A054272 -id:A054272 - cp's OEIS frontend

A283427 a(n) is the number of consecutive smallest prime totatives of primorial A002110(n).

0, 1, 7, 26, 34, 55, 65, 91, 137, 152, 208, 251, 270, 315, 394, 471, 502, 591, 656, 685, 790, 864, 977, 1139, 1227, 1268, 1354, 1395, 1494, 1847, 1945, 2109, 2157, 2455, 2512, 2693, 2878, 3005, 3202, 3396, 3471, 3826, 3902, 4045, 4119, 4581, 5059, 5226, 5307

Offset: 1

Jamie Morken and Michael De Vlieger, May 15 2017

Let p_n# = A002110(n) be the n-th primorial, and let t be a totative of p_n#, i.e., gcd(t, p_n#) = 1. Let q be the smallest prime totative of p_n#. We know q must be p_(n+1) by the definition of "primorial" as the product of the smallest n primes. This is the starting point of the range of primes we are considering. The ending point is the smallest composite totative, which is a square semiprime. This semiprime in fact must be q^2, since q is the smallest prime totative of p_n#. Stated in terms of prime n, the range we are considering are primes p_(n+1) <= t <= prevprime((p_(n+1))^2). For the smallest primorials, q^2 > p_n# with n <= 3. Thus a(n) < A054272(n) for n <= 3.

			a(2) = pi(min(prime(3)^2, p_2#)) - 2 = pi(min(25,6)) - 2 = 3 - 2 = 1.
a(4) = pi(min(prime(5)^2, p_4#)) - 4 = pi(min(121,210)) - 4 = 30 - 4 = 26.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000849, A054272, A283425.

Table[PrimePi[Min[Prime[n + 1]^2, Product[Prime@ i, {i, n}]]] - n, {n, 49}] (* Michael De Vlieger, May 16 2017 *)

a(n) = pi(min(prime(n+1)^2, Product_{k=1..n} ( prime(k) ) )) - n.

A308777 Number of twin primes between p and p^2 (inclusive) where p is the n-th prime.

Original entry on oeis.org

1, 3, 6, 9, 16, 19, 32, 35, 42, 58, 61, 82, 96, 101, 122, 148, 174, 183, 220, 242, 247, 276, 304, 332, 374, 404, 417, 436, 447, 468, 552, 576, 630, 641, 730, 749, 788, 822, 864, 910, 960, 985, 1082, 1095, 1134, 1149, 1252, 1370, 1416, 1433, 1464, 1528, 1545, 1636, 1702

Offset: 1

Michel Marcus, Jun 24 2019

nonn

Similar sequences given in cross-references have further information and references; in particular A273257 has much more efficient PARI code. - M. F. Hasler, Jun 27 2019

			There is a single twin prime (3) between 2 and 4, so a(1) = 1.
There are 3 twin primes (3, 5 and 7) between 3 and 9, so a(2) = 3.

Chai Wah Wu, Table of n, a(n) for n = 1..7944
Jon S. Birdsey, Geza Schay, A Sieve for Twin Primes, arXiv:1906.09220 [math.NT], 2019.

Cf. A001097 (twin primes), A054272, A057767 (twin pairs between p(n)^2 and p(n+1)^2), A088019.

Cf. A143738 (twin primes between n and n^2), A273257 (twin pairs between prime(n) and prime(n)^2).

a:= n-> (p-> add(`if`(isprime(j) and (isprime(j-2) or
        isprime(j+2)), 1, 0), j=p..p^2))(ithprime(n)):
seq(a(n), n=1..55);  # Alois P. Heinz, Jun 25 2019

a[n_] := With[{p = Prime[n]}, Sum[Boole[PrimeQ[k] && (PrimeQ[k-2] || PrimeQ[k+2])], {k, p, p^2}]];
Array[a, 55] (* Jean-François Alcover, Feb 29 2020 *)

a(n) = my(p=prime(n)); sum(k=p, p^2, isprime(k) && (isprime(k-2) || isprime(k+2)));

from sympy import prime, prevprime, nextprime
def A308777(n):
    if n == 1:
        return 1
    c, p = 0, prime(n)
    p2, x = p**2, [prevprime(p), p , nextprime(p)]
    while x[1] <= p2:
        if x[1] - x[0] == 2 or x[2] - x[1] == 2:
            c += 1
        x = x[1:] + [nextprime(x[2])]
    return c # Chai Wah Wu, Jun 25 2019

A359632 Sequence of gaps between deletions of multiples of 7 in step 4 of the sieve of Eratosthenes.

