A052248 -id:A052248 - cp's OEIS frontend

A080185 Primes p such that 5 is the largest of all prime factors of the numbers between p and the next prime (cf. A052248).

7, 29, 59, 149, 179, 239, 269, 599, 809, 1619, 2999, 4049, 4799, 8999, 9719, 15359, 21599, 23039, 33749, 138239, 179999, 281249, 345599, 737279, 3455999, 6143999, 6560999, 10124999, 13668749, 15551999, 17495999, 20995199, 22118399, 23999999

Offset: 1

Klaus Brockhaus, Feb 10 2003

nonn

The sequence consists of 7 and the lesser of twin primes q (A001359) such that q+1 is 5-smooth (A051037) but not 3-smooth (A003586, A080193).

			7 is a term since 8 = 2^3, 9 = 3^3, 10 = 2*5 are the numbers between 7 and the next prime 11; 149 is a term since 150 = 2*3*5^2 is the only number between 149 and the next prime 151.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A052248, A001359, A051037, A003586, A080193.

{forprime(p=2,24000000,q=nextprime(p+1); m=0; j=p+1; while(j

A114331 Observe that A052248(n) = greatest prime divisor q (say) of all composite numbers between p = prime(n) and next prime. There is only one composite number in this range which is divisible by q. Sequence lists these composite numbers.

Original entry on oeis.org

4, 6, 10, 12, 14, 18, 22, 26, 30, 34, 38, 42, 46, 51, 58, 60, 62, 69, 72, 74, 82, 86, 94, 99, 102, 106, 108, 111, 122, 129, 134, 138, 146, 150, 155, 158, 166, 172, 178, 180, 183, 192, 194, 198, 206, 218, 226, 228, 232, 237, 240, 249, 254, 262, 267, 270

Offset: 2

N. J. A. Sloane, based on correspondence from Leroy Quet and Hugo Pfoertner, Feb 22 2006

nonn

The uniqueness follows from generalization of Bertrand's Postulate. - Franklin T. Adams-Watters.

Robert Israel, Table of n, a(n) for n = 2..10000
Eric Weisstein's World of Mathematics, Bertrand's Postulate

Cf. A052248, A114349.

p:= 3: R:= NULL:
for i from 2 to 100 do
  pp:= p; p:= nextprime(p);
  rmax:= 0:
  for q from pp+1 to p-1 do
    r:= max(numtheory:-factorset(q));
    if r > rmax then rmax:= r; qmax:= q fi
  od;
  R:= R, qmax
od:
R; # Robert Israel, Apr 01 2021

A080186 Primes p such that 7 is the largest of all prime factors of the numbers between p and the next prime (cf. A052248).

Original entry on oeis.org

13, 41, 419, 881, 1049, 2267, 2687, 3359, 3527, 5879, 6299, 7349, 7559, 8231, 8819, 10499, 18521, 26249, 26879, 28349, 29399, 30869, 33599, 35279, 49391, 81647, 100799, 102059, 131249, 131711, 134399, 158759, 170099, 183707, 197567, 241919

Offset: 1

Klaus Brockhaus, Feb 10 2003

nonn

The sequence appears to consist of 13 and the lesser of twin primes q (A001359) such that q+1 is 7-smooth (A002473) but not 5-smooth (A051037, A080194).

			13 is a term since 14 = 2*7, 15 = 3*5, 16 = 2^4 are the numbers between 13 and the next prime 17; 419 is a term since 420 = 2^2*3*5*7 is the only number between 419 and the next prime 421.

Amiram Eldar, Table of n, a(n) for n = 1..200 (terms 1..100 from Harvey P. Dale)

Cf. A052248, A001359, A002473, A051037, A080194.

lpf7Q[n_]:=Max[Flatten[Transpose[FactorInteger[#]][[1]]&/@Range[ n+1, NextPrime[ n]-1]]]==7; Select[Prime[Range[22000]],lpf7Q] (* Harvey P. Dale, Sep 25 2015 *)

{forprime(p=2,250000,q=nextprime(p+1); m=0; j=p+1; while(j

A080187 Primes p such that 11 is the largest of all prime factors of the numbers between p and the next prime (cf. A052248).

Original entry on oeis.org

19, 97, 197, 461, 659, 1319, 1451, 2111, 2309, 2969, 3167, 3299, 4157, 5279, 7127, 9239, 10889, 11549, 15971, 16631, 22637, 25409, 26729, 29567, 30491, 34649, 34847, 55439, 55901, 64151, 87119, 92399, 98009, 110879, 118799, 152459, 164999, 176417

Offset: 1

Klaus Brockhaus, Feb 10 2003

nonn

The sequence appears to consist of 19, 97 and the lesser of twin primes q (A001359) such that q+1 is 11-smooth (A051038) but not 7-smooth (A002473, A080195).

			97 is a term since 98 = 2*7^2, 99 = 3^2*11, 100 = 2^2*5^2 are the numbers between 97 and the next prime 101;
461 is a term since 462 = 2*3*7*11 is the only number between 461 and the next prime 463.

