cp's OEIS Frontend

Numbers k such that 24k + 11 and 24k + 13 are both in A001122. See A319248 and A319249 for detailed information.

The highly composite numbers are listed in A068507, but their growth is such that one cannot list the terms beyond A002182(362), corresponding to a(17), in the DATA section.

The term a(21) corresponds to A002182(10535) = A108951(52900585920). - Daniel Suteu, Jun 27 2019

a(22) > 779674, if it exists. - Amiram Eldar, Dec 03 2020

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

Characteristic function of A002822. - Georg Fischer, Aug 04 2021

a(n) exists for every n>=2.

Conjecture: (i) a(n) > 0 for all n > 1. Also, any n > 12 can be written as k + m (k > 0 and m > 2) with 6*k - 1, 6*k + 1 and k + phi(m)/2 all prime.

(ii) Each integer n > 34 can be written as p + q (q > 0) with p and p + phi(q) both prime. Also, any integer n > 14 can be written as p + q (q > 2) with p, p + 6 and p + phi(q)/2 all prime.

Clearly, part (i) of the conjecture implies that any integer n > 1 can be written as p + m - phi(m), where p is a prime and m is a positive integer.

There are 146 terms below 10^3, 831 terms below 10^4, 5345 terms below 10^5, 37788 terms below 10^6 and 280140 terms below 10^7.

Prime pairs differing by 6 are called "sexy" primes. Other prime pairs with difference 6 are of the form 6n + 1 and 6n + 7.

Numbers in this sequence are those which are not 6cd + c - d - 1, 6cd + c - d, 6cd - c + d - 1 or 6cd - c + d, that is, they are not (6c - 1)d + c - 1, (6c - 1)d + c, (6c + 1)d - c - 1 or (6c + 1)d - c.

There are 138 such numbers between 1 and 1000.

Prime pairs that differ by 6 are called "sexy" primes. Other prime pairs that differ by 6 are of the form 6n - 1 and 6n + 5.

Numbers in this sequence are those which are not 6cd - c - d - 1, 6cd - c - d, 6cd + c + d - 1 or 6cd + c + d, that is, they are not (6c - 1)d - c - 1, (6c - 1)d - c, (6c + 1)d + c - 1 or (6c + 1)d + c.

There are 140 such numbers between 1 and 1000.

These numbers correspond to all the prime pairs which differ by 8 except 3 and 11.

Numbers in this sequence are those which are not 6cd - c - d - 1, 6cd + c - d, 6cd - c + d or 6cd + c + d - 1, that is, they are not (6c - 1)d - c - 1, (6c - 1)d + c, (6c + 1)d - c or (6c + 1)d + c - 1.

See A066272 for definition of anti-divisor.

The following conjectures have been proved by Bob Selcoe. - Michael Somos, Feb 28 2014

Additional conjectures suggested by computational experiments:

1) Numbers all of whose anti-divisors (AD's) are odd => {2^k} (A000079).

2) Numbers with AD 2, all other AD's odd => primes (A000040).

3) Numbers none of whose AD's are multiples of 3 => 3*2^k (A007283).

4) Numbers all of whose AD's are even => 3*A002822 = A040040 (except for a(0)=1), both related to twin prime pairs.

Calculations suggest the following conjecture. This sequence consists of all odd primes and nonnegative powers of 2 and no other terms. This has been verified for to n=100000. Robert G. Wilson v extended the conjecture out to 2^20.

From Bob Selcoe, Feb 24 2014: (Start)

The sequence consists of all odd primes and powers of two (>=2^2) and no other terms.

Proof: Denote the even anti-divisors of n as ADe(n). ADe(n) is defined as the set of numbers x satisfying the equation n(mod x)=x/2. Substitute x = 2n/y, since it can be shown that ADe(n) => 2n divided by the odd divisors of n when n>1 (This is because 2j anti-divides only numbers of the form 3j+2j*k; j>=1, k>=0. For example: j=7; 14 anti-divides only 21,35,49,63.... So in other words, even numbers anti-divide only odd multiples (>=3) of themselves, divided by 2). Therefore, ADe(n) is n(mod [2n/y])=n/y, and y must be an odd divisor of n and 2n, y>1. Since y is the only odd divisor of n when y>1 iff n is prime, then ADe(n) => 2 when n is prime. Since 2n has no odd divisors when n=2^k, then ADe(n) is null when n=2^k. Therefore, the only numbers whose anti-divisors are either 2 or odd must be primes and powers of 2.

Similarly, for odd anti-divisors (ADo(n)): Given 2j+1 (odd numbers) anti-divide only numbers of the forms [(3j+1)+(2j+1)*k] and [(3j+2)+(2j+1)*k]; j>=1, k>=0. (For example: j=6; 13 anti-divides only 19,20, 32,33, 45,46...). Since odd n divided by its odd divisors ARE its odd divisors, then ADo(n) => the divisors of 2n-1 and 2n+1 (except 1, 2n-1 and 2n+1).

By extension:

1) Numbers all of whose anti-divisors (AD's) are odd => {2^k} (A000079).

2) Numbers with ADe(n)=2, all other AD's odd => primes (A000040).

3) Numbers none of whose AD's are multiples of j => j*2^k.

4) When 2n-1 and 2n+1 are twin primes, (A040040, except for a(0)=1) then n has only even AD's.

(End)

If 1 and 2 are included, this sequence contains all positive integers not contained in A111774. - Bob Selcoe, Sep 09 2014 [corrected by Wolfdieter Lang, Nov 06 2020]