A060308 -id:A060308 - cp's OEIS frontend

A060265 Largest prime less than 2n.

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: 2

Labos Elemer, Mar 23 2001

a(n) = A007917(2*n) = A255313(n-1,1) = A255316(n-1,1) = A006530(A255427(n)). - Reinhard Zumkeller, Feb 22 2015

Harry J. Smith, Table of n, a(n) for n = 2..1000

Apart from initial term, same as A060308.

Cf. A020482, A049653, A088631, A007917, A005843, A255313.

a060265 = a007917 . (* 2)  -- Reinhard Zumkeller, Feb 22 2015

Maple
```
seq (prevprime(2*i+1), i=2..256);
```

Table[NextPrime[2 n, -1], {n, 2, 66}] (* Michael De Vlieger, Jul 04 2016 *)

a(n) = precprime(2*n-1) \\ Harry J. Smith, Jul 03 2009

A226078 Table read by rows: prime power factors of central binomial coefficients, cf. A000984.

Original entry on oeis.org

1, 2, 2, 3, 4, 5, 2, 5, 7, 4, 9, 7, 4, 3, 7, 11, 8, 3, 11, 13, 2, 9, 5, 11, 13, 4, 5, 11, 13, 17, 4, 11, 13, 17, 19, 8, 3, 7, 13, 17, 19, 4, 7, 13, 17, 19, 23, 8, 25, 7, 17, 19, 23, 8, 27, 25, 17, 19, 23, 16, 9, 5, 17, 19, 23, 29, 2, 9, 5, 17, 19, 23, 29, 31

Offset: 0

Reinhard Zumkeller, May 25 2013

			.   n        initial rows               A000984(n)   A226047(n)
.  ---+------------------------------+-------------+------------
.   0   [1]                                      1
.   1   [2]                                      2            2
.   2   [2,3]                                    6            3
.   3   [4,5]                                   20            5
.   4   [2,5,7]                                 70            7
.   5   [4,9,7]                                252            9
.   6   [4,3,7,11]                             924           11
.   7   [8,3,11,13]                           3432           13
.   8   [2,9,5,11,13]                        12870           13
.   9   [4,5,11,13,17]                       48620           17
.  10   [4,11,13,17,19]                     184756           19
.  11   [8,3,7,13,17,19]                    705432           19
.  12   [4,7,13,17,19,23]                  2704156           23
.  13   [8,25,7,17,19,23]                 10400600           25
.  14   [8,27,25,17,19,23]                40116600           27
.  15   [16,9,5,17,19,23,29]             155117520           29
.  16   [2,9,5,17,19,23,29,31]           601080390           31
.  17   [4,27,5,11,19,23,29,31]         2333606220           31
.  18   [4,3,25,7,11,19,23,29,31]       9075135300           31
.  19   [8,3,25,7,11,23,29,31,37]      35345263800           37
.  20   [4,9,5,7,11,13,23,29,31,37]   137846528820           37 .

Reinhard Zumkeller, Rows n = 0..250 of triangle, flattened

Cf. A067434 (row lengths), A001316 (left edge), A060308 (right edge), A226047 (row maxima), A226083 (row minima), A000984 (row products).

Cf. A267823.

a226078 n k = a226078_tabf !! n !! k
a226078_row n = a226078_tabf !! n
a226078_tabf = map a141809_row a000984_list

f:= n-> add(i[2]*x^i[1], i=ifactors(n)[2]):
b:= proc(n) local p;
      p:= add(f(n+i) -f(i), i=1..n);
      seq(`if`(coeff(p, x, i)>0,
             i^coeff(p, x, i), NULL), i=1..degree(p))
    end:
T:= n-> `if`(n=0, 1, b(n)):
seq(T(n), n=0..30);  # Alois P. Heinz, May 25 2013

Table[Power @@@ FactorInteger[(2n)!/n!^2] , {n, 0, 30}] // Flatten (* Jean-François Alcover, Jul 29 2015 *)

row(n)= if(n<1, [1], [ e[1]^e[2] |e<-Col(factor(binomial(2*n, n)))]); \\ Ruud H.G. van Tol, Nov 18 2024

T(n,k) = A141809(A000984(n),k) for k = 0..A067434(n)-1.

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.

A118752 a(n) = product[k=0..n] P(k), where P(k) is the smallest prime > 3*n. a(n) = product[k=0..n] A118751(k).

Original entry on oeis.org

2, 10, 70, 770, 10010, 170170, 3233230, 74364290, 2156564410, 62540367890, 1938751404590, 71733801969830, 2654150672883710, 108820177588232110, 4679267636293980730, 219925578905817094310, 11656055682008305998430

Offset: 0

Jonathan Vos Post, Apr 29 2006

Analogous to A118456 a(n) = product{k=1..n} P(k), where P(k) is the smallest prime >= k.

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A020482, A049653, A060266-A060268, A060264, A060265, A060308, A118455, A118456, A118748-51.