Amiram Eldar, Table of n, a(n) for n = 1..521 (terms below 10^11)

Cf. A052248, A001359, A051038, A002473, A080195.

maxPrime[n1_, n2_] := FactorInteger[#][[-1, 1]] & /@ Range[n1, n2]; Select[Range[180000], PrimeQ[#] && Max[maxPrime[# + 1, NextPrime[#] - 1]] == 11 &] (* Amiram Eldar, Feb 08 2020 *)

{forprime(p=2,180000,q=nextprime(p+1); m=0; j=p+1; while(j

A080188 Primes p such that 13 is the largest of all prime factors of the numbers between p and the next prime (cf. A052248).

Original entry on oeis.org

23, 311, 349, 857, 1091, 1871, 1949, 2027, 2339, 2729, 3119, 3821, 5849, 6551, 7487, 9437, 10139, 10529, 11699, 15287, 18251, 21059, 21839, 38609, 42899, 49919, 51479, 57329, 61151, 65519, 69497, 70199, 70979, 81899, 97499, 108107, 109199, 114659

Offset: 1

Klaus Brockhaus, Feb 10 2003

nonn

The sequence appears to consist of 23, 349 and the lesser of twin primes q (A001359) such that q+1 is 13-smooth (A080197) but not 11-smooth (A051038, A080196).

			349 is a term since 350 = 2*5^2*7, 351 = 3^3*13, 352 = 2^5*11 are the numbers between 349 and the next prime 353; 857 is a term since 858 = 2*3*11*13 is the only number between 857 and the next prime 859.

Amiram Eldar, Table of n, a(n) for n = 1..1092 (terms below 10^11)

Cf. A052248, A001359, A080197, A051038, A080196.

maxPrime[n1_, n2_] := FactorInteger[#][[-1, 1]] & /@ Range[n1, n2]; Select[Range[120000], PrimeQ[#] && Max[maxPrime[# + 1, NextPrime[#] - 1]] == 13 &] (* Amiram Eldar, Feb 08 2020 *)

{forprime(p=2,120000,q=nextprime(p+1); m=0; j=p+1; while(j

A080189 a(n) = k such that f^k(prime(n)) = 2, where f is the mapping of primes > 2 to primes defined by A052248.

Original entry on oeis.org

0, 1, 2, 3, 2, 4, 2, 3, 5, 3, 3, 4, 4, 6, 3, 4, 3, 4, 6, 2, 5, 5, 7, 4, 3, 3, 5, 2, 5, 5, 7, 7, 6, 6, 3, 4, 6, 8, 7, 5, 3, 5, 2, 4, 3, 6, 6, 6, 4, 4, 6, 3, 8, 8, 8, 5, 3, 7, 7, 4, 4, 5, 6, 5, 7, 9, 8, 6, 4, 5, 4, 6, 5, 4, 3, 5, 4, 7, 4, 7, 4, 7, 2, 6, 3, 7, 5, 5, 3, 7, 4, 9, 9, 8, 9, 8, 9, 4, 4, 8, 8, 5, 5, 4, 3

Offset: 1

Klaus Brockhaus, Feb 10 2003

nonn

Since the largest of all prime factors of the numbers between prime p and the next prime is smaller than p, we have p > f(p) > f^2(p) > ... > 2, so a(n) is finite for all n.

			prime(6) = 13, f(13) = 7, f(7) = 5, f(5) = 3, f(3) = 2, so f^4(13) = 2 and a(6) = 4.

Cf. A052248.

{forprime(k=2,580,c=0; p=k; while(p>2,q=nextprime(p+1); m=0; for(j=p+1,q-1,f=factor(j); a=f[matsize(f)[1],1]; if(m

A114349 Terms of A114331 divided by the appropriate prime (q) in A052248.

Original entry on oeis.org

2, 2, 2, 4, 2, 6, 2, 2, 6, 2, 2, 6, 2, 3, 2, 12, 2, 3, 24, 2, 2, 2, 2, 9, 6, 2, 3, 3, 2, 3, 2, 6, 2, 30, 5, 2, 2, 4, 2, 36, 3, 64, 2, 18, 2, 2, 2, 12, 8, 3, 48, 3, 2, 2, 3, 54, 2, 2, 6, 3, 2, 3, 24, 2, 2, 2, 2, 12, 27, 2, 2, 7, 8, 2, 2, 2, 2, 4, 3, 60, 2, 144, 4, 26, 2, 2, 2, 42, 2, 2, 2

Offset: 2

N. J. A. Sloane, based on correspondence from Leroy Quet and Hugo Pfoertner, Feb 22 2006

nonn

Cf. A052248, A114331.

A080184 Duplicate of A052248.

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

Offset: 2

dead

A080190 Smallest prime p such that n applications of f lead form p to 2, where f is the mapping of primes > 2 to primes defined by A052248.

