A052180 -id:A052180 - cp's OEIS frontend

A052238 Primes p from A031924 such that A052180(p) = 23.

941, 1217, 1907, 3607, 4391, 6047, 6367, 8117, 8713, 9127, 9221, 10093, 10601, 11981, 12577, 14741, 19571, 19753, 23203, 23893, 24677, 25367, 28723, 29921, 36131, 36313, 39857, 41143, 42937, 51907, 52183, 52691, 54667, 55633, 58211

Offset: 1

Labos Elemer, Feb 01 2000

nonn

Cf. A031924, A031925, A052180.

A052256 Last filtering prime (A052180) of primes p such that next prime is p+8 (A031926).

Original entry on oeis.org

7, 19, 17, 13, 11, 13, 17, 13, 19, 7, 7, 13, 11, 7, 23, 11, 7, 11, 7, 11, 11, 7, 19, 37, 31, 7, 7, 17, 7, 13, 43, 23, 43, 7, 17, 37, 41, 7, 43, 41, 23, 17, 13, 11, 7, 13, 19, 11, 61, 13, 7, 19, 13, 67, 7, 31, 31, 29, 11, 7, 41, 7, 11, 37, 29, 11, 7, 13, 13, 7, 11, 61, 7, 7, 67, 29, 7

Offset: 1

Labos Elemer, Feb 02 2000

A052180(A031926(n)).

A052257 Last filtering prime (A052180) of primes p such that next prime is p+10.

Original entry on oeis.org

11, 11, 13, 17, 11, 7, 7, 19, 11, 7, 17, 23, 13, 19, 7, 13, 13, 7, 7, 19, 11, 7, 23, 7, 23, 13, 11, 13, 7, 43, 19, 29, 7, 19, 7, 11, 7, 7, 47, 11, 37, 31, 11, 7, 13, 53, 7, 37, 47, 7, 43, 17, 7, 41, 19, 31, 17, 11, 61, 43, 17, 67, 13, 31, 17, 11, 29, 71, 73, 37, 7, 13, 11, 7

Offset: 1

Labos Elemer, Feb 02 2000

nonn

Cf. A031928, A052180.

A052180(A031928).

Corrected by Don Reble, May 07 2006

A052258 Last filtering prime (A052180) of primes p such that next prime is p+12.

Original entry on oeis.org

11, 13, 11, 11, 17, 23, 17, 19, 17, 29, 13, 13, 19, 37, 29, 29, 11, 23, 31, 31, 11, 11, 29, 31, 29, 43, 11, 29, 43, 37, 19, 31, 13, 47, 17, 31, 43, 19, 31, 13, 61, 11, 53, 11, 47, 43, 37, 17, 19, 13, 71, 11, 41, 23, 61, 37, 19, 17, 41, 61, 19, 17, 59, 13, 47, 79, 37, 13, 71

Offset: 1

Labos Elemer, Feb 02 2000

nonn

Cf. A031930, A052180.

A052259 Last filtering prime (A052180) of primes p such that next prime is p+14.

Original entry on oeis.org

11, 13, 17, 19, 29, 13, 31, 17, 37, 43, 23, 23, 11, 19, 31, 19, 17, 13, 59, 29, 19, 23, 53, 41, 11, 61, 67, 17, 13, 47, 19, 53, 19, 73, 53, 11, 23, 37, 23, 13, 71, 29, 83, 41, 17, 43, 79, 79, 41, 19, 83, 37, 53, 19, 79, 37, 13, 23, 83, 11, 43, 13, 59, 41, 37, 19, 43, 59, 83

Offset: 1

Labos Elemer, Feb 02 2000

nonn

Cf. A031932, A052180.

A052260 Last filtering prime (A052180) of primes p such that next prime is p+16 (A031934).

Original entry on oeis.org

19, 29, 29, 23, 37, 23, 53, 53, 31, 43, 19, 41, 31, 23, 23, 41, 47, 67, 43, 71, 61, 67, 41, 83, 41, 41, 53, 31, 41, 71, 19, 19, 47, 43, 67, 83, 97, 23, 41, 37, 23, 19, 37, 29, 59, 29, 61, 23, 89, 37, 113, 89, 59, 71, 127, 71, 23, 89, 23, 73, 37, 19, 17, 73, 97, 137, 107, 37

Offset: 1

Labos Elemer, Feb 02 2000

nonn

Cf. A031934, A052180.