Original entry on oeis.org

12, 7, 4, 7, 4, 7, 12, 3, 12, 7, 4, 7, 4, 7, 12, 3, 12, 7, 4, 7, 4, 7, 12, 3, 12, 7, 4, 7, 4, 7, 12, 3, 12, 7, 4, 7, 4, 7, 12, 3, 12, 7, 4, 7, 4, 7, 12, 3, 12, 7, 4, 7, 4, 7, 12, 3, 12, 7, 4, 7, 4, 7, 12, 3, 12, 7, 4, 7, 4, 7, 12, 3, 12, 7, 4, 7, 4, 7, 12, 3

Offset: 1

Alexandre Herrera, Jan 08 2023

This sequence is a repeating cycle 12, 7, 4, 7, 4, 7, 12, 3 of length A005867(4) = 8 = (prime(1)-1)*(prime(2)-1)*(prime(3)-1).
The mean of the cycle is prime(4) = 7.
The cycle is constructed from the sieve of Eratosthenes as follows.
In the first 2 steps of the sieve, the gaps between the deleted numbers are constant: gaps of 2 in step 1 when we delete multiples of 2, and gaps of 3 in step 2 when we delete multiples of 3.
In step 3, when we delete all multiples of 5, the gaps are alternately 7 and 3 (i.e., cycle [7,3]).
For this sequence, we look at the interesting cycle from step 4 (multiples of 7).
Excluding the final 3, the cycle has reflective symmetry: 12, 7, 4, 7, 4, 7, 12. This is true for every subsequent step of the sieve too.
The central element is 7 (BUT not all steps have their active prime number as the central element).
a(1) is A054272(4).
a(8) = 3, the first appearance of the last element of the cycle, corresponds to deletion of 217 = A002110(4)+7.

			After sieve step 3, multiples of 2,3,5 have been eliminated leaving
  7,11,13,17,19,23,29,31,37,41,43,47,49,53, ...
  ^                                   ^
The first two multiples of 7 are 7 itself and 49 and they are distance 12 apart in the list so that a(1) = 12.
For n = 2, a(n) = 7, because the third multiple of 7 that is not a multiple of 2, 3 or 5 is 77 = 7 * 11, which is located 7 numbers after 49 = 7*7 in the list of numbers without the multiples of 2, 3 and 5.

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,1).

Cf. A002110, A005867, A054272, A236175.

Equivalent sequences for steps 1..3: A007395, A010701, A010705 (without the initial 3).

PadRight[{}, 100, {12, 7, 4, 7, 4, 7, 12, 3}] (* Paolo Xausa, Jul 01 2024 *)

numbers = []
for i in range(2,880):
    numbers.append(i)
gaps = []
step = 4
current_step = 1
while current_step <= step:
    prime = numbers[0]
    new_numbers = []
    gaps = []
    gap = 0
    for i in range(1,len(numbers)):
        gap += 1
        if numbers[i] % prime != 0:
            new_numbers.append(numbers[i])
        else:
            gaps.append(gap)
            gap = 0
    current_step += 1
    numbers = new_numbers
print(gaps)

a(n) = A236175(n)+1. - Peter Munn, Jan 21 2023

A058020 Difference between lcm(1,..,n) and the smallest prime > lcm(1,...,n) + 1, where n runs over A000961, lcm(n) runs through A051451.

Original entry on oeis.org

3, 5, 5, 7, 11, 13, 11, 13, 31, 23, 19, 37, 41, 29, 31, 43, 53, 41, 53, 79, 59, 97, 59, 61, 113, 97, 179, 73, 73, 97, 103, 101, 109, 101, 229, 109, 139, 113, 227, 131, 191, 163, 139, 199, 151, 139, 181, 223, 229, 367, 239, 499, 251, 509, 251, 227, 373, 281, 233

Offset: 1

Labos Elemer, Nov 14 2000

nonn

Analogous to Fortunate numbers and like them so far proved to be primes. This holds for x<=421: if Q is the first follower prime, then Q(421)-lcm(1,...421) = 557. For first some cases when 1+LCM is also a prime, the 2nd primes give 3,5,5,7,11,11,.. deviations, i.e. give primes.

