The On-Line Encyclopedia of Integer Sequences (OEIS)

Revisions by Amiram Eldar (See also Amiram Eldar's wiki page
and changes approved by Amiram Eldar)

(Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
Showing entries 1-10 | older changes

A306616

Integers k such that phi(Catalan(n+1)) = 4*phi(Catalan(n)) where phi is A000010 and Catalan is A000108.
(history; published version)

#31 by Amiram Eldar at Wed Nov 27 00:35:48 EST 2024

STATUS

editing

proposed

A143268

a(n) = phi(n)*T(n), where phi(n) is Euler's totient function (A000010) and T(n) = n*(n+1)/2 is the n-th triangular number (A000217).
(history; published version)

#12 by Amiram Eldar at Wed Nov 27 00:35:47 EST 2024

STATUS

editing

proposed

A062624

Number of integers less than A000108(n) relatively prime to A000108(n).
(history; published version)

#28 by Amiram Eldar at Wed Nov 27 00:35:46 EST 2024

STATUS

editing

proposed

A060607

Number of iterations of phi(x) at prime(n) needed to reach 1.
(history; published version)

#16 by Amiram Eldar at Wed Nov 27 00:35:46 EST 2024

STATUS

editing

proposed

A060606

The n-th term is the sum of lengths of iteration chains to get fixed points(=1) for the Euler totient function from 1 to n.
(history; published version)

#23 by Amiram Eldar at Wed Nov 27 00:35:45 EST 2024

STATUS

editing

proposed

#22 by Amiram Eldar at Wed Nov 27 00:33:31 EST 2024

LINKS

Hartosh Singh Bal and Gaurav Bhatnagar, <a href="https://arxiv.org/abs/1903.09619">Prime number conjectures from the Shapiro class structure</a>, arXiv:1903.09619 [math.NT], 2019. See function S(n) , p. 2.

#21 by Amiram Eldar at Wed Nov 27 00:32:48 EST 2024

NAME

The n-th term is the sum of lengths of iteration chains to get fixed points (=1) for the Euler totient function from 1 to n.

LINKS

Hartosh Singh Bal, and Gaurav Bhatnagar, <a href="https://arxiv.org/abs/1903.09619">Prime number conjectures from the Shapiro class structure</a>, arXiv:1903.09619 [math.NT], 2019. See function S(n) p. 2.

Paul Erdos, Erdős, Andrew Granville, Carl Pomerance and Claudia Spiro, <a href="/A000010/a000010_1.pdf">On the normal behavior of the iterates of some arithmetic functions</a>, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204. [Annotated copy with A-numbers]

H. Harold Shapiro, <a href="https://www.jstor.org/stable/2303988">An arithmetic function arising from Phi-function</a>, American Math. Monthly , Vol. 50:, No. 1 (1943), pp. 18-30.

#20 by Amiram Eldar at Wed Nov 27 00:31:12 EST 2024

MATHEMATICA

f[1] = 0; f[n_] := f[n] = f[EulerPhi[n]] + 1; Accumulate[Array[f, 100]] (* Amiram Eldar, Nov 27 2024 *)

#19 by Amiram Eldar at Wed Nov 27 00:30:56 EST 2024

LINKS

Amiram Eldar, <a href="/A060606/b060606_1.txt">Table of n, a(n) for n = 0..10000</a>

STATUS

approved

editing

A060607

Number of iterations of phi(x) at prime(n) needed to reach 1.
(history; published version)

#15 by Amiram Eldar at Wed Nov 27 00:26:50 EST 2024

FORMULA

a(n) = A003434(A006093(n)) + 1. - _Amiram Eldar_, Nov 27 2024