A027856 -id:A027856 - cp's OEIS frontend

A385433 a(n) is the 2-adic valuation of A027856(n).

2, 1, 2, 1, 3, 2, 6, 4, 7, 5, 6, 3, 18, 12, 2, 18, 21, 24, 27, 30, 33, 43, 32, 36, 11, 31, 43, 32, 50, 63, 66, 79, 99, 57, 82, 148, 63, 56, 211, 275, 287, 90, 148, 298, 160, 363, 134, 49, 529, 264, 960, 541, 988, 1015, 1440, 1295, 979, 258, 637, 2320, 1036, 2815, 1063, 180, 888

Offset: 1

Ken Clements, Jul 31 2025

nonn

These are the exponents, i, of the prime factor 2 of the A027856 numbers m = 2^i * 3^j where m is the average of twin primes. After the second term, all sums of i+j are odd because even sums (where j>0) make either m-1 or m+1 divisible by 5, which precludes twin primes except for the case of 6, where m-1 is divisible by 5, but 5 is the only number divisible by 5 that is also prime.

			a(1) = 2 because A027856(1) = 4 = 2^2 * 3^0
a(2) = 1 because A027856(2) = 6 = 2^1 * 3^1
a(3) = 2 because A027856(3) = 12 = 2^2 * 3^1
a(4) = 1 because A027856(4) = 18 = 2^1 * 3^2
a(5) = 3 because A027856(5) = 72 = 2^3 * 3^2

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

Cf. A027856, A386730, A386731.

seq[max_] := 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]; 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, log_N))
    sorted_result = sorted(unsorted_result, key=lambda x: x[1])
    return sorted_result
print([i for i, _ in TP_pi_2_upto_sum(sum_limit) ])

A386730 a(n) is the 3-adic valuation of A027856(n).

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 1, 3, 2, 4, 7, 10, 1, 5, 15, 5, 4, 5, 4, 7, 8, 2, 9, 7, 24, 12, 8, 15, 9, 2, 25, 20, 10, 64, 63, 27, 88, 99, 2, 16, 10, 169, 135, 51, 141, 52, 231, 320, 44, 419, 143, 476, 207, 332, 97, 324, 738, 1493, 1320, 333, 1167, 188, 1440, 2251, 2033

Offset: 1

Ken Clements, Jul 31 2025

nonn

These are the exponents, j, of the prime factor 3 of the A027856 numbers m = 2^i * 3^j where m is the average of twin primes. Except for the first term, all are greater than zero because all other A027856 numbers have both 2 and 3 as prime factors. After the second term, all sums of i+j are odd because even sums make either m-1 or m+1 divisible by 5, which precludes twin primes except for the case of 6, where m-1 is divisible by 5, but 5 is the only number divisible by 5 that is also prime.

			a(1) = 0 because A027856(1) = 4 = 2^2 * 3^0
a(2) = 1 because A027856(2) = 6 = 2^1 * 3^1
a(3) = 1 because A027856(3) = 12 = 2^2 * 3^1
a(4) = 2 because A027856(4) = 18 = 2^1 * 3^2
a(5) = 2 because A027856(5) = 72 = 2^3 * 3^2

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

Cf. A027856, A385433, A386731.

seq[max_] := 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}] &], 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 = [(0, 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((j, log_N))
    sorted_result = sorted(unsorted_result, key=lambda x: x[1])
    return sorted_result
print([j for j, _ in TP_pi_2_upto_sum(sum_limit) ])

A059961 Duplicate of A027856.

Original entry on oeis.org

4, 6, 12, 18, 72, 108, 192, 432, 1152, 2592, 139968, 472392, 786432, 995328, 57395628, 63700992, 169869312, 4076863488, 10871635968, 2348273369088, 56358560858112, 79164837199872, 84537841287168, 150289495621632

Offset: 1

dead

A059960 Smaller term of a pair of twin primes such that prime factors of their average are only 2 and 3.

Original entry on oeis.org

5, 11, 17, 71, 107, 191, 431, 1151, 2591, 139967, 472391, 786431, 995327, 57395627, 63700991, 169869311, 4076863487, 10871635967, 2348273369087, 56358560858111, 79164837199871, 84537841287167, 150289495621631, 578415690713087, 1141260857376767

Offset: 1

Labos Elemer, Mar 02 2001

nonn

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

Primes p(k) such that the number of distinct prime divisors of all composite numbers between p(k) and p(k+1) is 2. - Amarnath Murthy, Sep 26 2002

			a(11)+1 = 2*2*2*3*3*3*3*3*3*3*3*3*3 = 472392.