Cf. A000961, A003418, A051451, A057019, A037153, A035346, A005235, A054272, A055211, A037155, A045493, A038710 etc

N=1; for(n=2,1e3, if(isprimepower(n,&p), N*=p; print1(nextprime(N+2)-N", "))) \\ Charles R Greathouse IV, Nov 18 2015

Name corrected by Charles R Greathouse IV, Nov 18 2015

A162541 Primes p such that a splitting of the cyclic group Zp by the perfect 3-shift code {+-1,+-2,+-3} exists.

Original entry on oeis.org

7, 37, 139, 163, 181, 241, 313, 337, 349, 379, 409, 421, 541, 571, 607, 631, 751, 859, 877, 937, 1033, 1087, 1123, 1171, 1291, 1297, 1447, 1453, 1483, 1693, 1741, 1747, 2011, 2161, 2239, 2311, 2371, 2473, 2539, 2647, 2677, 2707, 2719, 2857, 3169, 3361, 3433, 3511, 3547

Offset: 1

Ctibor O. Zizka, Jul 05 2009

nonn

This list was computed by S. Saidi.
From Travis Scott, Oct 04 2022: (Start)
These are also the p whose (phi/3)-th power residues have minimal bases at {1,2,3} (see under Example). Such covers {1a(n)-> {1,2,3}(n) =   7,  37,  139,  163,  181,  241, ... ~    (9*n)*log(n)
       {1,2,4}(n) =  13,  19,   61,   67,   73,   79, ... ~  (9*n/2)*log(n)
       {1,3,5}(n) =  31, 223,  229,  277,  283,  397, ... ~   (27*n)*log(n)
       {1,3,7}(n) =  43, 433,  457,  691, 1069, 1471, ... ~ (81*n/2)*log(n)
       {1,3,9}(n) = 109, 127,  157,  601,  733,  739, ... ~ (81*n/4)*log(n)
       {1,5,7}(n) = 307, 919, 1093, 2179, 2251, 3181, ... ~   (81*n)*log(n)
Note that the k-th q value takes A054272(k) x values and that a(n) = A040034(n) \ {1,2,4}(n). Following a result of Erdős (cf. A053760, A098990) the asymptotic means for q and x are Sum_{n>=1} prime(n)*2/3^n = 2.69463670741804726229622... and Sum_{n>=1} Sum_{prime(n) < k prime < prime(n)^2 OR k = prime(n)^2} D(prime(n),k)*k = 5.69767191389790422108748...
Subsequence of A040034 (2 is not a cubic residue modulo p) such that 3 is neither a residue nor in the same cubic power class as 2. (End)

			From _Travis Scott_, Oct 04 2022: (Start)
{1,2,3}^12 (mod 37) == {1,26,10} covers the 12th-power residues on Z/37Z.
{1,2,3}^14 (mod 43) == {1,1,36} misses 6. (End)

S. Saidi, Semicrosses and quadratic forms, European J. Combinatorics, vol.16, pp. 191-196, 1995. [This article has a typesetting error at the last term of this sequence on Table 1, p. 195. - _Travis Scott_, Oct 04 2022]

Subsequence of A040034.

Select[Prime@Range@497,Mod[#,3]==1&&DuplicateFreeQ@PowerMod[{1,2,3},(#-1)/3,#]&] (* Travis Scott, Oct 04 2022 *)

From Travis Scott, Oct 04 2022: (Start)
Primes of quadratic form 7x^2 +- 6xy + 36y^2 [from Saidi].
a(n) ~ 9*n*log(n). (End)

Incorrect term deleted and more terms from Travis Scott, Oct 04 2022

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A283427 a(n) is the number of consecutive smallest prime totatives of primorial A002110(n).

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A308777 Number of twin primes between p and p^2 (inclusive) where p is the n-th prime.

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A359632 Sequence of gaps between deletions of multiples of 7 in step 4 of the sieve of Eratosthenes.

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A058020 Difference between lcm(1,..,n) and the smallest prime > lcm(1,...,n) + 1, where n runs over A000961, lcm(n) runs through A051451.

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A162541 Primes p such that a splitting of the cyclic group Zp by the perfect 3-shift code {+-1,+-2,+-3} exists.

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