A080185 -id:A080185 - cp's OEIS frontend

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

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)).

A386252 Numbers m of the form 2^i * 3^j * 5^k such that i, j, k > 0 and m+1 and m-1 are both prime numbers.

Original entry on oeis.org

30, 60, 150, 180, 240, 270, 600, 810, 1620, 3000, 4050, 4800, 9000, 9720, 15360, 21600, 23040, 33750, 138240, 180000, 281250, 345600, 737280, 3456000, 6144000, 6561000, 10125000, 13668750, 15552000, 17496000, 20995200, 22118400, 24000000, 30000000, 54675000

Offset: 1

Ken Clements, Jul 16 2025

nonn

			a(1) = 2^1 * 3^1 * 5^1 = 30 where 29 and 31 are prime numbers.
a(2) = 2^2 * 3^1 * 5^1 = 60 where 59 and 61 are prime numbers.
a(3) = 2^1 * 3^1 * 5^2 = 150 where 149 and 151 are prime numbers.
a(4) = 2^2 * 3^2 * 5^1 = 180 where 179 and 181 are prime numbers.

Ken Clements, Table of n, a(n) for n = 1..926

Subsequence of A143207.

Cf. A027856, A384530, A080185.

seq[max_] := Select[Table[2^i*3^j*5^k, {i, 1, Log2[max]}, {j, 1, Log[3, max/2^i]}, {k, 1, Log[5, max/(2^i*3^j)]}] // Flatten // Sort, And @@ PrimeQ[# + {-1, 1}] &]; seq[10^8] (* Amiram Eldar, Jul 17 2025 *)

from math import log10
from gmpy2 import is_prime
l2, l3, l5 = log10(2), log10(3), log10(5)
upto_digits = 20
sum_limit = 3 + int((upto_digits - l3 - l5)/l2)
def TP_pi_3_upto_sum(limit): # Search all partitions up to the given exponent sum.
    unsorted_result = []
    for exponent_sum in range(3, limit+1):
        for i in range(1, exponent_sum -1):
             for j in range(1, exponent_sum - i):
                k = exponent_sum - i - j
                log_N = i*l2 + j*l3 + k*l5
                if log_N <= upto_digits:
                    N = 2**i * 3**j * 5**k
                    if is_prime(N-1) and is_prime(N+1):
                        unsorted_result.append((N, log_N))
    sorted_result = sorted(unsorted_result, key=lambda x: x[1])
    return sorted_result
print([n for n, _ in TP_pi_3_upto_sum(sum_limit) ])

A386498 a(n) is the 2-adic valuation of A386252(n).

Original entry on oeis.org

1, 2, 1, 2, 4, 1, 3, 1, 2, 3, 1, 6, 3, 3, 10, 5, 9, 1, 10, 5, 1, 9, 14, 10, 14, 3, 3, 1, 9, 6, 7, 15, 9, 7, 3, 1, 1, 3, 1, 17, 3, 13, 10, 16, 1, 4, 13, 11, 3, 5, 6, 8, 10, 15, 10, 3, 1, 3, 1, 9, 14, 10, 6, 7, 5, 2, 4, 2, 29, 26, 5, 15, 4, 2, 26, 15, 13, 17, 16

Offset: 1

Ken Clements, Jul 23 2025

nonn

			a(1) = 1 because A386252(1) = 2^1 * 3^1 * 5^1
a(2) = 2 because A386252(2) = 2^2 * 3^1 * 5^1
a(3) = 1 because A386252(3) = 2^1 * 3^1 * 5^2
a(4) = 2 because A386252(4) = 2^2 * 3^2 * 5^1

Ken Clements, Table of n, a(n) for n = 1..4000

Cf. A007814, A386252, A386499, A386500, A027856, A384530, A080185.

seq[max_] := IntegerExponent[Select[Table[2^i*3^j*5^k, {i, 1, Log2[max]}, {j, 1, Log[3, max/2^i]}, {k, 1, Log[5, max/(2^i*3^j)]}] // Flatten // Sort, And @@ PrimeQ[# + {-1, 1}] &], 2]; seq[10^12] (* Amiram Eldar, Jul 24 2025 *)

from math import log10
from gmpy2 import is_prime
l2, l3, l5 = log10(2), log10(3), log10(5)
upto_digits = 100
sum_limit = 3 + int((upto_digits - l3 - l5)/l2)
def TP_pi_3_upto_sum(limit): # Search all partitions up to the given exponent sum.
    unsorted_result = []
    for exponent_sum in range(3, limit+1):
        for i in range(1, exponent_sum -1):
             for j in range(1, exponent_sum - i):
                k = exponent_sum - i - j
                log_N = i*l2 + j*l3 + k*l5
                if log_N <= upto_digits:
                    N = 2**i * 3**j * 5**k
                    if is_prime(N-1) and is_prime(N+1):
                        unsorted_result.append((i, log_N))
    sorted_result = sorted(unsorted_result, key=lambda x: x[1])
    return sorted_result
print([i for i, _ in TP_pi_3_upto_sum(sum_limit) ])

a(n) = A007814(A386252(n)).

A386499 a(n) is the 5-adic valuation of A386252(n).