Ray Chandler, Table of n, a(n) for n = 1..61 (terms < 10^1000, first 49 terms from T. D. Noe)

Cf. A014574, A002822, A033845, A058383, A027856.
Cf. A052297, A075581, A075580, A075583, A075584, A075585, A075586, A075587, A075588, A075589.
Apart from initial terms, same as A078883.

nn=10^15; Sort[Reap[Do[n=2^i 3^j; If[n<=nn && PrimeQ[n-1] && PrimeQ[n+1], Sow[n-1]], {i, Log[2, nn]}, {j, Log[3, nn]}]][[2, 1]]]
Select[Select[Partition[Prime[Range[38*10^5]],2,1],#[[2]]-#[[1]]==2&][[All,1]],FactorInteger[#+1][[All,1]]=={2,3}&] (* The program generates the first 15 terms of the sequence. *)
seq[max_] := Select[Sort[Flatten[Table[2^i*3^j - 1, {i, 1, Floor[Log2[max]]}, {j, 1, Floor[Log[3, max/2^i]]}]]], And @@ PrimeQ[# + {0, 2}] &]; seq[2*10^15] (* Amiram Eldar, Aug 27 2024 *)

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

A060212 Primes q such that 6q-1 and 6q+1 are twin primes. Proper subset of A002822.

Original entry on oeis.org

2, 3, 5, 7, 17, 23, 47, 103, 107, 137, 283, 313, 347, 373, 397, 443, 467, 577, 593, 653, 773, 787, 907, 1033, 1117, 1423, 1433, 1613, 1823, 2027, 2063, 2137, 2153, 2203, 2287, 2293, 2333, 2347, 2677, 2903, 3257, 3307, 3407, 3413, 3593, 3623, 3673, 3923

Offset: 1

Labos Elemer, Mar 20 2001

nonn

Primes in A182521. Also all primes p for which A182481(p)=1. - Vladimir Shevelev, May 03 2012

Conjecture: a(n) ~ n*log(n)*log(n*log(n))*log(log(n)). - Carl R. White, Nov 16 2023

David Radcliffe, Table of n, a(n) for n = 1..10000

Cf. A001359, A002822, A014574, A027856, A058383, A059960, A182481, A182521, A294731.

lst={}; Do[p=Prime[n]; If[PrimeQ[6*p-1] && PrimeQ[6*p+1], AppendTo[lst,p]], {n,100}]; lst (* Vladimir Joseph Stephan Orlovsky, Aug 16 2009 *)

forprime(p=2, 9999, if(isprime(6*p+1) & isprime(6*p-1), print(p))) \\ David Radcliffe, Apr 02 2016

from sympy import *; print([p for p in primerange(2,9999) if isprime(6*p-1) and isprime(6*p+1)]) # David Radcliffe, Apr 02 2016

A060213 Lesser of twin primes whose average is 6 times a prime.

Original entry on oeis.org

11, 17, 29, 41, 101, 137, 281, 617, 641, 821, 1697, 1877, 2081, 2237, 2381, 2657, 2801, 3461, 3557, 3917, 4637, 4721, 5441, 6197, 6701, 8537, 8597, 9677, 10937, 12161, 12377, 12821, 12917, 13217, 13721, 13757, 13997, 14081, 16061, 17417

Offset: 1

Labos Elemer, Mar 20 2001

nonn

Lowest factor-density among all positive consecutive integer triples; for p > 41, last digit of p can be only 1 or 7 (see Alexandrov link, p. 15). - Lubomir Alexandrov, Nov 25 2001

			102197 is here because 102198 = 17033*6 and 17033 is prime.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Lubomir Alexandrov, On the nonasymptotic prime number distribution, arXiv:math/9811096 [math.NT], 1998, see p. 15.

Cf. A002822, A058383, A059960, A027856, A001454, A001359, A060212.

map(t -> 6*t-1, select(p -> isprime(p) and isprime(6*p-1) and isprime(6*p+1), [2,seq(i,i=3..10000,2)]));

Transpose[Select[Partition[Prime[Range[2500]],2,1],#[[2]]-#[[1]] == 2 && PrimeQ[Mean[#]/6]&]][[1]] (* Harvey P. Dale, May 04 2014 *)

isok(n) = isprime(n) && isprime(n+2) && !((n+1) % 6) && isprime((n+1)/6); \\ Michel Marcus, Dec 14 2013

a(n) = 6 * A060212(n) - 1. - Sean A. Irvine, Oct 31 2022

Offset changed to 1 by Michel Marcus, Dec 14 2013

A384530 Intersection of A055932 and A014574.

Original entry on oeis.org

4, 6, 12, 18, 30, 60, 72, 108, 150, 180, 192, 240, 270, 420, 432, 600, 810, 1050, 1152, 1620, 2310, 2592, 3000, 3360, 4050, 4800, 5880, 6300, 7350, 7560, 8820, 9000, 9240, 9720, 10500, 11550, 15360, 21600, 23040, 25410, 26250, 26880, 28350, 29400, 30870, 33600

Offset: 1

Ken Clements, Jun 01 2025

nonn

Numbers that have a complete set of prime factors from 2 to their greatest prime factor (A055932) that are bracketed by twin primes (in A014574).