Original entry on oeis.org

2, 3, 5, 7, 13, 23, 43, 83, 163, 317, 631, 1259, 2503, 5003, 9973, 19937, 39869, 119617, 239233, 480023, 960031, 1920049, 3840091, 7680181, 15360361, 30720719, 61441379, 122882741, 245765449, 491530873, 983061713, 1966123417

Offset: 0

Klaus Brockhaus, Feb 10 2003

nonn

RECORDS transform of A080189; prime p sets a new record for the number of applications of f that are required to reach 2. - a(n) = prime preceding 2*a(n-1) as long as a(n-1) is a term of A080191; if however a(n-1) is a term of A080192, then a(n) > 2*a(n-1). - Next term a(32) > 3932600000, presumably a(32) = 5274863189, a(33) = 10549726367. - The sequence coincides with A006992 (Bertrand primes: a(n) is largest prime < 2*a(n-1)) for the first 17 terms; first divergence occurs after term 39869 because this is the first term which belongs to A080192.

			f(23) = 13, f(13) = 7, f(7) = 5, f(5) = 3, f(3) = 2; five applications of f are required to reach 2 and for all primes < 23 at most four applications are required, so a(5) = 23.

Cf. A052248, A080189, A080191, A080192, A006992.

f^n(p) = 2.

A052180 Last filtering prime for n-th prime p: find smallest prime factor of each of the composite numbers between p and next prime; take maximal value.

Original entry on oeis.org

2, 2, 3, 2, 3, 2, 3, 5, 2, 5, 3, 2, 3, 7, 5, 2, 5, 3, 2, 7, 3, 5, 7, 3, 2, 3, 2, 3, 11, 3, 7, 2, 11, 2, 5, 7, 3, 13, 5, 2, 11, 2, 3, 2, 11, 13, 3, 2, 3, 5, 2, 13, 11, 7, 5, 2, 5, 3, 2, 17, 13, 3, 2, 3, 17, 5, 11, 2, 3, 5, 19, 7, 13, 3, 5, 17, 3, 13, 7, 2, 7, 2, 19, 3, 5, 11, 3, 2, 3, 11, 13, 3, 17

Offset: 2

Labos Elemer, Feb 05 2000

A000879(n) is the least index i such that a(i) = prime(n). - Labos Elemer, May 14 2003

			For n=9, n-th prime is 23, composites between 23 and next prime are 24 25 26 27 28, smallest prime divisors are 2 5 2 3 2; maximal value is 5, so a(9)=5.

T. D. Noe, Table of n, a(n) for n=2..10000

Cf. A052248, A020639, A000720, A083269, A000879.

Cf. A010051.

a052180 n = a052180_list !! (n-2)
a052180_list = f [4..] where
   f ws = (maximum $ map a020639 us) : f vs where
     (us, _:vs) = span  ((== 0) . a010051) ws
-- Reinhard Zumkeller, Dec 27 2012

ffi[x_] := Flatten[FactorInteger[x]];
lf[x_] := Length[FactorInteger[x]];
ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}];
mi[x_] := Min[ba[x]];
Table[Max[Table[mi[ba[w]], {w, Prime[j]+1, -1+Prime[j+1]}]], {j, 2, 256}]
(* Second program: *)
mpf[{a_,b_}] := Max[FactorInteger[#][[1,1]]& /@ Range[a+1,b-1]];
mpf/@ Partition[ Prime[Range[2,100]],2,1] (* Harvey P. Dale, Apr 30 2013 *)

a(n) = {my(p = prime(n), amax = 0); forcomposite(c = p, nextprime(p+1), amax = max(factor(c)[1,1], amax);); amax;} \\ Michel Marcus, Apr 21 2018

from sympy import prime, nextprime, primefactors
def a(n):
  p = prime(n); q = nextprime(p)
  return max(min(primefactors(m)) for m in range(p+1, q))
print([a(n) for n in range(2, 95)]) # Michael S. Branicky, Feb 02 2021

a(n) = Max_{j=1+prime(n)..prime(n+1)-1} A020639(j) where A020639(j) is the least prime dividing j.

cp's OEIS Frontend

A080185 Primes p such that 5 is the largest of all prime factors of the numbers between p and the next prime (cf. A052248).

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A114331 Observe that A052248(n) = greatest prime divisor q (say) of all composite numbers between p = prime(n) and next prime. There is only one composite number in this range which is divisible by q. Sequence lists these composite numbers.

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A080186 Primes p such that 7 is the largest of all prime factors of the numbers between p and the next prime (cf. A052248).

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Mathematica

PARI

A080187 Primes p such that 11 is the largest of all prime factors of the numbers between p and the next prime (cf. A052248).

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A080188 Primes p such that 13 is the largest of all prime factors of the numbers between p and the next prime (cf. A052248).

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PARI

A080189 a(n) = k such that f^k(prime(n)) = 2, where f is the mapping of primes > 2 to primes defined by A052248.

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PARI

A114349 Terms of A114331 divided by the appropriate prime (q) in A052248.

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A080184 Duplicate of A052248.

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A080190 Smallest prime p such that n applications of f lead form p to 2, where f is the mapping of primes > 2 to primes defined by A052248.

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A052180 Last filtering prime for n-th prime p: find smallest prime factor of each of the composite numbers between p and next prime; take maximal value.