A027856 -id:A027856 - cp's OEIS frontend

A060211 Larger term of a pair of twin primes such that the prime factors of their average are only 2 and 3. Proper subset of A058383.

7, 13, 19, 73, 109, 193, 433, 1153, 2593, 139969, 472393, 786433, 995329, 57395629, 63700993, 169869313, 4076863489, 10871635969, 2348273369089, 56358560858113, 79164837199873, 84537841287169, 150289495621633, 578415690713089, 1141260857376769, 57711166318706689

Offset: 1

Labos Elemer, Mar 20 2001

nonn

Larger of twin primes p such that p-1 = (2^u)*(3^w), u,w >= 1.

			a(4) = 73, {71,73} are twin primes and (71 + 73)/2 = 72 = 2*2*2*3*3.

Ray Chandler, Table of n, a(n) for n = 1..61 (terms < 10^1000)
Harsh Aggarwal, Table of n, a(n) for n = 62..91 (terms from 10^1000 to 10^20000)

Cf. A001454, A002822, A027856, A033845, A058383, A059960.

Take[Select[Sort[Flatten[Table[2^a 3^b,{a,250},{b,250}]]],AllTrue[#+{1,-1},PrimeQ]&]+1,23] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Apr 17 2019 *)

isok(p) = isprime(p) && isprime(p-2) && (vecmax(factor(p-1)[,1]) == 3); \\ Michel Marcus, Sep 05 2017

a(n) = A027856(n+1) + 1. - Amiram Eldar, Mar 17 2025

Name corrected by Sean A. Irvine, Oct 31 2022

A078884 Greater member p of a twin prime pair such that p-1 is 3-smooth.

Original entry on oeis.org

5, 7, 13, 19, 73, 109, 193, 433, 1153, 2593, 139969, 472393, 786433, 995329, 57395629, 63700993, 169869313, 4076863489, 10871635969, 2348273369089, 56358560858113, 79164837199873, 84537841287169, 150289495621633

Offset: 1

Reinhard Zumkeller, Dec 11 2002

nonn

			A000040(21)=73 and 73-1=72=2^3*3^2=A003586(17) and 73-2=71=A000040(20), therefore 73 is a term.

Ray Chandler, Table of n, a(n) for n = 1..62 (terms < 10^1000, first 56 terms from Robert Israel)

Cf. A006512, A014574, A003586, A027856.

Essentially the same as A060211.

N:= 10^100:
sort(select(t -> isprime(t) and isprime(t-2),
[seq(seq(1+2^i*3^j,i=1..ilog2(floor(N/3^j))),j=0..floor(log[3](N)))])); # Robert Israel, May 14 2018

1 + Select[With[{n = 10^15}, Sort@ Flatten@ Table[2^p * 3^q, {p, 0, Log2@ n}, {q, 0, Log[3, n/(2^p)]}] ], AllTrue[# + {-1, 1}, PrimeQ] &] (* Michael De Vlieger, May 14 2018 *)

a(n) = A027856(n) + 1 = A078883(n) + 2.

A325204 Numbers k such that k(k+1)(k+2) has exactly 4 distinct prime factors.

Original entry on oeis.org

5, 9, 10, 11, 12, 14, 15, 17, 18, 22, 23, 24, 25, 26, 27, 30, 31, 32, 36, 46, 47, 48, 52, 62, 71, 72, 79, 80, 81, 96, 106, 107, 126, 127, 162, 191, 192, 241, 242, 256, 382, 431, 486, 512, 576, 862, 1151, 1152, 2186, 2591, 2592, 2916, 4372, 8191, 8746, 131071, 131072, 139967, 472391, 524287, 786431, 995326, 995327

Offset: 1

J. M. Bergot and Robert Israel, Sep 05 2019

nonn

Contains 2^p-1 for p in A107360 except 3.
Contains all terms of A325255 except 2 and 4.
Contains k-1 for k in A027856 except 4.
Contains k-2 for k in A327240 except 6 and 8. - Ray Chandler, Sep 14 2019

			a(3)=10 is in the sequence because 10*11*12 has four distinct prime factors: 2, 3, 5, 11.

Ray Chandler, Table of n, a(n) for n = 1..178 (terms < 10^1000; first 114 terms from Robert Israel)
Ray Chandler, Mathematica code used to compute b-file.
Math StackExchange, Three consecutive numbers with exactly different four prime factors.

Cf. A027856, A107360, A325255, A327240.

select(t -> nops(numtheory:-factorset(t) union numtheory:-factorset(t+1) union numtheory:-factorset(t+2))=4, [$1..10^6]);

select(k->4==omega(k*(k+1)*(k+2)), [1..10000]) \\ Andrew Howroyd, Sep 05 2019

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)).