The density of twin prime numbers around terms appears to be enhanced, since the blocks of prime factors of a(n) "sweep" out possible low prime factors from a(n)-1 and a(n)+1.

			4 is a term since 4 = prime(1)# * 2 is in A055932 and 4-1 = 3 and 4+1 = 5 are both prime.
6 is a term since 6 = prime(2)# is in A055932 and 6-1 = 5 and 6+1 = 7 are both prime.
30 is a term since 30 = prime(3)# is in A055932 and 30-1 = 29 and 30+1 = 31 are both prime.
420 is a term since 420 = prime(4)# * 2 is in A055932 and 420-1 = 419 and 420+1 = 421 are both prime.

Alois P. Heinz, Table of n, a(n) for n = 1..3719

Supersequence of A027856.

Cf. A014574, A055932.

q:= n-> andmap(isprime, [n-1, n+1]) and (s-> nops(s)=
        numtheory[pi](max(s)))({ifactors(n)[2][.., 1][]}):
select(q, [$1..40000])[];  # Alois P. Heinz, Jun 01 2025

from sympy import factorint, prime, isprime
def is_pi_complete(n):  # n is a term of A055932
    if n <= 1:
        return False
    factors = factorint(n)
    primes = list(factors.keys())
    max_prime, r = max(primes), len(primes)
    return max_prime == prime(r)
def is_twin_prime_bracketed(n):  # n is a term of A014574
    return isprime(n-1) and isprime(n+1)
def aupto(limit):
    return [n for n in range(4, limit+1, 2) if is_twin_prime_bracketed(n) and is_pi_complete(n)]
print(aupto(40_000))

A060210 Largest prime factor of 1+smaller term of twin primes.

Original entry on oeis.org

2, 3, 3, 3, 5, 7, 5, 3, 17, 3, 23, 5, 5, 3, 11, 19, 5, 5, 47, 13, 29, 7, 3, 11, 29, 19, 5, 103, 107, 11, 5, 137, 23, 13, 7, 17, 43, 7, 59, 13, 3, 41, 71, 43, 31, 11, 17, 11, 19, 31, 67, 5, 139, 283, 41, 149, 13, 313, 23, 13, 37, 13, 347, 29, 11, 71, 17, 373, 7, 11, 13, 397, 17

Offset: 1

Labos Elemer, Mar 20 2001

nonn

Also: Largest prime factor of the average, or the sum, of twin prime pairs. - M. F. Hasler, Jan 03 2011

			101 is the 9th lesser twin, 102 = 2*3*17, and its max p factor is 17=a(9).

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

Cf. A001359, A001454, A002822, A059960, A027856.

FactorInteger[1+#][[-1,1]]&/@Select[Partition[Prime[Range[500]],2,1], #[[2]]- #[[1]]==2&][[All,1]] (* Harvey P. Dale, Jan 16 2017 *)

p=3; for(n=1,1e3, until(o+2==p,p=nextprime(2+o=p)); print1(vecmax(factor(p-1)[,1])","))  \\ M. F. Hasler, Jan 03 2011

A078883 Lesser member p of a twin prime pair such that p+1 is 3-smooth.

Original entry on oeis.org

3, 5, 11, 17, 71, 107, 191, 431, 1151, 2591, 139967, 472391, 786431, 995327, 57395627, 63700991, 169869311, 4076863487, 10871635967, 2348273369087, 56358560858111, 79164837199871, 84537841287167, 150289495621631, 578415690713087, 1141260857376767

Offset: 1

Reinhard Zumkeller, Dec 11 2002

nonn

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

Ray Chandler, Table of n, a(n) for n = 1..62 (terms < 10^1000)

Cf. A000040, A001359, A014574, A003586, A027856, A078884.

Apart from initial terms, same as A059960.

seq[max_] := Select[Sort[Flatten[Table[2^i*3^j - 1, {i, 1, Floor[Log2[max]]}, {j, 0, Floor[Log[3, max/2^i]]}]]], And @@ PrimeQ[# + {0, 2}] &]; seq[2*10^15] (* Amiram Eldar, Aug 27 2024 *)

a(n) = A027856(n)-1 = A078884(n)-2.

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) ])

cp's OEIS Frontend

A385433 a(n) is the 2-adic valuation of A027856(n).

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A386730 a(n) is the 3-adic valuation of A027856(n).

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A059961 Duplicate of A027856.

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A059960 Smaller term of a pair of twin primes such that prime factors of their average are only 2 and 3.

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A060212 Primes q such that 6*q-1 and 6*q+1 are twin primes. Proper subset of A002822.

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A060213 Lesser of twin primes whose average is 6 times a prime.

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A384530 Intersection of A055932 and A014574.

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A060210 Largest prime factor of 1+smaller term of twin primes.

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A060212 Primes q such that 6q-1 and 6q+1 are twin primes. Proper subset of A002822.