A378637 -id:A378637 - cp's OEIS frontend

A378636 Irregular triangle read by rows: row n lists all m <= n such that phi(m) divides n, where phi is the Euler totient function (A000010).

1, 1, 2, 1, 2, 1, 2, 3, 4, 1, 2, 1, 2, 3, 4, 6, 1, 2, 1, 2, 3, 4, 5, 6, 8, 1, 2, 1, 2, 3, 4, 6, 1, 2, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 1, 2, 1, 2, 3, 4, 6, 1, 2, 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 1, 2, 1, 2, 3, 4, 6, 7, 9, 14, 18, 1, 2, 1, 2, 3, 4, 5, 6, 8, 10, 11, 12

Offset: 1

Paolo Xausa, Dec 02 2024

If n = 2 or an odd number >= 3, row n is {1, 2}.

If n is an even number >= 4, row n begins with {1, 2, 3, 4}.

			Triangle begins:
  n\k| 1  2  3  4  5  6  7   8   9  10  11 ...
  --------------------------------------------
   1 | 1;
   2 | 1, 2;
   3 | 1, 2;
   4 | 1, 2, 3, 4;
   5 | 1, 2;
   6 | 1, 2, 3, 4, 6;
   7 | 1, 2;
   8 | 1, 2, 3, 4, 5, 6, 8;
   9 | 1, 2;
  10 | 1, 2, 3, 4, 6;
  11 | 1, 2;
  12 | 1, 2, 3, 4, 5, 6, 7,  8,  9, 10, 12;
  13 | 1, 2;
  14 | 1, 2, 3, 4, 6;
  15 | 1, 2;
  16 | 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16;
  17 | 1, 2;
  18 | 1, 2, 3, 4, 6, 7, 9, 14, 18;
  19 | 1, 2;
  20 | 1, 2, 3, 4, 5, 6, 8, 10, 11, 12;
  ...

Paolo Xausa, Table of n, a(n) for n = 1..11226 (rows 1..1000 of triangle, flattened).

Cf. A069932 (row lengths), A362469 (row sums), A378637 (right border).
Subsequence of A378638.
Cf. A000010.

With[{nmax = 25}, Table[If[OddQ[n] && n > 2, {1, 2}, PositionIndex[Divisible[n, #[[;; n]]]][True]], {n, nmax}] & [EulerPhi[Range[nmax]]]]

row(n) = select(x->!(n % eulerphi(x)), [1..n]); \\ Michel Marcus, Dec 05 2024

A378641 Largest m <= n such that phi(m) does not divide n, or -1 if no such m exists, where phi is the Euler totient function (A000010).

Original entry on oeis.org

-1, -1, 3, -1, 5, 5, 7, 7, 9, 10, 11, 11, 13, 14, 15, 14, 17, 17, 19, 20, 21, 22, 23, 23, 25, 26, 27, 28, 29, 30, 31, 31, 33, 34, 35, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 47, 49, 50, 51, 52, 53, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 63, 65, 66, 67, 68, 69

Offset: 1

Paolo Xausa, Dec 05 2024

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A000010, A378636, A378637, A378640, A378642.

A378641[n_] := Module[{m = n}, While[m > 0 && Divisible[n, EulerPhi[m]], m--]; If[m == 0, -1, m]];
Array[A378641, 100]

a(n) = n if n is an odd number >= 3.

A378642 Number of numbers m <= n such that phi(m) does not divide n, where phi is the Euler totient function (A000010).

Original entry on oeis.org

0, 0, 1, 0, 3, 1, 5, 1, 7, 5, 9, 1, 11, 9, 13, 5, 15, 9, 17, 10, 19, 17, 21, 5, 23, 21, 25, 19, 27, 19, 29, 16, 31, 29, 33, 16, 35, 33, 37, 22, 39, 33, 41, 34, 43, 41, 45, 16, 47, 43, 49, 43, 51, 41, 53, 41, 55, 53, 57, 34, 59, 57, 61, 42, 63, 55, 65, 59, 67, 63

Offset: 1

Paolo Xausa, Dec 05 2024

Paolo Xausa, Table of n, a(n) for n = 1..10000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000010, A069932, A378637, A378641.

Table[Count[Divisible[n, #[[;;n]]], False], {n, Length[#]}] & [EulerPhi[Range[100]]]

a(n) = n - sumdiv(n, d, #select(x -> x<=n, invphi(d))); \\ Amiram Eldar, Dec 10 2024, using Max Alekseyev's invphi.gp

a(n) = n - A069932(n).

cp's OEIS Frontend

A378636 Irregular triangle read by rows: row n lists all m <= n such that phi(m) divides n, where phi is the Euler totient function (A000010).

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A378641 Largest m <= n such that phi(m) does not divide n, or -1 if no such m exists, where phi is the Euler totient function (A000010).

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A378642 Number of numbers m <= n such that phi(m) does not divide n, where phi is the Euler totient function (A000010).

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