A007917 -id:A007917 - cp's OEIS frontend

A083722 Product of primes greater than the greatest prime factor of n but not greater than n.

1, 1, 1, 3, 1, 5, 1, 105, 35, 7, 1, 385, 1, 143, 1001, 15015, 1, 85085, 1, 323323, 46189, 4199, 1, 37182145, 7436429, 7429, 37182145, 1062347, 1, 215656441, 1, 100280245065, 86822723, 392863, 955049953, 33426748355, 1, 765049, 247110827, 247357937827, 1, 1448810778701, 1

Offset: 1

Reinhard Zumkeller, May 04 2003

nonn

a(n) = 1 iff n is prime or n is 1.

Apart from 1-terms, the other duplicated terms are: a(24) = a(27), a(120) = a(125), a(140) = a(147), a(528) = a(539), etc, whose positions are listed by A293893 and A293894. - Antti Karttunen, Nov 01 2017

Antti Karttunen, Table of n, a(n) for n = 1..2001

Cf. A006530, A083721, A083720.

Cf. A293892 (restricted growth sequence transform), A293893, A293894.

Array[Times @@ Select[Prime@ Range[#1, #1 + #2], Function[p, p <= #3]] & @@ {PrimePi@ NextPrime[FactorInteger[#][[-1, 1]]], PrimePi@ #, #} &, 43] (* Michael De Vlieger, Nov 01 2017 *)

a(n) = {if (n==1, return (1)); my(gpf = vecmax(factor(n)[,1])); my(pp = 1); forprime(p=gpf+1, n, pp *= p;); pp;} \\ Michel Marcus, Jun 26 2016

a(n) = A002110(A049084(A007917(n)))/A002110(A049084(A006530(n))).

More terms from Michel Marcus, Jun 26 2016

A126990 Largest prime preceding geometric mean of prime(n) and prime(n+2).

Original entry on oeis.org

3, 3, 7, 7, 13, 13, 19, 23, 23, 31, 31, 37, 43, 47, 47, 53, 61, 61, 67, 73, 73, 83, 89, 89, 97, 103, 103, 109, 113, 113, 131, 131, 139, 139, 151, 151, 157, 167, 167, 173, 181, 181, 193, 193, 199, 199, 211, 223, 229, 233, 233, 241, 241, 251, 257, 263, 271, 271, 277

Offset: 1

Artur Jasinski, Jan 01 2007

nonn

With duplicates removed, seems to be a subsequence of A105399 and A105792. - M. F. Hasler, Jun 14 2007

P. Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A007917, A073273.

<< NumberTheory`NumberTheoryFunctions` a = {}; Do[AppendTo[a,PreviousPrime[Sqrt[(Prime[x])*(Prime[x + 2])]]], {x, 1, 100}]; a

A126990(n)={ n=sqrtint(prime(n)*prime(n+2)); if( 0==n%2, n--); while(!isprime(n), n-=2); n } /* then vector(50,n,A126990(n)) displays a list of values, M. F. Hasler, Jun 14 2007 */

a(n)= precprime(sqrtint(prime(n)*prime(n+2))); \\ Michel Marcus, Nov 07 2013

a(n) = A007917(A073273(n)). - Michel Marcus, Nov 07 2013

Edited by M. F. Hasler, Jun 14 2007

Definition changed so that offset is now 1 by Michel Marcus, Nov 07 2013

A245396 Largest prime not exceeding prime(n)^(1 + 1/n).

Original entry on oeis.org

3, 5, 7, 11, 17, 19, 23, 23, 31, 37, 41, 47, 53, 53, 59, 67, 73, 73, 83, 83, 89, 89, 97, 107, 113, 113, 113, 113, 127, 131, 139, 151, 157, 157, 167, 173, 179, 181, 181, 193, 199, 199, 211, 211, 211, 223, 233, 241, 251, 251, 257, 263, 263, 277, 283, 283, 293, 293, 293, 307, 307, 317, 331, 337

Offset: 1

M. F. Hasler, Nov 03 2014

nonn

Firoozbakht's conjecture, prime(n+1) < prime(n)^(1 + 1/n), is equivalent to a(n) > prime(n). See also A182134.
Here prime(n) = A000040(n). The conjecture is also equivalent to a(n) - prime(n) >= A001223(n), the n-th gap between primes. See also A246778(n) = floor(prime(n)^(1 + 1/n)) - prime(n).
It is also conjectured that the equality a(n) - prime(n) = A001223(n) holds only for n in the set {1, 2, 3, 4, 8}, see A246782. a(n) is also largest prime less than prime(n)^(1 + 1/n), since prime(n)^(1 + 1/n) is never prime. - Farideh Firoozbakht, Nov 03 2014
a(n) = A007917(A249669(n)) = A244365(n,A182134(n)) = A006530(A245722(n)). - Reinhard Zumkeller, Nov 18 2014

