A291785 -id:A291785

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A291784

a(n) = (psi(n) + phi(n))/2.

+10
15

1, 2, 3, 4, 5, 7, 7, 8, 9, 11, 11, 14, 13, 15, 16, 16, 17, 21, 19, 22, 22, 23, 23, 28, 25, 27, 27, 30, 29, 40, 31, 32, 34, 35, 36, 42, 37, 39, 40, 44, 41, 54, 43, 46, 48, 47, 47, 56, 49, 55, 52, 54, 53, 63, 56, 60, 58

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

This is (A001615 + A000010)/2. It is easy to see that this is always an integer.

If n is a power of a prime (including 1 and primes), then a(n) = n, and in any other case a(n) > n. - M. F. Hasler, Sep 09 2017

If n is in A006881, then a(n)=n+1. - Robert Israel, Feb 10 2019

REFERENCES

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004. See Section B41 (page 96 of 2nd ed., pages 147ff of 3rd ed.).

LINKS

Hugo Pfoertner, Table of n, a(n) for n = 1..10000

Marcin Mazur and Bogdan V. Petrenko, Generalizations of Arnold's version of Euler's theorem for matrices, Japanese Journal of Mathematics, 5:183-189, 2010.

N. J. A. Sloane, Three (No, 8) Lovely Problems from the OEIS, Experimental Mathematics Seminar, Rutgers University, Oct 05 2017, Part I, Part 2, Slides. (Mentions this sequence)

FORMULA

Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c = 21/(4*Pi^2) = 0.531936... . - Amiram Eldar, Dec 05 2023

MAPLE

f:= proc(n) local P, p;

P:= numtheory:-factorset(n);

n*(mul((p-1)/p, p=P) + mul((p+1)/p, p=P))/2

end proc:

map(f, [$1..100]); # Robert Israel, Feb 10 2019

MATHEMATICA

psi[n_] := If[n == 1, 1, n*Times @@ (1 + 1/FactorInteger[n][[All, 1]])];

a[n_] := (psi[n] + EulerPhi[n])/2;

Array[a, 100] (* Jean-François Alcover, Feb 25 2019 *)

PROG

(PARI) A291784(n)=(eulerphi(n)+n*sumdivmult(n, d, issquarefree(d)/d))\2 \\ M. F. Hasler, Sep 03 2017

CROSSREFS

Cf. A000010, A001615, A006881, A291785, A291786, A291787, A291788.

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, Sep 02 2017

STATUS

approved

A291787

Trajectory of 45 under repeated application of the map k -> A291784(k).

+10
8

45, 48, 56, 60, 80, 88, 92, 94, 95, 96, 112, 120, 160, 176, 184, 188, 190, 216, 252, 324, 378, 486, 567, 594, 738, 876, 1032, 1224, 1488, 1776, 2112, 2624, 2656, 2672, 2680, 2976, 3552, 4224, 5248, 5312, 5344, 5360, 5952, 7104, 8448, 10496, 10624

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

It may be that every trajectory under iteration of the map k -> A291784(k) which increases indefinitely will eventually merge with this sequence. This is certainly true for the terms 45 through 152 of A291788. - N. J. A. Sloane, Sep 24 2017

REFERENCES

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004. See Section B41, p. 147.

LINKS

N. J. A. Sloane, Table of n, a(n) for n = 0..10000

N. J. A. Sloane, Three (No, 8) Lovely Problems from the OEIS, Experimental Mathematics Seminar, Rutgers University, Oct 05 2017, Part I, Part 2, Slides. (Mentions this sequence)

C. R. Wall, Unbounded sequences of Euler-Dedekind means, Amer. Math. Monthly, 92 (1985), 587.

FORMULA

a(n) = 2*a(n-7) for n >= 35, which proves this is unbounded. [Guy, Wall]

CROSSREFS

Cf. A000010, A001615, A291784, A291785, A291786, A291788.

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Sep 02 2017

EXTENSIONS

More terms from Hugo Pfoertner, Sep 03 2017

STATUS

approved

A291786

a(n) = number of iterations of k -> (psi(k)+phi(k))/2 (A291784) needed to reach a prime or a power of a prime or 1, or -1 if that doesn't happen.

+10
5

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 3, 0, 2, 1, 0, 0, 3, 0, 2, 2, 1, 0, 6, 0, 1, 0, 5, 0, 4, 0, 0, 9, 8, 7, 6, 0, 5, 4, 3, 0, 5, 0, 2, -1, 1, 0, -1, 0, -1, 6, 5, 0, 4, -1, -1, 2, 1, 0, -1, 0, 4, 3, 0, 3, 2, 0, -1, -1, -1, 0, -1, 0

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,12

COMMENTS

Primes and prime powers are fixed points under the map f(k) = (psi(k)+phi(k))/2, so in that case we take a(n)=0. (If n = p^k, then psi(n) = p^k(1+1/p), phi(n) = p^k(1-1/p), and their average is p^k, so n is a fixed point under the map.)

Since f(n)>n if n is not a prime power, there can be no nontrivial cycles.

Wall (1985) observes that the trajectories of 45 and 50 are unbounded, so a(45) = a(50) = -1. See A291787, A291788.

REFERENCES

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004. See Section B41, p. 147.

LINKS

Table of n, a(n) for n=1..73.

C. R. Wall, Unbounded sequences of Euler-Dedekind means, Amer. Math. Monthly, 92 (1985), 587.

FORMULA

a(n) = 0 iff n is in A000961. - M. F. Hasler, Sep 03 2017

PROG

(PARI) A291786(n, L=n)=n>1&&for(i=0, L, isprimepower(n)&&return(i); n=A291784(n)); -(n>1) \\ The suggested search limit L=n is only empirical and might require revision. The code also currently assumes that the prime powers are the only cycles. - M. F. Hasler, Sep 03 2017

CROSSREFS

Cf. A000010, A001615, A291784, A291785, A291787, A291788.

KEYWORD

sign,more

AUTHOR

N. J. A. Sloane, Sep 02 2017

EXTENSIONS

Initial terms corrected and more terms from M. F. Hasler, Sep 03 2017

STATUS

approved

A291788

Numbers n whose trajectory under the map k -> (psi(k)+phi(k))/2 (A291784) grows without limit.

+10
4

45, 48, 50, 55, 56, 60, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 102, 104, 105, 108, 111, 112, 115, 116, 117, 118, 119, 120, 122, 123, 124, 126, 133, 134, 135, 136, 140, 141, 142, 143, 144, 145, 146, 147, 152

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

See A291787 (where A291787(m) = 2*A291787(m-7) for m >= 35) for the trajectory of 45.

There is a similar proof that all the terms from 48 though 152 have a trajectory that merges with the trajectory of 45, and so doubles every 7 steps after a certain point. For example, the trajectory of 152 reaches 2^106*33 at step 390, is 2^107*33 at step 397, and thereafter doubles every 7 steps.- N. J. A. Sloane, Sep 24 2017

LINKS

Table of n, a(n) for n=1..59.

CROSSREFS

Cf. A291784, A291785, A291786, A291787.

KEYWORD

nonn,more

AUTHOR

N. J. A. Sloane, Sep 03 2017, based on data supplied by Hans Havermann.

EXTENSIONS

Terms 104 to 152 added by N. J. A. Sloane, Sep 24 2017

STATUS

approved

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