset: 1

Labos Elemer, Mar 27 2001

a(n) is the smallest k such that C(2n,n) divides k!. - Benoit Cloitre, May 30 2002
a(n) is largest prime factor of C(2n,n) = (2n)!/(n!)^2. - Alexander Adamchuk, Jul 11 2006
a(n) is also the largest prime in the interval [n,2n]. - Peter Luschny, Mar 04 2011
Odd prime p repeats (q-p)/2 times, where q > p is the next prime. In particular, every lesser of twin primes (A001359) occurs 1 time,  every lesser more than 3 of cousin primes (A023200) occurs 2 times, etc. - Vladimir Shevelev, Mar 12 2012

			n=1, 2n=2, p(1) = 2 = a(1) is the largest prime not exceeding 2.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Apart from initial term, same as A060265.
Cf. A007917 (largest prime <= n), A005843 (2n).
Cf. A020482, A049653, A060266, A060267, A060268, A060264.

a060308 = a007917 . a005843 -- Reinhard Zumkeller, May 25 2013

[NthPrime(#PrimesUpTo(2*n)): n in [2..100]]; // Vincenzo Librandi, Nov 25 2015

seq (prevprime(2*i+1), i=1..256);
seq(max(op(select(isprime,[$n..2*n]))),n=1..66); # Peter Luschny, Mar 04 2011

Table[Max[FactorInteger[(2n)!/(n!)^2]],{n,1,100}] (* Alexander Adamchuk, Jul 11 2006 *)
NextPrime[2*Range[80]+1,-1] (* Harvey P. Dale, Apr 23 2017 *)

a(n)=precprime(2*n) \\ Charles R Greathouse IV, May 24 2013

a(n) = Max[FactorInteger[(2n)!/(n!)^2]]. - Alexander Adamchuk, Jul 11 2006
a(n) = A006530(A000142(2*n)) and a(n) = A006530(A056040(2*n)). - Peter Luschny, Mar 04 2011
a(n) ~ 2*n as n tends to infinity. - Vladimir Shevelev, Mar 12 2012
a(n) = A007917(A005843(n)) = A226078(n, A067434(n)). - Reinhard Zumkeller, May 25 2013

More terms from Alexander Adamchuk, Jul 11 2006

A060266 Difference between 2n and the following prime.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 5, 3, 1, 1, 5, 3, 1, 3, 1, 1, 3, 1, 5, 3, 1, 5, 3, 1, 1, 5, 3, 1, 3, 1, 1, 5, 3, 1, 3, 1, 5, 3, 1, 7, 5, 3, 1, 3, 1, 1, 3, 1, 1, 3, 1, 13, 11, 9, 7, 5, 3, 1, 3, 1, 5, 3, 1, 1, 9, 7, 5, 3, 1, 1, 5, 3, 1, 5, 3, 1, 3, 1, 5, 3, 1, 5, 3, 1, 1, 9, 7, 5, 3, 1, 1, 3, 1, 1, 11, 9, 7, 5

Offset: 1

Labos Elemer, Mar 23 2001

nonn

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

Cf. A020482, A049653, A060267, A060268, A060264.

with(numtheory): [seq(nextprime(2*i)-2*i,i=1..256)];

d2n[n_]:=Module[{c=2n},NextPrime[c]-c]; Array[d2n,120] (* Harvey P. Dale, May 14 2011 *)
Table[NextPrime@ # - # &[2 n], {n, 120}] (* Michael De Vlieger, Feb 18 2017 *)

a(n) = nextprime(2*n+1) - 2*n; \\ Michel Marcus, Feb 19 2017

Conjecture: Limit_{n->oo} (Sum_{k=1..n} a(k)) / (Sum_{k=1..n} log(2*k)) = 1. - Alain Rocchelli, Oct 24 2023

A118750 a(n) = product[k=1..n] P(k), where P(k) is the largest prime <= 3*k. a(n) = product[k=1..n] A118749(k).