A052248 Greatest prime divisor of all composite numbers between p and next prime.

Original entry on oeis.org

2, 3, 5, 3, 7, 3, 11, 13, 5, 17, 19, 7, 23, 17, 29, 5, 31, 23, 3, 37, 41, 43, 47, 11, 17, 53, 3, 37, 61, 43, 67, 23, 73, 5, 31, 79, 83, 43, 89, 5, 61, 3, 97, 11, 103, 109, 113, 19, 29, 79, 5, 83, 127, 131, 89, 5, 137, 139, 47, 97, 151, 103, 13, 157, 163, 167, 173, 29, 13

Offset: 2

N. J. A. Sloane

Or, largest of all prime factors of the numbers between prime(n) and prime(n+1).
a(n) = 3, 5, 7, 11, 13 iff prime(n) is in A059960, A080185, A080186, A080187, A080188 respectively. This sequence defines a mapping f of primes > 2 to primes (cf. A080189) and f(p) < p holds for all p > 2. - Klaus Brockhaus, Feb 10 2003
a(n) = A006530(A061214(n)). - Reinhard Zumkeller, Jun 22 2011

			a(8) = 11 since 20 = 2*2*5, 21 = 3*7, 22 = 2*11 are the numbers between prime(8) = 19 and prime(9) = 23.
For n=9, n-th prime is 23, composites between 23 and next prime are 24 25 26 27 29 of which largest prime divisor is 13, so a(9)=13.

T. D. Noe, Table of n, a(n) for n = 2..1000

Cf. A006530, A059960, A080185, A080186, A080187, A080188, A080189, A052180, A065091.

a052248 n = a052248_list !! (n-2)
a052248_list = f a065091_list where
   f (p:ps'@(p':ps)) = (maximum $ map a006530 [p+1..p'-1]) : f ps'
-- Reinhard Zumkeller, Jun 22 2011

g[n_] := Block[{t = Range[Prime[n] + 1, Prime[n + 1] - 1]}, Max[First /@ Flatten[ FactorInteger@t, 1]]]; Table[ g[n], {n, 2, 72}] (* Robert G. Wilson v, Feb 08 2006 *)
cmp[{a_,b_}]:=Max[Flatten[FactorInteger/@Range[a+1,b-1],1][[All,1]]]; cmp/@ Partition[ Prime[Range[2,80]],2,1] (* Harvey P. Dale, May 16 2020 *)

forprime(p=3,360,q=nextprime(p+1); m=0; for(j=p+1,q-1,f=factor(j); a=f[matsize(f)[1],1]; if(m

a(n) = max(prime(n) < k < prime(n+1), A006530(k)).

A000879 Number of primes < prime(n)^2.

Original entry on oeis.org

2, 4, 9, 15, 30, 39, 61, 72, 99, 146, 162, 219, 263, 283, 329, 409, 487, 519, 609, 675, 705, 811, 886, 1000, 1163, 1252, 1294, 1381, 1423, 1523, 1877, 1976, 2141, 2190, 2489, 2547, 2729, 2915, 3043, 3241, 3436, 3512, 3868, 3945, 4089, 4164, 4627, 5106

Offset: 1

gandalf(AT)hrn.office.ssi.net (James D. Ausfahl)

nonn

a(n) is the least index i such that A052180(i) = prime(n). - Labos Elemer, May 14 2003
Number of primes determined at the n-th step of the sieve of Eratosthenes. - Jean-Christophe Hervé, Oct 21 2013
There are only 3 squares in the current data: 4, 9, 7745089. - Michel Marcus, Apr 07 2018
There are no other squares up to a(780000). - Giovanni Resta, Apr 09 2018

Donovan Johnson, Table of n, a(n) for n = 1..10000

Cf. A050216 (first differences), A089609, A052180, A000720, A001248, A000885, A054270 (primes of rank a(n)).

