cp's OEIS Frontend

From Omar E. Pol, Dec 21 2021: (Start)

Also triangle read by rows: T(n,j) = A000040(n-j+1)*A000203(j), 1 <= j <= n.

For a visualization of T(n,j) first consider a tower (a polycube) in which the terraces are the symmetric representation of sigma(j), for j = 1 to n, starting from the top, and the heights of the terraces are the first n prime numbers respectively starting from the base. Then T(n,j) can be represented with a set of A237271(j) right prisms of height A000040(n-j+1) since T(n,j) is also the total number of cubes that are exactly below the parts of the symmetric representation of sigma(j) in the tower.

The sum of the n-th row of triangle is A086718(n) equaling the volume of the tower whose largest side of the base is n and its total height is the n-th prime.

The tower is an member of the family of the stepped pyramids described in A245092 and of the towers described in A221529. That is an infinite family of symmetric polycubes whose volumes represent the convolution of A000203 with any other integer sequence. (End)

An idempotent factorization of n is a way of writing n = p*q such that b^(k(p-1)(q-1)+1) is congruent to b mod n for any integer k >= 0 and any b in Z_n. For example, p = 19, q = 15 is an idempotent factorization of n = 285. All factorizations of semiprimes are idempotent, so this sequence is restricted to n with >= 3 factors. Idempotent factorizations have the property that p and q generate correctly functioning RSA keys, even if one or both are composite.

We show in the reference below that a bipartite factorization of a squarefree integer n = pq is idempotent if and only if lambda(pq) divides (p-1)(q-1).

(p and q are not required to be primes. - N. J. A. Sloane, Feb 08 2019)

The lowermost conspicuous horizontal line in the scatter plot (at about log 3) is caused by value 1020, which corresponds to the prime signature 30 (p*q*r) and deficiency -12 packed together with the pairing function (as A002260(1020) = 30 and A004736(1020) = 16, A286449(24) = 16 and A033879(24) = -12). This value occurs in this sequence (at least) in the positions given by A138636, from its third term 30 onward.

Fully composite idempotent factorizations are bipartite factorizations n=p*q such that p and q are composite numbers with the property that for any b in Z_n, b^(k(p-1)(q-1)+1) is congruent to b mod n for any integer k >= 0. Idempotent factorizations have the property that p and q generate correctly functioning RSA keys, even if one or both are composite.

2730 has more than one fully composite idempotent factorization (10*273, 21*130). It is the smallest positive integer with that property. 7770 and 8463 are similar.

T(n, k) is the k-superdiagonal sum of an n X n Toeplitz matrix M(n) whose first row consists of successive prime numbers prime(1), ..., prime(n).

The h-th subdiagonal of the triangle T gives the primes multiplied by (h + 1).

The k-th column of the triangle T gives the multiples of prime(1 + k).

Also array A(n, k) = n*prime(1 + k) read by ascending antidiagonals, with 0 <= k < n. - Michel Marcus, Jul 15 2019

An integer n has an idempotent factorization n=pq if b^(k(p-1)(q-1)+1) is congruent to b mod n for any integer k >= 0 and any b in Z_n (see A306330). An integer is maximally idempotent if all its bipartite factorizations n=pq are idempotent.

There are 15506 maximally idempotent integers less than 2^30. 15189 have three factors, 315 have four, two have five. The smallest maximally idempotent integer with four factors is 63973=7*13*19*37, a Carmichael number. The two with five factors are 13*19*37*73*109 and 11*31*41*101*151. The smallest maximally idempotent integer with six factors is 11*31*41*61*101*151.

Let m, n >= 1 and let f(m) denote number of Bernoulli numbers less than or equal to 10^m having denominator divisible by a(n). For any n, f(m) = floor(10^m/(a(n)/6 - 1)). It appears that the fraction of even Bernoulli numbers with denominator 6 is not so close to 1/6.