A386731 a(n) = A385433(n) + A386730(n).

Original entry on oeis.org

2, 2, 3, 3, 5, 5, 7, 7, 9, 9, 13, 13, 19, 17, 17, 23, 25, 29, 31, 37, 41, 45, 41, 43, 35, 43, 51, 47, 59, 65, 91, 99, 109, 121, 145, 175, 151, 155, 213, 291, 297, 259, 283, 349, 301, 415, 365, 369, 573, 683, 1103, 1017, 1195, 1347, 1537, 1619, 1717, 1751, 1957

Offset: 1

Ken Clements, Jul 31 2025

nonn

These numbers are sum of the exponents of 2 and 3 for the averages of twin primes in A027856. An interesting aspect is that after the first 2 terms, all of these are odd numbers. For all of those, the sum cannot be even because then for m = 2^i * 3^j, m-1 or m+1 would be divisible by 5.

			a(1) = A385433(1) + A386730(1) = 2
a(2) = A385433(2) + A386730(2) = 2
a(3) = A385433(3) + A386730(3) = 3
a(4) = A385433(4) + A386730(4) = 3
a(5) = A385433(5) + A386730(5) = 5

Ken Clements, Table of n, a(n) for n = 1..82

Cf. A385433, A386730, A027856.

seq[max_] := Total[IntegerExponent[Select[Sort[Flatten[Table[2^i*3^j, {i, 1, Floor[Log2[max]]}, {j, 0, Floor[Log[3, max/2^i]]}]]], And @@ PrimeQ[# + {-1, 1}] &], #] & /@ {2, 3}]; seq[10^250] (* Amiram Eldar, Aug 01 2025 *)

from math import log10
from gmpy2 import is_prime
l2, l3 = log10(2), log10(3)
upto_digits = 200
sum_limit = 2 + int((upto_digits - l3)/l2)
def TP_pi_2_upto_sum(limit): # Search all partitions up to the given exponent sum.
    unsorted_result = [(2, log10(4)), (1, log10(6))]
    for exponent_sum in range(3, limit+1, 2):
        for i in range(1, exponent_sum):
            j = exponent_sum - i
            log_N = i*l2 + j*l3
            if log_N <= upto_digits:
                N = 2**i * 3**j
                if is_prime(N-1) and is_prime(N+1):
                     unsorted_result.append((i+j, log_N))
    sorted_result = sorted(unsorted_result, key=lambda x: x[1])
    return sorted_result
print([s for s, _ in TP_pi_2_upto_sum(sum_limit) ])

A360844 a(n) is the least k-full number that is sandwiched between twin primes.

Original entry on oeis.org

4, 432, 2592, 139968, 139968, 174960000000, 56358560858112, 84537841287168, 578415690713088, 578415690713088, 1141260857376768, 61628086298345472, 61628086298345472, 61628086298345472, 322850407500000000000000000000, 322850407500000000000000000000, 62518864539857068333550694039552

Offset: 2

Amiram Eldar, Feb 23 2023

nonn

k-full number is a number m such that if a prime p divides m then so does p^k. All the exponents in the canonical prime factorization of a k-full number are not smaller than k.
a(2)-a(15) are the terms below 3*10^19. Except for a(7) = 174960000000, they are all 3-smooth numbers (A003586, and thus they are terms of A027856). Are there other terms that are not 3-smooth?
a(168) = 2^176 * 3^173 * 7^168 is the first term that is not 5-smooth. - Bert Dobbelaere, Feb 24 2023

			The first 3 terms, their factorizations and the corresponding twin primes are:
  n |   a(n)  prime factorization  A051904(a(n))  {a(n)-1, a(n)+1}
  ----------------------------------------------------------------
  2 |     4                  2^2              2             {3, 5}
  3 |   432            2^4 * 3^3              3         {431, 433}
  4 |  2592            2^5 * 3^4              4       {2591, 2593}

Bert Dobbelaere, Table of n, a(n) for n = 2..200
Eric Weisstein's World of Mathematics, Twin Primes.
Wikipedia, Powerful number: Generalization.
Index entries for sequences related to powerful numbers.

Cf. A001097, A001694, A014574, A036966, A036967, A069492, A069493.
Cf. A113839, A360840, A360841, A360842, A360843.
Cf. A027856, A003586, A051904.