A. Kourbatov, Verification of the Firoozbakht conjecture for primes up to four quintillion, arXiv:1503.01744 [math.NT], 2015
A. Kourbatov, Upper bounds for prime gaps related to Firoozbakht's conjecture, J. Int. Seq. 18 (2015) 15.11.2
Nilotpal Kanti Sinha, On a new property of primes that leads to a generalization of Cramer's conjecture, arXiv:1010.1399 [math.NT]
Wikipedia, Firoozbakht's conjecture

Cf. A000040, A001223, A007917, A182134, A246778, A249669.

Cf. A244365.

a245396 n = a244365 n (a182134 n)  -- Reinhard Zumkeller, Nov 16 2014

seq(prevprime(ceil(ithprime(n)^(1+1/n))),n=1..100); # Robert Israel, Nov 03 2014

Table[NextPrime[Prime[n]^(1 + 1/n), -1], {n, 64}] (* Farideh Firoozbakht, Nov 03 2014 *)

PARI
```
a(n)=precprime(prime(n)^(1+1/n))
```

a(n)=precprime(sqrtnint(prime(n)^(n+1),n)) \\ Charles R Greathouse IV, Oct 29 2018

A245396 = A007917 o A249669, i.e., a(n) = A007917(A249669(n)). Although one could say "less than" in the definition of this sequence, one cannot use A151799 in this formula because for n = 2 and n = 4, one has a(n) = A249669(n).

A378457 Difference between n and the greatest prime power <= n, allowing 1.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 2, 0, 0, 1, 0, 1, 2, 3, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 2, 3, 4, 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, 0, 1, 2, 0, 1, 2, 3, 0, 1, 0, 1, 2, 3, 4, 5, 0, 1, 0, 1, 0, 1, 2, 3, 4

Offset: 1

Gus Wiseman, Nov 29 2024

nonn

Prime powers allowing 1 are listed by A000961.

			The greatest prime power <= 6 is 5, so a(6) = 1.

Sequences obtained by subtracting each term from n are placed in parentheses below.
For nonprime we have A010051 (almost) (A179278).
Subtracting from n gives (A031218).
For prime we have A064722 (A007917).
For perfect power we have A069584 (A081676).
For squarefree we have (A070321).
Adding one gives A276781.
For nonsquarefree we have (A378033).
For non perfect power we have (A378363).
For non prime power we have A378366 (A378367).
The opposite is A378370 = A377282-1.
A000015 gives the least prime power >= n.
A000040 lists the primes, differences A001223.
A000961 and A246655 list the prime powers, differences A057820.
A024619 and A361102 list the non prime powers, differences A375708 and A375735.
A151800 gives the least prime > n, weak version A007918.
Prime powers between primes: A053607, A080101, A304521, A366833, A377057.
Cf. A001597, A007920, A013632, A065514, A074984, A377051, A377054, A377281, A377289, A377468, A378357, A378371.

Table[n-NestWhile[#-1&,n,#>1&&!PrimePowerQ[#]&],{n,100}]

a(n) = n - A031218(n).

a(n) = A276781(n) - 1.

A083716 a(n) = integer part of (greatest prime <= n)/(greatest prime factor of n); a(1) = 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 3, 1, 1, 2, 6, 1, 5, 1, 3, 2, 1, 1, 7, 4, 1, 7, 3, 1, 5, 1, 15, 2, 1, 4, 10, 1, 1, 2, 7, 1, 5, 1, 3, 8, 1, 1, 15, 6, 9, 2, 3, 1, 17, 4, 7, 2, 1, 1, 11, 1, 1, 8, 30, 4, 5, 1, 3, 2, 9, 1, 23, 1, 1, 14, 3, 6, 5, 1, 15, 26, 1, 1, 11, 4, 1, 2, 7, 1, 17, 6, 3, 2, 1, 4, 29, 1, 13, 8

Offset: 1

Reinhard Zumkeller, May 04 2003

a(n) = floor(A007917(n)/A006530(n));

a(p) = 1 for primes p.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A006530, A007917.

Cf. A083715, A083717, A083714, A083718.

