A007614 -id:A007614 - cp's OEIS frontend

A064275 Inverse of sequence A032447 considered as a permutation of the positive integers.

1, 2, 3, 4, 6, 5, 10, 7, 11, 8, 19, 9, 21, 12, 14, 15, 27, 13, 33, 16, 22, 20, 42, 17, 37, 23, 34, 24, 54, 18, 56, 28, 38, 29, 44, 25, 65, 35, 45, 30, 73, 26, 82, 39, 46, 43, 89, 31, 83, 40, 58, 47, 102, 36, 74, 48, 66, 55, 109, 32, 111, 57, 67, 59, 91, 41, 128, 60, 86, 49, 130, 50, 132, 68, 75, 69, 112, 51, 149, 61

Offset: 1

Howard A. Landman, Sep 23 2001

Cf. A032447, A007614.

import Data.List (elemIndex); import Data.Maybe (fromJust)
a064275 = (+ 1) . fromJust . (`elemIndex` a032447_list)
-- Reinhard Zumkeller, Nov 22 2015

Edited by N. J. A. Sloane, Nov 05 2007

More terms from Alois P. Heinz, Jan 20 2011

A262406 Squarefree k such that phi(k) is a perfect square.

Original entry on oeis.org

1, 2, 5, 10, 17, 34, 37, 57, 74, 85, 101, 114, 170, 185, 197, 202, 219, 257, 273, 285, 370, 394, 401, 438, 451, 489, 505, 514, 546, 570, 577, 629, 677, 679, 802, 902, 969, 978, 985, 1010, 1057, 1095, 1154, 1258, 1285, 1297, 1354, 1358, 1365

Offset: 1

Charles R Greathouse IV, Oct 13 2015

nonn

The subsequence of primes is A002496 (primes of the form k^2+1). - Michel Marcus, Oct 14 2015

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
W. D. Banks, J. B. Friedlander, C. Pomerance and I. E. Shparlinski, Multiplicative structure of values of the Euler function, in High Primes and Misdemeanours: Lectures in Honour of the Sixtieth Birthday of Hugh Cowie Williams (A. Van der Poorten, ed.), Fields Inst. Comm. 41 (2004), pp. 29-47.

Intersection of A039770 and A005117.

Cf. A000010, A002496, A007614, A068560, A004171, A114063.

[n: n in [1..1400] | IsSquarefree(n) and IsSquare(EulerPhi(n))]; // Vincenzo Librandi, May 05 2016

Select[Range[1500], SquareFreeQ[#] && IntegerQ @ Sqrt @ EulerPhi[#] &] (* Amiram Eldar, Jul 16 2022 *)

is(n)=my(f=factor(n)); issquare(eulerphi(f)) && (n==1 || vecmax(f[,2])==1)

Banks, Friedlander, Pomerance, and Shparlinski show that a(n) = O(n^1.421).

A253215 a(n) is the greatest positive integer m such that phi(m) <= n where phi is Euler's totient function.

Original entry on oeis.org

2, 6, 6, 12, 12, 18, 18, 30, 30, 30, 30, 42, 42, 42, 42, 60, 60, 60, 60, 66, 66, 66, 66, 90, 90, 90, 90, 90, 90, 90, 90, 120, 120, 120, 120, 126, 126, 126, 126, 150, 150, 150, 150, 150, 150, 150, 150, 210, 210, 210, 210, 210, 210, 210, 210

Offset: 1

Jean-François Alcover, Jan 08 2015

nonn

If all duplicates are removed the result is A036913. The indices where a(n) takes a new value are A036912. - Jeppe Stig Nielsen, Sep 28 2021

Jean-François Alcover, Table of n, a(n) for n = 1..1000
MathOverflow, The maximum of the preimage of [1,x] through Euler's totient function
Eric Weisstein's World of Mathematics, Totient Function
Wikipedia, Euler's totient function

Cf. A000010, A007614, A032447, A036912, A036913, A057635.

inversePhi[m_?EvenQ] := Module[{p, nmax, n, nn}, p = Select[Divisors[m]+1, PrimeQ]; nmax = m*Times @@ (p/(p-1)); n = m; nn = {}; While[n <= nmax, If[EulerPhi[n] == m, AppendTo[nn, n]]; n++]; nn]; a[1] = 2; a[n_?OddQ] := a[n-1]; a[n_] := a[n] = Module[{m}, m = inversePhi[n] // Max; If[m > a[n-1], m, a[n-1]]]; Table[a[n], {n, 1, 100}]

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A064275 Inverse of sequence A032447 considered as a permutation of the positive integers.

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A262406 Squarefree k such that phi(k) is a perfect square.

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A253215 a(n) is the greatest positive integer m such that phi(m) <= n where phi is Euler's totient function.

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