cp's OEIS Frontend

No further terms < 254731536.

The status of 254731536 is unknown, but conjectured not a term.

Additional terms include 2142720000, 5033164800, 150493593600, 3852635996160.

See further cycles in the linked document here below which contains 422 cycles. It includes the 80 cycles complied by Jud McCrainie in a linked document at A095955.

It is interesting to note that this sum is very close in value to 1/3 of Product_{n>=2} zeta(n)^n, A375887, where that factor's first 30 digits are: 0.333319653211135001436063576617.

It is interesting to note that this product is very close in value to 3 * Sum_{n>=2} (zeta(n)^n-1), A375920, where that factor's first 30 digits are: 3.00012312615292744064909403341.

This is the third sequence in a family defined by this same condition, when generalized as (30*k)^(2^m).

A111175 and A138220 are the first and second sequences in this family, with m = 0 and m = 1, respectively.

All values of a(n), except {1, 22}, equal 118 mod 120.

Corresponding values of p = sqrt(c+3)begin {2,5,11,19,41,59,71,79,101...}.

This relates to a comment at A047222 regarding c values for the general case of p^2-c and p^2+c both being positive primes.

All values of a(n), except {2,3}, are equal to 1 mod 30.

These are also primes p such that both p^2+c and p^2-c are positive primes, for some c, when c is a square, since that requires c = (p-1)^2. Corresponding c values begin {1, 4, 900, 108900, ...}. This relates to a comment at A047222.

m^2 = Sum_{i=k..j} prime(i)^2 is a square, for some k,j, j > k.

The 30 numbers given above are the only m values for all possible summations where the resulting m^2 < 10^14 (m <10^7). This requires searching from k values up to ~482,000, but with decreasing j-k ranges for efficiency.

Values of k that yield results begin: 13, 37, 101, 183, 235, 588, 805, 891, 1066, ... but do not correspond fully to the order of the m values shown.

Number of sequential summands (i.e., j-k+1) vary widely, with the smallest being 28 and largest being 10360, for those m values listed above.

Also note j-k+1 mod 8 = {0,1,4}, as expected, since prime(i)^2 mod 24 = 1, for i > 2.

Conjecture: For any odd prime p there is at most one odd prime q, with q < p, for which p*q-1 is square.

(Note: If p were not restricted to being prime, there could be multiple primes q which make p*q-1 a square, with increasing multiplicities for larger p. The upper limit on the multiplicity of solutions, for prime and nonprime p and q values, is given by A006278. Note, however that those limits allow for solutions where q=2, and therefore two solutions when p and q are prime.)

a(n) is a subsequence of the Pythagorean Primes (A002144), which are of the form 4k+1.

The density of a(n) values among the primes declines with increasing n. For example, a(n) is about 22% of the first 1000 primes, and drops to about 15% of "incremental" primes around prime(10000). The density continues to fall among even larger primes. Twice those percentages apply as a portion of A002144.

All values of q also belong to A002144. It appears the set of q values "intends to" fully comprise A002144. This is notable because p values comprise an increasingly sparse subsequence within A002144, and each p value has just one q value.

The ability to fully comprise A002144 with q values is further challenged by the fact that for any given q value (i.e., any term of A002144) multiple values of p > q can be found such that p*q-1 is square. Thus q values are "promiscuous", and apparently without bounds on the number of p values they can serve.

Contrast this with primes p and q such that p*q+1 is square. The result are the Twin Primes (A001359 and A006512), arranged in a simple one-to-one correspondence, with p = q+2.

A set of seven primes defined by the rules above may be called a "Prime Septet". For a given "a" value, from a(n) above, there may be multiple such septets.

These septets are found by searching all combinations of three distinct primes for a given candidate "a" value.

Once found, the following three properties of these septets are observed without exception for "a" values up to prime(800), resulting in these conjectures:

1. a-b-c = +-3.

2. b > a/2.

3. 5 <= c < a/2.

4. If a prime p >= 3 belongs to one Prime Septet, then it belongs to an infinite number of Prime Septets.

From the first conjecture it follows that no values for "a", "b" or "c" belong to A172256.

The first three conjectures can be used to accelerate the search for prime septets. Further observations below are from accelerated searches.

There are a total of 43 "missing" values in a(n), compared to the full odd prime complement of A172256, starting with 3, 5, 7, 11, 29, 31, 41, 71, ...., 10677.

There are a total of 25 missing "b" values compared to the full odd prime complement of A172256. The last missing value is 3067.

It appears the values of "c" comprise the full odd prime complement of A172256.

The three sets of values for a+b+c, a+b-c and a-b+c, by contrast, split their membership between A172256 and its odd prime complement.

Positive vs. negative values for a-b-c occur with approximately equal frequency, with positive values at 48.05% of the total for all 1106 septets with a <= prime(1000); reaching 48.30% of the total for all 65821 septets with a <= prime(10000); and reaching the 48.97% of the total for all 195359 septets with a <= prime(20000). No other members of these septets are negative.

In the first 1000 primes, 502 primes play the role of an "a" value, with an average of 4 Prime Septets subordinate to that "a" value.

In the 1000 primes from prime(9001) to prime(10000), 376 primes play the role of an "a" value, with an average of 26 Prime Septets subordinate to each "a" value, with none having only 1,2,3,4 or 5 septets.

It appears the highest "a" value having only one subordinate septet is prime(2127)=18587, where {a,b,c} = {18587, 18427, 157}. This was checked for candidate "a" values up to prime(20000). By prime(10^6) it is typical for each "a" value to support 500 to 1500 Prime Septets.

It might seem that all odd primes should belong to at least one Prime Septet. This is not the case.

The primes not included in any Prime Septet may be called "Lonely Primes". The first few such odd primes are: 151, 179, 239, 293, 313. They are a subset of A172256.

Within the first 1000 odd primes there are 259 primes that are "Lonely". Within the first 10000 odd primes there are 3978 such primes. In the 1000 primes from prime(9001) to prime(10000), 455 of them are "Lonely".

It appears likely that "Lonely Primes" are a majority among all primes.

Denominators are given by A130190, which in that entry are associated with the denominators of the z-sequence for a certain Sheffer matrix (triangle), but also apply here. See also the formula given there for the denominator of a certain sum involving the Stirling2 numbers.

Note that a(0) = 2 uses 0^0 := 1. For n >= 1 use triangle A079618 and the formula (1/e)*Sum_{k>=0} ((k+1)^n)/k! = Bell(n+1) = A000110(n+1). - Wolfdieter Lang, Feb 03 2015

If instead of Sum_{j=0..k} j^n one uses the sum with falling factorials, namely F(k, n) := Sum_{j=0..k} A008279(j, n) = A008279(k+1, n+1)/(n+1) the result for the rationals R(n) = (1/e)*Sum_{k>=0} (1/k!)*F(k, n) becomes very simple, namely R(n) = (n+2)/(n+1), n >= 0. - Wolfdieter Lang, Feb 03 2015