PrimePi[Prime[Range[50]]^2] (* Harvey P. Dale, Jan 16 2013 *)

a(n) = primepi(prime(n)^2); \\ Michel Marcus, Oct 28 2013

a(n) = A000720(A001248(n)). - Michel Marcus, Apr 07 2018

Edited by Ralf Stephan, Aug 24 2004

A052297 Number of distinct prime factors of all composite numbers between n-th and (n+1)st primes.

Original entry on oeis.org

0, 1, 2, 3, 2, 4, 2, 5, 5, 3, 6, 5, 3, 5, 6, 7, 3, 7, 6, 2, 8, 4, 8, 9, 5, 3, 6, 2, 6, 14, 5, 8, 3, 11, 3, 9, 7, 6, 8, 8, 3, 13, 2, 6, 3, 14, 15, 5, 3, 7, 9, 3, 11, 8, 9, 9, 3, 9, 6, 3, 13, 16, 7, 3, 6, 16, 8, 13, 3, 6, 9, 10, 9, 9, 6, 8, 11, 6, 12, 14, 4, 14, 2, 10, 7, 8, 11, 6, 4, 6, 16, 10, 6, 13

Offset: 1

Labos Elemer, Feb 09 2000

nonn

From Lei Zhou, Mar 18 2014: (Start)
This is also the number of primes such that the (n+1)-th prime (mod i-th prime) is smaller than the (n+1)-th prime (mod n-th prime) for 1 <= i < n.
Proof: We denote the n-th prime number as P_n. Suppose P_(n+1) mod P_i = k; we can write P_(n+1) = m*P_i + k. Setting l = P_(n+1) - P_n, the composite numbers between P_n and P_(n+1) will be consecutively m*P_i + C, where C = k-l+1, k-l+2, ..., k-1. If k < l, there must be a value at which C equals zero since k-1 > 0 and k-l+1 <= 0, so P_i is a factor of a composite number between P_n and P_(n+1). If k >= l, all C values are greater than zero, thus P_i cannot be a factor of a composite number between P_n and P_(n+1). (End)

			n=30, p(30)=113, the next prime is 127. Between them are 13 composites: {114, 115, ..., 126}. Factorizing all and collecting prime factors, the set {2,3,5,7,11,13,17,19,23,29,31,41,59,61} is obtained, consisting of 14 primes, so a(30)=14.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A052180, A052248, A061214, A077218.

Length[Union[Flatten[Table[Transpose[FactorInteger[n]][[1]],{n, First[#]+ 1, Last[#]-1}]]]]&/@Partition[Prime[Range[100]],2,1] (* Harvey P. Dale, Jan 19 2012 *)

A052298 Maximal number of distinct prime factors of any composite number between n-th and (n+1)-st primes.

Original entry on oeis.org

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

Offset: 1

Labos Elemer, Feb 09 2000

nonn

			For n = 46, prime(46) = 199, next prime is 211. In between, the number of prime factors for {200,201,...,210} is {2,2,2,2,3,2,2,2,2,2,4} of which the maximum is 4, which arises at 210. So a(46) = 4. [Corrected by _Sean A. Irvine_, Nov 04 2021]

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

Cf. A052180, A052248, A052299.

Join[{0}, Max[PrimeNu[Range[First[#]+1, Last[#]-1]]]&/@Partition[ Prime[ Range[ 2, 110]], 2, 1]] (* Harvey P. Dale, Sep 26 2014 *)

Missing a(1)=0 inserted by Sean A. Irvine, Nov 04 2021

cp's OEIS Frontend

A052238 Primes p from A031924 such that A052180(p) = 23.

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A052256 Last filtering prime (A052180) of primes p such that next prime is p+8 (A031926).

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A052257 Last filtering prime (A052180) of primes p such that next prime is p+10.

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A052258 Last filtering prime (A052180) of primes p such that next prime is p+12.

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A052259 Last filtering prime (A052180) of primes p such that next prime is p+14.

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A052260 Last filtering prime (A052180) of primes p such that next prime is p+16 (A031934).

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A052248 Greatest prime divisor of all composite numbers between p and next prime.

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A000879 Number of primes < prime(n)^2.

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A052297 Number of distinct prime factors of all composite numbers between n-th and (n+1)st primes.

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Mathematica

A052298 Maximal number of distinct prime factors of any composite number between n-th and (n+1)-st primes.

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Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Extensions