Rest[FoldList[Times,1,Table[NextPrime[3n],{n,0,20}]]] (* Harvey P. Dale, Mar 09 2014 *)

Definition corrected by Harvey P. Dale, Mar 09 2014

A120303 Largest prime factor of Catalan number A000108(n).

Original entry on oeis.org

2, 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, 131

Offset: 2

Alexander Adamchuk, Jul 13 2006

nonn

All prime numbers (except 3) are present in this sequence in their natural order with repetition. The number of repetitions is equal to A028334(n): differences between consecutive primes, divided by 2. - Alexander Adamchuk, Jul 30 2006
For p>3 a((p+1)/2) = p and all a(n) = p for n >= (p+1)/2 until the first occurrence of the next prime q = NextPrime(p) at a((q+1)/2) = q. - Alexander Adamchuk, Dec 27 2013
For n>2, a(n) is the largest prime less than 2*n. - Gennady Eremin, Mar 02 2021

			G.f. = 2*x^2 + 5*x^3 + 7*x^4 + 7*x^5 + 11*x^6 + 13*x^7 + 13*x^8 + 17*x^9 + ...

Gennady Eremin, Table of n, a(n) for n = 2..800
Gennady Eremin, Factoring Catalan numbers, arXiv:1908.03752 [math.NT], 2019.

Cf. A000108, A006530, A020482, A028334, A060265, A060308, A152765.

Table[Max[FactorInteger[(2n)!/n!/(n+1)! ]],{n,2,100}]
FactorInteger[CatalanNumber[#]][[-1,1]]&/@Range[2,70] (* Harvey P. Dale, May 02 2017 *)

a(n) = vecmax(factor(binomial(2*n, n)/(n+1))[,1]); \\ Michel Marcus, Nov 14 2015

a(n)=if(n>2,precprime(2*n),2) \\ Charles R Greathouse IV, Nov 17 2015

from gmpy2 import is_prime
A120303 = [2]
for n in range(3, 801):
    for k in range(2*n-1, n, -2):
        if is_prime(k, n):
            A120303.append(k)
            break
for n in range(len(A120303)):
    print(n+2, A120303[n])  # Gennady Eremin, Mar 17 2021

a(n) = A060308(n) = A060265(n) for n>2.
a(n) = A006530(A000108(n)). - Michel Marcus, Nov 14 2015
G.f.: A(x) - x^2, where A(x) is the g.f. of A060265. - Gennady Eremin, Mar 02 2021

A224911 Greatest prime dividing A190339(n).

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, 131

Offset: 0

Paul Curtz, Apr 19 2013

nonn

It appears that a(n) = A060308(n+1), verified for n <=420. - R. J. Mathar, Apr 28 2013
This appears to be a sequence of nondecreasing primes containing each prime at least once.
We might also consider a sequence b(n) defined by 2 followed by A006094(n): 2, 6, 15, 35, 77, 143, 221, ... . A190339(n) is also divisible by a stuttered version of b(n), namely by the sequence 2, 6, 15, 35, 35, 77, 143, 143, ... .

			a(0) = 6/2 = 3, a(1) = 15/3 = 5, a(2) = 105/15 = 7, a(3) = 105/15 = 7, a(4) = 231/21 = 11.

Cf. A006530, A060308, A065091, A190339, A120303, A060265.

A224911 := proc(n)
    A006530(A190339(n)) ;
end proc: # R. J. Mathar, Apr 25 2013

nmax = 67; b[n_] := BernoulliB[n]; b[1] = 1/2; bb = Table[b[n], {n, 0, 2*nmax-1}]; diff = Table[Differences[bb, n], {n, 1, nmax}]; FactorInteger[#][[-1, 1]]& /@ Denominator[Diagonal[diff]] (* Jean-François Alcover, Mar 03 2014 *)

a(n) = A006530(A190339(n)).

A226047 Largest prime power dividing binomial(2n, n).

Original entry on oeis.org

2, 3, 5, 7, 9, 11, 13, 13, 17, 19, 19, 23, 25, 27, 29, 31, 31, 31, 37, 37, 41, 43, 43, 47, 49, 49, 53, 53, 53, 59, 61, 61, 61, 67, 67, 71, 73, 73, 73, 79, 81, 83, 83, 83, 89, 89, 89, 89, 97, 97, 101, 103, 103, 107, 109, 109, 113, 113, 113, 113

Offset: 1

Charles R Greathouse IV, May 24 2013

The first indices n for which a(n) < a(n-1) are 123, 315, 366, 671, 1095, 1098, 1204, 1565, 6095, 7326, 9843, 39065, 58828, 88575, 88578, 195315, 195320, 265722, 265725 and 709937. - Giovanni Resta, May 24 2013

a(n) = maximum of n-th row in A226078. - Reinhard Zumkeller, May 25 2013

			Binomial(10, 5) = 2^2 * 3^2 * 7 and so a(5) = max({2^2, 3^2, 7}) = 3^2.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
P. Erdős, Beweis eines Satzes von Tschebyschef (in German), Acta Litt. Sci. Szeged 5 (1932), pp. 194-198.