More terms from Bert Dobbelaere, Feb 24 2023

A384835 The exponents (j, k) of the numbers 2^j*3^k that are averages of twin primes, with both j and k > 0, in the order of their sum, and then by j.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 2, 3, 3, 2, 4, 3, 6, 1, 5, 4, 7, 2, 3, 10, 6, 7, 2, 15, 12, 5, 18, 1, 18, 5, 21, 4, 24, 5, 27, 4, 11, 24, 30, 7, 32, 9, 33, 8, 31, 12, 36, 7, 43, 2, 32, 15, 43, 8, 50, 9, 63, 2, 66, 25, 79, 20, 99, 10, 57, 64, 82, 63, 63, 88, 56, 99, 148, 27

Offset: 1

Ken Clements, Jun 10 2025

These are the (j,k) exponents of the numbers 2^j*3^k that are averages of twin primes, ordered by j+k, j. They are remarkable in structure because except for the first pair (2^1*3^1), j+k is always an odd number. I have proof of this, and the reason it is not the case for the first pair is that 6-1=5 is the only number divisible by 5 that is prime.

			2^a(1) * 3^a(2) = 6.
2^a(3) * 3^a(4) = 18.
2^a(5) * 3^a(6) = 12.
2^a(7) * 3^a(8) = 108.
2^a(9) * 3^a(10) = 72.

Ken Clements, Table of n, a(n) for n = 1..162

Cf. A027856, A384639 (ordered by value of 2^j*3^k).

seq[max_] := Flatten@ Transpose[IntegerExponent[Select[Flatten[Table[2^j*3^(m-j), {m, 2, max}, {j, 1, m-1}]], And @@ PrimeQ[# + {-1, 1}] &], #] & /@ {2, 3}]; seq[200] (* Amiram Eldar, Jun 26 2025 *)

from sympy import isprime
def is_TP_pi_2(j, k):
    N = 2**j * 3**k
    return isprime(N-1) and isprime(N+1)
def aupto(limit):
    result = [1, 1]
    for exponent_sum in range(3, limit+1, 2):
        for j in range(1, exponent_sum):
             k = exponent_sum - j
             if is_TP_pi_2(j, k):
                  result.append(j)
                  result.append(k)
    return result
print(aupto(10_000))

import heapq
from gmpy2 import is_prime
from itertools import islice
def agen(): # generator of terms
    v, oldv, h = 1, 0, [(2, 1, 1, 6)]
    while True:
        s, e2, e3, v = heapq.heappop(h)
        if v != oldv:
            if is_prime(v-1) and is_prime(v+1):
                yield from (e2, e3)
            oldv = v
            heapq.heappush(h, (s+1, e2+1, e3, 2*v))
            heapq.heappush(h, (s+1, e2, e3+1, 3*v))
print(list(islice(agen(), 70))) # Michael S. Branicky, Jun 26 2025

A387060 Numbers k such that 16 * 3^k + 1 is prime.

Original entry on oeis.org

0, 3, 4, 5, 12, 24, 36, 77, 195, 296, 297, 533, 545, 644, 884, 932, 1409, 2061, 2453, 2985, 3381, 4980, 5393, 11733, 13631, 14516, 21004, 27663, 32645, 39453, 67055, 90543

Offset: 1

Ken Clements, Aug 15 2025

a(33) > 10^5.
Conjecture: The only intersection with A385115 is at k = 3 where 2^4 * 3^3 = 432 = A027856(8).
Idea: For odd k > 3, covering systems ensure mutual exclusion:
 If k = 1, 9, 13, 19, 25, 31, 37, 39, 43, 49, 55 (mod 60), then 7 or 31 divides (16*3^k+1).
 If k = 5, 7, 11, 17, 23, 27, 29, 35, 41, 47, 53, 57, 59 (mod 60), then 11 or 13 divides (16*3^k-1).
 If k = 15, 21, 33, 45, 51 (mod 60), various primes including {11,31,43,109,277,433,...} ensure at least one of 16*3^k +- 1 is composite.
 If k = 3 (mod 60) and k > 3, the probability of intersection becomes vanishingly small.
 Only k = 3 escapes all divisibility conditions. Verified to k = 10^5.

Ken Clements, Python program to calculate covering system.

Cf. A027856, A005537, A005538, A385115.

Select[Range[0, 4000], PrimeQ[16*3^# + 1] &] (* Amiram Eldar, Aug 16 2025 *)

from gmpy2 import is_prime
print([k for k in range(4_000) if is_prime(16 * 3**k + 1)])

cp's OEIS Frontend

A060211 Larger term of a pair of twin primes such that the prime factors of their average are only 2 and 3. Proper subset of A058383.

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A078884 Greater member p of a twin prime pair such that p-1 is 3-smooth.

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A325204 Numbers k such that k*(k+1)*(k+2) has exactly 4 distinct prime factors.

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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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A386731 a(n) = A385433(n) + A386730(n).

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A325204 Numbers k such that k(k+1)(k+2) has exactly 4 distinct prime factors.