Original entry on oeis.org

1, 1, 2, 1, 1, 1, 2, 1, 1, 3, 2, 2, 3, 1, 1, 2, 1, 4, 1, 4, 6, 2, 1, 3, 3, 3, 6, 5, 3, 3, 2, 2, 6, 7, 5, 9, 7, 3, 8, 4, 8, 4, 6, 5, 6, 2, 3, 6, 4, 10, 9, 2, 4, 6, 3, 2, 3, 9, 8, 2, 6, 1, 11, 2, 5, 3, 9, 1, 1, 3, 10, 3, 3, 8, 2, 2, 7, 2, 8, 8, 5, 7, 11, 3, 5, 14

Offset: 1

Ken Clements, Jul 23 2025

nonn

			a(1) = 1 because A386252(1) = 2^1 * 3^1 * 5^1
a(2) = 1 because A386252(2) = 2^2 * 3^1 * 5^1
a(3) = 2 because A386252(3) = 2^1 * 3^1 * 5^2
a(4) = 1 because A386252(4) = 2^2 * 3^2 * 5^1

Ken Clements, Table of n, a(n) for n = 1..4000

Cf. A112765, A386252, A386498, A386500, A027856, A384530, A080185.

seq[max_] := IntegerExponent[Select[Table[2^i*3^j*5^k, {i, 1, Log2[max]}, {j, 1, Log[3, max/2^i]}, {k, 1, Log[5, max/(2^i*3^j)]}] // Flatten // Sort, And @@ PrimeQ[# + {-1, 1}] &], 5]; seq[10^12] (* Amiram Eldar, Jul 24 2025 *)

from math import log10
from gmpy2 import is_prime
l2, l3, l5 = log10(2), log10(3), log10(5)
upto_digits = 100
sum_limit = 3 + int((upto_digits - l3 - l5)/l2)
def TP_pi_3_upto_sum(limit): # Search all partitions up to the given exponent sum.
    unsorted_result = []
    for exponent_sum in range(3, limit+1):
        for i in range(1, exponent_sum -1):
             for j in range(1, exponent_sum - i):
                k = exponent_sum - i - j
                log_N = i*l2 + j*l3 + k*l5
                if log_N <= upto_digits:
                    N = 2**i * 3**j * 5**k
                    if is_prime(N-1) and is_prime(N+1):
                        unsorted_result.append((k, log_N))
    sorted_result = sorted(unsorted_result, key=lambda x: x[1])
    return sorted_result
print([k for k, _ in TP_pi_3_upto_sum(sum_limit) ])

a(n) = A112765(A386252(n)).

A386500 a(n) is the 3-adic valuation of A386252(n).

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 4, 4, 1, 4, 1, 2, 5, 1, 3, 2, 3, 3, 2, 2, 3, 2, 3, 1, 8, 4, 7, 5, 7, 8, 3, 1, 1, 7, 3, 6, 11, 5, 1, 4, 4, 3, 1, 9, 13, 6, 3, 11, 1, 2, 11, 7, 1, 9, 15, 15, 5, 8, 12, 3, 13, 1, 14, 11, 16, 6, 19, 2, 1, 4, 8, 15, 9, 3, 10, 4, 9, 1, 8, 3, 7, 7

Offset: 1

Ken Clements, Jul 23 2025

nonn

			a(1) = 1 because A386252(1) = 2^1 * 3^1 * 5^1
a(2) = 1 because A386252(2) = 2^2 * 3^1 * 5^1
a(3) = 1 because A386252(3) = 2^1 * 3^1 * 5^2
a(4) = 2 because A386252(4) = 2^2 * 3^2 * 5^1

Ken Clements, Table of n, a(n) for n = 1..4000

Cf. A007949, A386252, A386498, A386499, A027856, A384530, A080185.

seq[max_] := IntegerExponent[Select[Table[2^i*3^j*5^k, {i, 1, Log2[max]}, {j, 1, Log[3, max/2^i]}, {k, 1, Log[5, max/(2^i*3^j)]}] // Flatten // Sort, And @@ PrimeQ[# + {-1, 1}] &], 3]; seq[10^12] (* Amiram Eldar, Jul 24 2025 *)

from math import log10
from gmpy2 import is_prime
l2, l3, l5 = log10(2), log10(3), log10(5)
upto_digits = 100
sum_limit = 3 + int((upto_digits - l3 - l5)/l2)
def TP_pi_3_upto_sum(limit): # Search all partitions up to the given exponent sum.
    unsorted_result = []
    for exponent_sum in range(3, limit+1):
        for i in range(1, exponent_sum -1):
             for j in range(1, exponent_sum - i):
                k = exponent_sum - i - j
                log_N = i*l2 + j*l3 + k*l5
                if log_N <= upto_digits:
                    N = 2**i * 3**j * 5**k
                    if is_prime(N-1) and is_prime(N+1):
                        unsorted_result.append((j, log_N))
    sorted_result = sorted(unsorted_result, key=lambda x: x[1])
    return sorted_result
print([j for j, _ in TP_pi_3_upto_sum(sum_limit) ])

a(n) = A007949(A386252(n)).

cp's OEIS Frontend

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

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A386252 Numbers m of the form 2^i * 3^j * 5^k such that i, j, k > 0 and m+1 and m-1 are both prime numbers.

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A386498 a(n) is the 2-adic valuation of A386252(n).

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A386499 a(n) is the 5-adic valuation of A386252(n).

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A386500 a(n) is the 3-adic valuation of A386252(n).

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