Cf. A000961, A000984, A060308, A067434, A226083.

a226047 = maximum . a226078_row  -- Reinhard Zumkeller, May 25 2013

f:= n-> add(i[2]*x^i[1], i=ifactors(n)[2]):
a:= proc(n) local p;
      p:= add(f(n+i) -f(i), i=1..n);
      max(seq(i^coeff(p, x, i), i=1..degree(p)))
    end:
seq(a(n), n=1..60);  # Alois P. Heinz, May 24 2013

cnt[n_, p_] := (n - Total@IntegerDigits[n, p])/(p-1); a[n_] := Block[{k = 2*n, p, e}, While[! PrimePowerQ[k] || ({p, e} = FactorInteger[k][[1]]; cnt[2*n , p] - 2 cnt[n, p] != e), k--]; k]; Array[a, 60] (* Giovanni Resta, May 24 2013 *)
Table[Max[Select[Divisors[Binomial[2 n,n]],PrimePowerQ]],{n,60}] (* Harvey P. Dale, Feb 26 2024 *)

ord(n,p)=my(s);while(n\=p,s+=n);s
a(n)=my(p=precprime(2*n));forstep(k=2*n,p+1,-1, my(q,e=isprimepower(k, &q)); if(e && e == ord(2*n,q)-2*ord(n,q), return(k)));p /* requires PARI v.2.5 or later */

A226047(n)={for(k=2,#n=factor(binomial(2*n,n))~,factorback(n[,k-1]~)>factorback(n[,k]~) && n[,k]=n[,k-1]);factorback(n[,#n]~)} \\ highly unoptimized, not suitable for n>>10^4. - M. F. Hasler, May 24 2013

Erdős proved that a(n) <= 2n.

A118751 Smallest prime >= 3*n.

Original entry on oeis.org

2, 5, 7, 11, 13, 17, 19, 23, 29, 29, 31, 37, 37, 41, 43, 47, 53, 53, 59, 59, 61, 67, 67, 71, 73, 79, 79, 83, 89, 89, 97, 97, 97, 101, 103, 107, 109, 113, 127, 127, 127, 127, 127, 131, 137, 137, 139, 149, 149, 149, 151, 157, 157, 163, 163, 167, 173, 173, 179, 179, 181

Offset: 0

Jonathan Vos Post, Apr 29 2006

Analogous to A060264 = first prime after 2n.

Cf. A060308, A060264, A118750, A118752.

A118749 Largest prime <= 3*n.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 23, 23, 29, 31, 31, 37, 41, 43, 47, 47, 53, 53, 59, 61, 61, 67, 71, 73, 73, 79, 83, 83, 89, 89, 89, 97, 101, 103, 107, 109, 113, 113, 113, 113, 113, 127, 131, 131, 137, 139, 139, 139, 149, 151, 151, 157, 157, 163, 167, 167, 173, 173, 179, 181

Offset: 1

Jonathan Vos Post, Apr 29 2006

Analogous to A060308 largest prime <= 2*k.

Cf. A007917 (largest prime <= n), A008585 (3n).
Cf. A060308, A118750, A118751, A118752.
Essentially the same as A081259.

[NthPrime(#PrimesUpTo(3*n)): n in [1..100]]; // Vincenzo Librandi, Nov 25 2015

Table[Max[FactorInteger[(3 n)!/(n!)^3]], {n, 1, 70}] (* Vincenzo Librandi, Nov 25 2015 *)
NextPrime[3*Range[70]+1,-1] (* Harvey P. Dale, Nov 12 2017 *)

vector(100, n, precprime(3*n)) \\ Altug Alkan, Nov 25 2015

a(n) = A007917(A008585(n)). - Michel Marcus, Nov 25 2015

A118754 Smallest prime >= 5*n.

Original entry on oeis.org

2, 5, 11, 17, 23, 29, 31, 37, 41, 47, 53, 59, 61, 67, 71, 79, 83, 89, 97, 97, 101, 107, 113, 127, 127, 127, 131, 137, 149, 149, 151, 157, 163, 167, 173, 179, 181, 191, 191, 197, 211, 211, 211, 223, 223, 227, 233, 239, 241, 251, 251, 257, 263, 269, 271, 277, 281

Offset: 0

Jonathan Vos Post, Apr 29 2006

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

Cf. A007918, A008587, A060308, A060264, A118747-A118755.

Table[If[PrimeQ[5n],5n,NextPrime[5n]],{n,0,60}] (* Harvey P. Dale, Nov 29 2024 *)

a(n) = nextprime(5*n); \\ Michel Marcus, Feb 13 2021

a(n) = A007918(A008587(n)). - Michel Marcus, Feb 13 2021

cp's OEIS Frontend

A060265 Largest prime less than 2n.

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A226078 Table read by rows: prime power factors of central binomial coefficients, cf. A000984.

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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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A118752 a(n) = product[k=0..n] P(k), where P(k) is the smallest prime > 3*n. a(n) = product[k=0..n] A118751(k).

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A120303 Largest prime factor of Catalan number A000108(n).

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A224911 Greatest prime dividing A190339(n).

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A226047 Largest prime power dividing binomial(2n, n).

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