A007917 -id:A007917 - cp's OEIS frontend

A249151 Largest m such that m! divides the product of elements on row n of Pascal's triangle: a(n) = A055881(A001142(n)).

1, 1, 2, 1, 4, 2, 6, 1, 2, 4, 10, 7, 12, 6, 4, 1, 16, 2, 18, 4, 6, 10, 22, 11, 4, 12, 2, 6, 28, 25, 30, 1, 10, 16, 6, 36, 36, 18, 12, 40, 40, 6, 42, 10, 23, 22, 46, 19, 6, 4, 16, 12, 52, 2, 10, 35, 18, 28, 58, 47, 60, 30, 63, 1, 12, 10, 66, 16, 22, 49, 70, 41, 72, 36, 4, 18, 10, 12, 78, 80, 2

Offset: 0

Antti Karttunen, Oct 25 2014

A000225 gives the positions of ones.

A006093 seems to give all such k, that a(k) = k.

			              Binomial coeff.   Their product  Largest k!
                 A007318          A001142(n)   which divides
Row 0                1                    1        1!
Row 1              1   1                  1        1!
Row 2            1   2   1                2        2!
Row 3          1   3   3   1              9        1!
Row 4        1   4   6   4   1           96        4! (96 = 4*24)
Row 5      1   5  10  10   5   1       2500        2! (2500 = 1250*2)
Row 6    1   6  15  20  15   6   1   162000        6! (162000 = 225*720)

Chai Wah Wu, Table of n, a(n) for n = 0..10000 (terms 0..4096 from Antti Karttunen)

One more than A249150.
Cf. A249423 (numbers k such that a(k) = k+1).
Cf. A249429 (numbers k such that a(k) > k).
Cf. A249433 (numbers k such that a(k) < k).
Cf. A249434 (numbers k such that a(k) >= k).
Cf. A249424 (numbers k such that a(k) = (k-1)/2).
Cf. A249428 (and the corresponding values, i.e. numbers n such that A249151(2n+1) = n).
Cf. A249425 (record positions).
Cf. A249427 (record values).
Cf. A001142, A006093, A000225, A007917, A055881, A187059, A249346, A249421, A249430, A249431, A249432.

A249151(n) = { my(uplim,padicvals,b); uplim = (n+3); padicvals = vector(uplim); for(k=0, n, b = binomial(n, k); for(i=1, uplim, padicvals[i] += valuation(b, prime(i)))); k = 1; while(k>0, for(i=1, uplim, if((padicvals[i] -= valuation(k, prime(i))) < 0, return(k-1))); k++); };
\\ Alternative implementation:
A001142(n) = prod(k=1, n, k^((k+k)-1-n));
A055881(n) = { my(i); i=2; while((0 == (n%i)), n = n/i; i++); return(i-1); }
A249151(n) = A055881(A001142(n));
for(n=0, 4096, write("b249151.txt", n, " ", A249151(n)));

from itertools import count
from collections import Counter
from math import comb
from sympy import factorint
def A249151(n):
    p = sum((Counter(factorint(comb(n,i))) for i in range(n+1)),start=Counter())
    for m in count(1):
        f = Counter(factorint(m))
        if not f<=p:
            return m-1
        p -= f # Chai Wah Wu, Aug 19 2025

(define (A249151 n) (A055881 (A001142 n)))

a(n) = A055881(A001142(n)).

A064722 a(1) = 0; for n >= 2, a(n) = n - (largest prime <= n).

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Oct 13 2001

			a(26) = 26 - 23 = 3, a(37) = 37 - 37 = 0.

Harry J. Smith, Table of n, a(n) for n=1..1000

Cf. A000040, A000720, A007917, A007920, A010051, A049711.

0, seq(n - prevprime(n+1), n=2..100); # Robert Israel, Aug 25 2014

Join[{0},Table[n-NextPrime[n+1,-1],{n,2,110}]] (* Harvey P. Dale, Aug 23 2011 *)

{ for (n = 1, 1000, if (n>1, a=n - precprime(n), a=0); write("b064722.txt", n, " ", a) ) } \\ Harry J. Smith, Sep 23 2009

a(n) = n - A007917(n).
a(n) = 0 iff n is 1 or a prime.
Computable also as a "commutator": pi(prime(m)) - prime(pi(m)) = A000720(A000040(m))-A000040(A000720(m)). Labels position of composites between 2 consecutive primes. - Labos Elemer, Oct 19 2001
a(n) = a(n-1)*0^A010051(n) + 1 - A010051(n), a(1) = 0. - Reinhard Zumkeller, Mar 23 2006
a(n) = n mod A007917(n). - Michel Marcus, Aug 22 2014
a(n) = A049711(n+1) - 1 for n >= 2. - Pontus von Brömssen, Jul 31 2022

A006990 Largest prime <= n!.

Original entry on oeis.org

2, 5, 23, 113, 719, 5039, 40289, 362867, 3628789, 39916787, 479001599, 6227020777, 87178291199, 1307674367953, 20922789887947, 355687428095941, 6402373705727959, 121645100408831899, 2432902008176639969, 51090942171709439969, 1124000727777607679927

Offset: 2

N. J. A. Sloane, Robert G. Wilson v

nonn

Conjecture: For n >= 2, n! - a(n) is 1 or a prime, see A033933. - Amarnath Murthy, Mar 19 2002

a(n) is the largest prime divisor of (n!)! of the sequence A000197. - Stanislav Sykora, Jul 14 2014

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Donovan Johnson, Table of n, a(n) for n = 2..400
R. G. Wilson v, Letter to N. J. A. Sloane, Oct. 1993

Cf. A000142, A007917.