Original entry on oeis.org

3, 15, 105, 1155, 15015, 255255, 4849845, 111546435, 2565568005, 74401472145, 2306445636495, 71499814731345, 2645493145059765, 108465218947450365, 4664004414740365695, 219208207492797187665, 10302785752161467820255

Offset: 1

Jonathan Vos Post, Apr 29 2006

Differs from (after first term) A048599 "Partial products of the sequence (A001097) of twin primes" after 8th term. Differs from (after first term) A070826 "One half of product of first n primes A000040" after 9th term. Analogous to A118455 a(1)=1. a(n) = product{k=1..n} P(k), where P(k) is the largest prime <= k.

Cf. A020482, A049653, A060266-A060268, A060264, A060265, A060308, A118455, A118456, A118748.

A097050 Smallest prime > n(n+1)/2.

Original entry on oeis.org

2, 2, 5, 7, 11, 17, 23, 29, 37, 47, 59, 67, 79, 97, 107, 127, 137, 157, 173, 191, 211, 233, 257, 277, 307, 331, 353, 379, 409, 439, 467, 499, 541, 563, 599, 631, 673, 709, 743, 787, 821, 863, 907, 947, 991, 1039, 1087, 1129, 1181, 1229, 1277, 1327, 1381, 1433, 1487, 1543

Offset: 0

N. J. A. Sloane, Sep 15 2004, Nov 21 2008

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A000217, A151800.

Cf. A060264. See A065383 for another version.

NextPrime@Table[n(n+1)/2,{n,0,60}]  (* Vladimir Joseph Stephan Orlovsky, Apr 11 2011 *)
NextPrime[Accumulate[Range[0,60]]] (* Harvey P. Dale, Aug 23 2023 *)

a(n) = nextprime(n*(n+1)/2+1); \\ Michel Marcus, Nov 13 2015

a(n) = A151800(A000217(n)). - Michel Marcus, Nov 13 2015

A060267 Difference between 2 closest primes surrounding 2n.

Original entry on oeis.org

2, 2, 4, 4, 2, 4, 4, 2, 4, 4, 6, 6, 6, 2, 6, 6, 6, 4, 4, 2, 4, 4, 6, 6, 6, 6, 6, 6, 2, 6, 6, 6, 4, 4, 2, 6, 6, 6, 4, 4, 6, 6, 6, 8, 8, 8, 8, 4, 4, 2, 4, 4, 2, 4, 4, 14, 14, 14, 14, 14, 14, 14, 4, 4, 6, 6, 6, 2, 10, 10, 10, 10, 10, 2, 6, 6, 6, 6, 6, 6, 4, 4, 6, 6, 6, 6, 6, 6, 2, 10, 10, 10, 10, 10, 2, 4

Offset: 2

Labos Elemer, Mar 23 2001

nonn

			a(3) = 2 because the closest primes to 2*3 = 6 are (5,7) and the difference between these is 2. - _Michael De Vlieger_, Nov 02 2017

Michael De Vlieger, Table of n, a(n) for n = 2..10000

Cf. A020482, A049653, A060264, A060265, A060266, A060268.

with(numtheory): [seq(nextprime(2*i)-prevprime(2*i),i=2..256)];

Array[Subtract @@ NextPrime[#, {1, -1}] &[2 #] &, 96, 2] (* Michael De Vlieger, Nov 02 2017 *)
NextPrime[#]-NextPrime[#,-1]&/@(2*Range[2,100]) (* Harvey P. Dale, Nov 07 2017 *)

a(n) = nextprime(2*n+1) - precprime(2*n-1); \\ Michel Marcus, Sep 16 2020

A230504 Smallest prime in r(k) = r(k-1) + gcd(k,r(k-1)) with r(1) = n.