1,seq(floor(prevprime(n+1)/max(numtheory:-factorset(n))), n=2..100); # Robert Israel, Jun 13 2017

a[n_] := If[n == 1 || PrimeQ[n], 1, Floor[NextPrime[n, -1]/ FactorInteger[n][[-1, 1]]]];
Array[a, 100] (* Jean-François Alcover, Mar 04 2019 *)

A088631 Largest number m < n such that m+n is a prime.

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 5, 8, 9, 8, 11, 10, 9, 14, 15, 14, 13, 18, 17, 20, 21, 20, 23, 22, 21, 26, 25, 24, 29, 30, 29, 28, 33, 32, 35, 36, 35, 34, 39, 38, 41, 40, 39, 44, 43, 42, 41, 48, 47, 50, 51, 50, 53, 54, 53, 56, 55, 54, 53, 52, 51, 50, 63, 62, 65, 64, 63, 68, 69, 68, 67, 66, 65, 74, 75

Offset: 2

N. J. A. Sloane, Nov 24 2003

			Adding 1,2,3,2,5 to 2,3,4,5,6 we get the primes 3,5,7,7,11.

Reinhard Zumkeller, Table of n, a(n) for n = 2..10000

Cf. A088633. Second column of A088643.

Cf. A010051, A060265, A007917.

a088631 n = a060265 n - n  -- Reinhard Zumkeller, Feb 22 2015

with(numtheory); A088631 := n->prevprime(2*n)-n;

a(n) = p-n where p = largest prime <= 2n-1.

a(n) = A060265(n) - n. - Reinhard Zumkeller, Feb 22 2015

A113523 a(n) = largest composite nonnegative integer <= n.

Original entry on oeis.org

0, 0, 0, 4, 4, 6, 6, 8, 9, 10, 10, 12, 12, 14, 15, 16, 16, 18, 18, 20, 21, 22, 22, 24, 25, 26, 27, 28, 28, 30, 30, 32, 33, 34, 35, 36, 36, 38, 39, 40, 40, 42, 42, 44, 45, 46, 46, 48, 49, 50, 51, 52, 52, 54, 55, 56, 57, 58, 58, 60, 60, 62, 63, 64, 65, 66, 66, 68, 69, 70, 70, 72

Offset: 1

Leroy Quet, Jan 12 2006

nonn

For n > 3: a(n) = A179278(n). [From Reinhard Zumkeller, Jul 08 2010]

Cf. A007917, A014684.

nnci[n_]:=Module[{k=0},While[PrimeQ[n-k],k++];n-k]/.{1->0}; Array[nnci,80] (* Harvey P. Dale, Jul 19 2012 *)

a(1)=a(2)=a(3) = 0. For n >= 4, a(n) = A014684(n).

A284374 a(1) = a(2) = 1; a(n) is the largest prime <= (a(n-a(n-1)) + a(n-a(n-2))) for n > 2.

Original entry on oeis.org

1, 1, 2, 3, 3, 3, 5, 5, 5, 5, 5, 7, 7, 7, 7, 7, 7, 7, 13, 11, 7, 11, 13, 11, 13, 13, 13, 13, 13, 13, 13, 23, 13, 11, 19, 19, 13, 23, 19, 13, 23, 23, 19, 19, 23, 23, 19, 23, 23, 23, 23, 23, 23, 23, 43, 19, 29, 31, 23, 23, 43, 31, 31, 23, 31, 37, 31, 23, 43, 31, 23, 43, 31

Offset: 1

Altug Alkan, Mar 25 2017

nonn

			a(4) = 3 because a(4 - a(3)) + a(4 - a(2)) = a(2) + a(3) = 1 + 2 = 3 and A007917(3) = 3.

Altug Alkan, Table of n, a(n) for n = 1..10000
Altug Alkan, Alternative scatterplot of A284374
Altug Alkan, Alternative graph of A284374

Cf. A005185, A007917, A284383, A284397.

a[1] = a[2] = 1; a[n_] := a[n] = Prime@ PrimePi[a[n - a[n - 1]] + a[n - a[n - 2]]]; Array[a, 73] (* Michael De Vlieger, Mar 25 2017 *)

a=vector(1000); a[1]=a[2]=1; for(n=3, #a, a[n] = precprime(a[n-a[n-1]]+a[n-a[n-2]])); a

a(1) = a(2) = 1; a(n) = A007917(a(n-a(n-1)) + a(n-a(n-2))) for n > 2.