PrevPrime[ n_Integer ] := Module[ {k = n - 1}, While[ ! PrimeQ[ k ], k-- ]; k ]; Table[ PrevPrime[ n! ], {n, 3, 25} ]
Join[{2},NextPrime[Range[3,30]!,-1]] (* Harvey P. Dale, Jan 24 2014 *)

More terms from Jud McCranie; also from Robert G. Wilson v, Jan 03 2001

A060265 Largest prime less than 2n.

Original entry on oeis.org

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

A007409 Denominators of Sum_{k=1..n} 1/k^3.

Original entry on oeis.org

1, 8, 216, 1728, 216000, 24000, 8232000, 65856000, 16003008000, 16003008000, 21300003648000, 21300003648000, 46796108014656000, 46796108014656000, 46796108014656000, 374368864117248000, 1839274229408039424000

Offset: 1

N. J. A. Sloane, Mira Bernstein

Largest prime factor in A007409(n) (n > 1) is A007917(n), occurring always to the power 3. - M. F. Hasler, Nov 10 2006

D. Y. Savio, E. A. Lamagna and S.-M. Liu, Summation of harmonic numbers, pp. 12-20 of E. Kaltofen and S. M. Watt, editors, Computers and Mathematics, Springer-Verlag, NY, 1989.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=1..200

Cf. A007408, A007917.

A007409:= n->denom(sum(1/k^3,k=1..n)); # M. F. Hasler, Nov 10 2006

Table[Denominator[Sum[1/k^3, {k, n}]], {n, 10}] (* Alonso del Arte, Dec 30 2012 *)

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

A055500 a(0)=1, a(1)=1, a(n) = largest prime <= a(n-1) + a(n-2).

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 17, 23, 37, 59, 89, 139, 227, 359, 577, 929, 1499, 2423, 3919, 6337, 10253, 16573, 26821, 43391, 70207, 113591, 183797, 297377, 481171, 778541, 1259701, 2038217, 3297913, 5336129, 8633983, 13970093, 22604069, 36574151, 59178199, 95752333

Offset: 0

N. J. A. Sloane, Jul 08 2000

Or might be called Ishikawa primes, as he proved that prime(n+2) < prime(n) + prime(n+1) for n > 1. This improves on Bertrand's Postulate (Chebyshev's theorem), which says prime(n+2) < prime(n+1) + prime(n+1). - Jonathan Sondow, Sep 21 2013

			a(8) = 23 because 23 is largest prime <= a(7) + a(6) = 17 + 11 = 28.

Michael De Vlieger, Table of n, a(n) for n = 0..1000 (first 100 terms from Zak Seidov)
Heihachiro Ishikawa, Über die Verteilung der Primzahlen, Sci. Rep. Tokyo Bunrika Daigaku, Sect. A 2 (1934), 27-40.

Cf. A007917, A055498, A055499, A055400, A055401, A055502, A065435.

a055500 n = a055500_list !! n
a055500_list = 1 : 1 : map a007917
               (zipWith (+) a055500_list $ tail a055500_list)
-- Reinhard Zumkeller, May 01 2013

PrevPrim[n_] := Block[ {k = n}, While[ !PrimeQ[k], k-- ]; Return[k]]; a[1] = a[2] = 1; a[n_] := a[n] = PrevPrim[ a[n - 1] + a[n - 2]]; Table[ a[n], {n, 1, 42} ]
(* Or, if version >= 6 : *)a[0] = a[1] = 1; a[n_] := a[n] = NextPrime[ a[n-1] + a[n-2] + 1, -1]; Table[a[n], {n, 0, 100}](* Jean-François Alcover, Jan 12 2012 *)
nxt[{a_,b_}]:={b,NextPrime[a+b+1,-1]}; Transpose[NestList[nxt,{1,1},40]] [[1]] (* Harvey P. Dale, Jul 15 2013 *)

from sympy import prevprime; L = [1, 1]
for _ in range(36): L.append(prevprime(L[-2] + L[-1] + 1))
print(*L, sep = ", ")  # Ya-Ping Lu, May 05 2023

a(n) is asymptotic to C*phi^n where phi = (1+sqrt(5))/2 and C = 0.41845009129953131631777132510164822489... - Benoit Cloitre, Apr 21 2003
a(n) = A007917(a(n-1) + a(n-2)) for n > 1. - Reinhard Zumkeller, May 01 2013
a(n) >= prime(n-1) for n > 1, by Ishikawa's theorem. - Jonathan Sondow, Sep 21 2013

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

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

cp's OEIS Frontend

A249151 Largest m such that m! divides the product of elements on row n of Pascal's triangle: a(n) = A055881(A001142(n)).

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A064722 a(1) = 0; for n >= 2, a(n) = n - (largest prime <= n).

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Maple

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A006990 Largest prime <= n!.

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A060265 Largest prime less than 2n.

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Haskell

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A007409 Denominators of Sum_{k=1..n} 1/k^3.

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Maple

Mathematica

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

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Mathematica

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A055500 a(0)=1, a(1)=1, a(n) = largest prime <= a(n-1) + a(n-2).

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