Original entry on oeis.org

2, 2, 3, 19, 5, 19, 7, 11, 11, 17, 11, 17, 13, 17, 17, 23, 17, 23, 19, 23, 23, 29, 23, 29, 29, 29, 29, 37, 29, 37, 31, 37, 37, 53, 53, 53, 37, 41, 41, 47, 41, 47, 43, 47, 47, 53, 47, 53, 53, 53, 53, 59, 53, 59, 59, 59, 59, 67, 59, 67, 61, 67, 67, 79, 79, 79

Offset: 1

Reinhard Zumkeller, Nov 15 2013

nonn

a(p) = p, p prime;

a(2*n-1) = A060264(n-1).

			n = 1 -> 1 + GCD(1,2) = 1+1 = 2 = prime(1) = a(1);
n = 2 = prime(1) = a(2);
n = 3 = prime(2) = a(3);
n = 4 -> 4+GCD(4,2) = 4+2 = 6 -> 6+GCD(6,3) = 6+3 = 9 -> 9+GCD(9,4) = 9+1 = 10 -> 10+GCD(10,5) = 10+5 = 15 -> 15+GCD(15,6) = 15+3 = 18 -> 18+GCD(18,7) = 18+1 = 19 = prime(8) = a(4) = A084662(7);
n = 5 = prime(3) = a(5) = A134736(1);
n = 6 -> 6+GCD(6,2) = 6+2 = 8 -> 8+GCD(8,3) = 8+1 = 9 -> 9+GCD(9,4) = 9+1 = 10 -> 10+GCD(10,5) = 10+5 = 15 -> 15+GCD(15,6) = 15+3 = 18 -> 18+GCD(18,7) = 18+1 = 19 = prime(8) = a(6);
n = 7 = prime(4) = a(7) = A106108(1);
n = 8 -> 8+GCD(8,2) = 8+2 = 10 -> 10+GCD(10,3) = 10+1 = 11 = prime(5) = a(8) = A084663(3);
n = 9 -> 9+GCD(9,2) = 9+2 = 11 = prime(5) = a(9);
n = 10 -> 10+GCD(10,2) = 10+2 = 12 -> 12+GCD(12,3) = 12+3 = 15 -> 15+GCD(15,4) = 15+1 = 16 -> 16+GCD(16,5) = 16+1 = 17 = prime(7) = a(10);
n = 11 = prime(5) = a(11);
n = 12 -> 12+GCD(12,2) = 12+2 = 14 -> 14+GCD(14,3) = 14+1 = 15 -> 15+GCD(15,4) = 15+1 = 16 -> 16+GCD(16,5) = 16+1 = 17 = prime(7) = a(10).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric S. Rowland, A simple prime-generating recurrence, arXiv:0710.3217 [math.NT], 2007-2008.

Cf. A084662, A134736, A106108, A084663.

a230504 n = head $ filter ((== 1) . a010051') rs where
                   rs = n : zipWith (+) rs (zipWith gcd rs [2..])

a[n_] := Module[{r}, If[PrimeQ[n], n, r[1]=n; r[k_] := r[k] = r[k-1] + GCD[k, r[k-1]]; For[k=1, True, k++, If[PrimeQ[r[k]], Return[r[k]]]]]];
Array[a, 66] (* Jean-François Alcover, Dec 03 2018 *)

from math import gcd
from itertools import count, accumulate
from sympy import isprime
def A230504(n): return next(filter(isprime,accumulate(count(2),lambda x,y:x+gcd(x,y),initial=n))) # Chai Wah Wu, Mar 15 2023

cp's OEIS Frontend

A164960 The minimum number of steps needed to generate prime(n) under the map x -> A060264(x) starting from any x taken from {2,3} or from A164333.

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A118753 First prime after 4n. Smallest prime >= 4*n. Bisection of A060264.

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A055496 a(1) = 2; a(n) is smallest prime > 2*a(n-1).

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A020482 Greatest p with p, q both prime, p+q = 2n.

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A060308 Largest prime <= 2n.

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A060266 Difference between 2n and the following prime.

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A118750 a(n) = product[k=1..n] P(k), where P(k) is the largest prime <= 3*k. a(n) = product[k=1..n] A118749(k).

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