cp's OEIS Frontend

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.

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A139795 Least m such that k>=m implies phi(k)>=n (where phi is the Euler totient function, sequence A000010).

Original entry on oeis.org

1, 3, 7, 7, 13, 13, 19, 19, 31, 31, 31, 31, 43, 43, 43, 43, 61, 61, 61, 61, 67, 67, 67, 67, 91, 91, 91, 91, 91, 91, 91, 91, 121, 121, 121, 121, 127, 127, 127, 127, 151, 151, 151, 151, 151, 151, 151, 151, 211, 211, 211, 211, 211, 211, 211, 211, 211, 211, 211, 211, 211
Offset: 1

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Author

Benoit Jubin, May 21 2008

Keywords

Comments

Define b(n)=A006511(m)+1 where m is the unique integer such that A002202(m)A002202(m+1) (with the convention A002202(0)=A006511(0)=0). Then a(1)=b(1) and a(n+1)=max(a(n),b(n+1)).
The sequence a(n) without the repetitions is 1+A036913(n).

Examples

			a(5)=13 because if k>=13, then phi(k)>=5, but phi(12)=4.

Links

Crossrefs

Different from A137315 (see Comments in that entry).

Programs

  • PARI
    {m=0;for(n=1,100,print1(m+1,",");trap(,0,m=max(m,vecmax(invphi(n)))))}

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

Views

Author

Jean-François Alcover, Jan 08 2015

Keywords

Comments

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

Links

Crossrefs

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

Programs

  • Mathematica
    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}]

A066362 a(n) = least k > n such that phi(k) < phi(n), if such a k exists; otherwise a(n) = 0.

Original entry on oeis.org

0, 0, 0, 0, 6, 0, 8, 0, 10, 0, 12, 0, 14, 0, 18, 18, 18, 0, 20, 0, 22, 24, 24, 0, 26, 30, 28, 30, 30, 0, 32, 36, 34, 36, 36, 0, 38, 40, 40, 42, 42, 0, 44, 48, 46, 48, 48, 0, 50, 54, 52, 54, 54, 60, 56, 60, 58, 60, 60, 0, 62, 66, 64, 66, 66, 0, 68, 70, 70, 0, 72, 0, 74, 78, 76, 78, 78, 0
Offset: 1

Views

Author

Joseph L. Pe, Dec 20 2001

Keywords

Comments

If a(n) = 0, then from n onwards, phi will not go below its value at n.
The first odd term in this sequence is a(314) = 315. - Franklin T. Adams-Watters, Oct 25 2006

Examples

			a(2) = 0 since there is no k > 2 for which phi(k) < 1 = phi(2). a(5) = 6 since for k = 6, phi(6) = 2 < 4 = phi(5).

Crossrefs

Cf. A000010, A036912, A036913.

Extensions

More terms from Franklin T. Adams-Watters, Oct 25 2006

A122957 Numbers n such that if m > n, phi(m) >= phi(n).

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 10, 12, 14, 18, 20, 24, 30, 36, 42, 48, 60, 66, 70, 72, 78, 84, 90, 96, 102, 120, 126, 132, 150, 156, 168, 180, 210, 240, 252, 270, 300, 330, 336, 360, 390, 420, 450, 462, 480, 510, 540, 546, 570, 630, 660, 690, 714, 720, 780, 840, 870, 900, 924
Offset: 1

Views

Author

Franklin T. Adams-Watters, Oct 25 2006

Keywords

Comments

Conjecture: except for 1,2,3,4, this is the nonzero values of A066362.

Examples

			10 is in the sequence because phi(10) = 4 and no larger m has phi(m) < 4. phi(12) = 4; it is not smaller.

Links

Crossrefs

Cf. A036913, A000010, zeros of A066362.
Previous Showing 11-14 of 14 results.

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