This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A309287 #55 Oct 27 2019 12:01:23
%S A309287 1,1,1,2,2,1,2,3,2,1,4,3,3,2,1,2,5,4,3,2,1,6,6,5,4,3,2,1,4,7,6,5,4,3,
%T A309287 2,1,6,6,7,6,5,4,3,2,1,4,8,7,7,6,5,4,3,2,1,10,10,9,8,7,6,5,4,3,2,1,4,
%U A309287 11,10,9,8,7,6,5,4,3,2,1,12,9,11,10,9,8,7,6,5,4,3,2,1,6,13,12,11,10,9,8,7,6,5,4,3,2,1
%N A309287 Square array T(v, m), read by antidiagonals, for the Rogel-Klee arithmetic function: number of positive integers h in the set [m] for which gcd(h, m) is v-th-power-free, i.e., gcd(h, m) is not divisible by any v-th power of an integer > 1 (with v, m >= 1).
%C A309287 For fixed v >= 1, T(v, .) is a multiplicative arithmetic function with T(v, 1) = 1; T(v, p^e) = p^e, if e < v; and T(v, p^e) = p^e - p^(e-v) if e >= v (where p is a prime >= 2).
%C A309287 Here, T(v=1, m) = phi(m) is the number of arithmetic progressions (s*m + k: s >= 0), k = 1, ..., m, that contain infinitely many primes (by Dirichlet's theorem). For v >= 2, T(v, m) is the number of these arithmetic progressions that contain infinitely many v-th-power-free numbers.
%C A309287 In Section 6 of his paper, Cohen (1959) mentions that this function was introduced by Rogel (1900) in an article published in a Bohemian journal. Roger's (1900) paper is a continuation of Rogel (1897) and the two should be read together.
%C A309287 McCarthy (1958) uses the asymptotic result given in the FORMULA section below to prove that the probability that the GCD of two positive integers is v-th-power-free is 1/zeta(2*v).
%D A309287 Paul J. McCarthy, Introduction to Arithmetical Functions, Springer-Verlag, 1986; see pp. 38-40 and 69.
%H A309287 Eckford Cohen, <a href="https://msp.org/pjm/1959/9-1/pjm-v9-n1-s.pdf">A class of residue systems (mod r) and related arithmetical functions. I. A generalization of the Moebius function</a>, Pacific J. Math. 9(1) (1959), 13-24; see Section 6 where T(v, m) = Phi_v(m).
%H A309287 Eckford Cohen, <a href="https://www.jstor.org/stable/2688817">A generalized Euler phi-function</a>, Math. Mag. 41 (1968), 276-279; here T(v, m) = phi_v(m).
%H A309287 E. K. Haviland, <a href="https://projecteuclid.org/euclid.dmj/1077472821">An analogue of Euler's phi-function</a>, Duke Math. J. 11 (1944), 869-872; here T(v=2, m) = rho(m).
%H A309287 V. L. Klee, Jr., <a href="https://www.jstor.org/stable/2304963">A generalization of Euler's phi function</a>, Amer. Math. Monthly, 55(6) (1948), 358-359; here T(v, m) = Phi_v(m).
%H A309287 Paul J. McCarthy, <a href="https://www.jstor.org/stable/2309112">On a certain family of arithmetic functions</a>, Amer. Math. Monthly 65 (1958), 586-590; here, T(v, m) = T_v(m).
%H A309287 Franz Rogel, <a href="https://www.zobodat.at/pdf/SB-Ges-Wiss-Prag_1896_2_0001-0687.pdf">Entwicklung einiger zahlentheoreticher Funktionen in unendliche Reihen</a>, S.-B. Kgl. Bohmischen Ges. Wiss. Article XLVI/XLIV (1897), Prague (26 pages). [This paper deals with arithmetic functions, especially the Euler phi function. It was continued three years later with the next paper, which contains his function phi_k(n). As stated at the end of the volume, in the table of contents, there is a mistake in numbering the article, so two Roman numerals appear in the literature for labeling this article!]
%H A309287 Franz Rogel, <a href="https://archive.org/details/vestnkkrlovs1900kn/page/n863">Entwicklung einiger zahlentheoreticher Funktionen in unendliche Reihen</a>, S.-B. Kgl. Bohmischen Ges. Wiss. Article XXX (1900), Prague (9 pages). [This is a continuation of the previous article, which was written three years earlier and has the same title. The numbering of the equations continues from the previous paper, but this paper is the one that introduces the function phi_k(n). In our notation, T(v, m) = phi_v(m). Cohen (1959) refers to this paper and correctly attributes this function to F. Rogel.]
%F A309287 T(v, m) = m * Product_{p prime and p^v|m} (1 - p^(-v)) for v, m >= 1.
%F A309287 T(v, m) = Sum_{n >= 1} mu(n) * [m, n^v] * (m/n^v), where [m, n^v] = 1 when m is a multiple of n^v, and = 0 otherwise. [This is Eq. (53) in Rogel (1900) and Eq. (6.1) in Cohen (1959).]
%F A309287 Dirichlet g.f. for row v: Sum_{m >= 1} T(v, m)/m^s = zeta(s-1)/zeta(v*s) for Re(s) > 1.
%F A309287 Asymptotics: Sum_{m = 1..n} T(v, m) = n^2/(2*zeta(2*v)) + O(n) for v >= 2 and = n^2/(2*zeta(2)) + O(n*log(n)) for v = 1 (for Euler's phi-function).
%F A309287 Analog of Fermat's theorem: if gcd(a, m) = 1 with a >= 1, then m/gcd(a^T(v, m) - 1, m) is v-th-power-free. (For v = 1, this means m/gcd(a^T(v=1, m) - 1, m) = 1.)
%F A309287 T(v, m^v)/m^v = Sum_{d|m} mu(d)/d^v for m, v >= 1. (It generalizes the formula phi(m)/m = Sum_{d|m} mu(d)/d since phi(m) = T(v=1, m).)
%e A309287 Table for T(v, m) (with rows v >= 1 and columns m >= 1) begins as follows:
%e A309287 v=1: 1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4, 12, 6, 8, 8, ...
%e A309287 v=2: 1, 2, 3, 3, 5, 6, 7, 6, 8, 10, 11, 9, 13, 14, 15, 12, ...
%e A309287 v=3: 1, 2, 3, 4, 5, 6, 7, 7, 9, 10, 11, 12, 13, 14, 15, 14, ...
%e A309287 v=4: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 15, ...
%e A309287 v=5: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, ...
%e A309287 v=6: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, ...
%e A309287 v=7: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, ...
%e A309287 ...
%e A309287 Clearly, lim_{v -> infinity} T(v, m) = m.
%o A309287 (PARI) /* Modification of _Michel Marcus_'s program from sequence A254926: */
%o A309287 T(v, m) = {f = factor(m); for (i=1, #f~, if ((e=f[i, 2])>=v, f[i, 1] = f[i, 1]^e - f[i, 1]^(e-v); f[i, 2]=1); ); factorback(f); }
%o A309287 /* Print the first 40 terms of each of the first 10 rows: */
%o A309287 { for (v=1, 10, for (m=1, 40, print1(T(v, m), ", "); ); print(); ); }
%Y A309287 A000010 (row v = 1 is Euler's phi function), A063659 (row v = 2 is Haviland's function), A254926 (row v = 3).
%K A309287 nonn,tabl
%O A309287 1,4
%A A309287 _Petros Hadjicostas_, Jul 21 2019