A064924 If n is prime then a(n) = n; for the subsequent nonprime positions a(n + k) = (k+1)*n; then at the next prime position a new subsequence begins.

Original entry on oeis.org

2, 3, 6, 5, 10, 7, 14, 21, 28, 11, 22, 13, 26, 39, 52, 17, 34, 19, 38, 57, 76, 23, 46, 69, 92, 115, 138, 29, 58, 31, 62, 93, 124, 155, 186, 37, 74, 111, 148, 41, 82, 43, 86, 129, 172, 47, 94, 141, 188, 235, 282, 53, 106, 159, 212, 265, 318, 59, 118, 61, 122, 183, 244

Offset: 2

Reinhard Zumkeller, Oct 14 2001

A064920(a(n)) = n.

			a(7) = A007917(7) * (A064722(7) + 1) = 7 * (0 + 1) = 7; a(8) = A007917(8) * (A064722(8) + 1) = 7 * (1 + 1) = 14; a(9) = A007917(9) * (A064722(9) + 1) = 7 * (2 + 1) = 21; a(10) = A007917(10) * (A064722(10) + 1) = 7 * (3 + 1) = 28; a(11) = 11.

R. Zumkeller, Table of n, a(n) for n = 2..10000

Cf. A064920, A064923, A007917, A064722, A064919.

import Data.List (genericTake)
a064924 n = a064924_list !! (n-1)
a064924_list = concat $ zipWith (\p g -> genericTake g [p, 2 * p ..])
   a000040_list $ zipWith (-) (tail a000040_list) a000040_list
-- Reinhard Zumkeller, Jul 05 2013

a[n_?PrimeQ] := n; a[n_] := NextPrime[n, -1]*(n - NextPrime[n, -1] + 1); Table[a[n], {n, 2, 64}] (* Jean-François Alcover, Sep 19 2011 *)
Flatten[First[#]Range[Last[#]-First[#]]&/@Partition[Prime[Range[20]],2,1]] (* Harvey P. Dale, May 03 2012 *)

{ for (n=2, 10000, if (isprime(n), a=m=n; k=2, a=k*m; k++); write("b064924.txt", n, " ", a) ) } \\ Harry J. Smith, Sep 29 2009

a(n) = A007917(n) * (A064722(n) + 1)

A072680 Difference between (least prime >= n) and (largest prime <= n).

Original entry on oeis.org

0, 0, 2, 0, 2, 0, 4, 4, 4, 0, 2, 0, 4, 4, 4, 0, 2, 0, 4, 4, 4, 0, 6, 6, 6, 6, 6, 0, 2, 0, 6, 6, 6, 6, 6, 0, 4, 4, 4, 0, 2, 0, 4, 4, 4, 0, 6, 6, 6, 6, 6, 0, 6, 6, 6, 6, 6, 0, 2, 0, 6, 6, 6, 6, 6, 0, 4, 4, 4, 0, 2, 0, 6, 6, 6, 6, 6, 0, 4, 4, 4, 0, 6, 6, 6, 6, 6, 0, 8, 8, 8, 8, 8, 8, 8, 0, 4, 4, 4, 0, 2, 0, 4, 4, 4

Offset: 2

Reinhard Zumkeller, Jul 01 2002

nonn

a(n) = 0 iff n is prime.

Antti Karttunen, Table of n, a(n) for n = 2..65537

Cf. A007917, A007918, A013633, A072681, A097106.

f[n_]:=If[PrimeQ[n],0,NextPrime[n]-NextPrime[n,-1]];Array[f,110,2] (* Harvey P. Dale, Sep 22 2011 *)

numlib::prevprime(i)*(-1)-nextprime(i)*(-1)$ i = 2..106 // Zerinvary Lajos, Feb 26 2007

A072680(n) = (nextprime(n) - precprime(n)); \\ Antti Karttunen, Sep 23 2018

a(n) = A007918(n) - A007917(n).

a(n) = A057427(n - A007917(n)) * A001223(A049084(A007917(n))).

cp's OEIS Frontend

A083722 Product of primes greater than the greatest prime factor of n but not greater than n.

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A126990 Largest prime preceding geometric mean of prime(n) and prime(n+2).

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A245396 Largest prime not exceeding prime(n)^(1 + 1/n).

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A378457 Difference between n and the greatest prime power <= n, allowing 1.

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A083716 a(n) = integer part of (greatest prime <= n)/(greatest prime factor of n); a(1) = 1.

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A088631 Largest number m < n such that m+n is a prime.

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A113523 a(n) = largest composite nonnegative integer <= n.

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