1,1

Analogous to Niven's constant for shifted primes p-1 and shifted primes p+1.

The formula for calculating this constant (the third in the Formula section) can be derived from the asymptotic density given by Mirsky (1949, Theorem 2, p. 19), in a similar way in which Niven's constant is calculated (Niven, 1969).

In general, for shifted primes p + k or p - k, where k is any positive integer, the asymptotic mean can be calculating in the same way, with restricting the product to be over all the primes that do not divide k.

Let c(k) be the asymptotic mean corresponding to primes shifted by k (either p-k or p+k). Then, c(k) attains its maximal value, this constant, at k = 1. c(k) depends only on the distinct primes dividing k, i.e., c(k) = c(rad(k)), where rad = A007947 is the squarefree kernel function. The values of k in which c(k) attains record low values are the primorials (A002110). The values of k in which c(k) attains record values of proximity to 2 are the half primorials (A070826).

Amiram Eldar, Analogs of Niven's Constant for Shifted Primes, includes Pari code and various values of c(k).

Leon Mirsky, The number of representations of an integer as the sum of a prime and a k-free integer, American Mathematical Monthly, Vol. 56, No. 1 (1949), pp. 17-19.

Ivan Niven, Averages of Exponents in Factoring Integers, Proc. Amer. Math. Soc., Vol. 22, No. 2 (1969), pp. 356-360.

Equals lim_{m->oo} (1/m) * Sum_{k=1..m} A023504(k).

Equals lim_{m->oo} (1/m) * Sum_{k=1..m} A023510(k).

Equals 1 + Sum_{k>=1} (1 - Product_{p prime} (1 - 1/((p-1)*p^k))).

2.20635569921538156547280650209370983414264310952472...

(PARI) f(k) = 1 - prodeulerrat(1 - 1/((p-1)*p^k));

1 + sumpos(k = 1, f(k))

Cf. A005117, A002110, A007947, A023504, A023510, A033150, A051903, A070826.

Sequence in context: A127467 A338001 A271708 * A284983 A140333 A182062

Adjacent sequences: A392815 A392816 A392817 * A392819 A392820 A392821

nonn,cons

Amiram Eldar, Jan 24 2026

approved