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These are points at which the second differences (A376596) are zero.

Inclusive means 1 is a prime-power. For the exclusive version, subtract 1 and shift left.

The name has been edited to clarify that the indices k refer to A000961 ("powers of primes" = {1} U A246655) and not to the list A246655 of proper prime powers. - M. F. Hasler, Jun 16 2021

These are points at which the second differences (A376596) are nonzero.

Inclusive means 1 is a prime-power. For the exclusive version, subtract 1 from all terms.

The length of row n is (A000961(n) + 1)*A335866(n) = {2, 6, 16, 10, 60, 96, ...}. Every representative difference set begins with 0, 1, ... .

A simple difference set of Singer type of order m in the additive group (Z_v(m),+), with complete residue system modulo v(m) chosen as RS(v(m)) = {0, 1, ..., v(m)-1}, where v(m) := m^2 + m + 1, is denoted by (v(m), m+1, 1), provided m = m(n) = A000961(n) (powers of primes), for n >= 1. It is defined by a set of m+1 integers {a_0, a_1, ..., a_m}, with a_j from RS(v(m)) such that the m*(m+1) differences d_{i, j} := a_i - a_j (mod v(m)), with i not j, give (in some order) the nonzero members of RS(v(m)) i.e., {1,2, ..., (m+1)*m} exactly once. (v is a notation used in block designs, originating from 'variety'; see the Stinson reference, p. 2.)

A representative difference set of this type is one which uses 0 and 1 as elements. By adding each element of a given representative difference set by 1, 2, ..., (m+1)*m, modulo m^2 + m + 1, all m^2 + m + 1 members of a class of difference sets are obtained. This equivalence class is called Dev(D), the development of a difference set $D$. The number of representative difference sets, hence the number of classes, has been conjectured by Singer, and is given in A335866(n). The difference sets will here be ordered increasingly. The set of all A335866(n) representative difference sets of this type will here be denoted by Sr(m) (also increasingly ordered). See the W. Lang link for these representatives of order m = m(n) = A000961(n), for n = 1, 2, ..., 11. The set of all difference sets of order m will be denoted by S(m).

A symmetric BIBD (Balanced Incomplete Block Design) is a block design (X, A) (X a set of points, A a set of nonempty subsets of A, called blocks), denoted by (v, k, lambda) with v = |A|, k = |B_i|, v > k >= 2, for each block B_i from A, and each pair of distinct points of X appears exactly in lambda blocks. The number of appearances of each point of X (the replication number) is r = lambda*(v-1)/(k-1), and the number of blocks is b = v*r/k. For symmetric BIBDs v = b. See, e.g., the Stinson reference, p. 2 and pp. 41 - 58.

A symmetric and simple (lambda = 1) BIBD (symsBIBD) (m^2+m+1, m+1, 1), with k = r = m+1 and b = v = m*(m+1) + 1, is called a projective plane of order m, if m >= 2. The trivial case (3, 2, 1) for m = 1 (a triangle) is not regarded as a projective plane. m = 2 gives the Fano plane (7, 3, 1). See Stinson, p. 27, and the links. Not all m values allow such a symsBIBD. Singer proved that for m a power of a prime (including 1) such symsBIBD exist.

See the Singer reference, Theorem, pp. 380-381, where this is called a perfect difference set of order m + 1 (not m like here, and in Stinson). There only one representative is given for allowed m values. The other ones can be obtained by using certain multipliers M from the restricted residue system RRS(v = m^2+m+1), and omitting powers of divisors of m, applied to each entry, taken modulo v(m). There are A335866(n) - 1 other representative difference sets. For more details see Stinson, sect. 3.4., pp. 54-58. In the W. Lang link all representative difference sets for m = 1, 2, 3, 4, 5, 7, 8, 9, 11, 13 and 16 are given, with explanations on how to find them using Stinson's approach.

A general card game Dobble (see the Goertz link) can use v = m*(m+1) + 1 cards with k = m+1 distinct symbols from a repertoire of v distinct symbols, where m is a power of a prime. The task is to find out the one common symbol of any pair of cards. E.g., m = 7, v = 57, k = 8. One can compose 12 = A335866(6) possible such 57 card decks with different distributions of the 8 from 57 symbols.

Except for the initial term a(1)=2, indices k such that A020513(k)=Phi[k](-1) is prime, where Phi is a cyclotomic polynomial.

This is illustrated by the PARI code, although it is probably more efficient to calculate a(n) as 2*A000961(n).

{ a(n)/2 ; n>1 } are also the indices for which A020500(k)=Phi[k](1) is prime.

A188666(k) = A000961(k+1) for k: a(k) <= k < a(k+1), k > 0;

A188666(a(n)) = A000961(n+1). [Reinhard Zumkeller, Apr 25 2011]

{T(n,k): k = 1..A073093(n)} subset of {A027750(n,k): k = 1..A000005(n)} for all n.

The subsequence of prime partial sums of prime powers begins: 3, 79, 163, 499, 947, 1279, 5297. What is the smallest value which is a prime power p^k for k>1? [Jonathan Vos Post, Feb 11 2010]