A362469 -id:A362469 - 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

A378639 a(n) = sum of all m such that phi(m) divides n, where phi is the Euler totient function (A000010).

Original entry on oeis.org

3, 16, 3, 51, 3, 64, 3, 156, 3, 49, 3, 265, 3, 16, 3, 387, 3, 202, 3, 302, 3, 85, 3, 991, 3, 16, 3, 138, 3, 190, 3, 968, 3, 16, 3, 1058, 3, 16, 3, 1240, 3, 340, 3, 419, 3, 157, 3, 2736, 3, 49, 3, 210, 3, 445, 3, 620, 3, 193, 3, 1723, 3, 16, 3, 2283, 3, 334, 3, 51, 3, 262

Offset: 1

Paolo Xausa, Dec 03 2024

nonn

Paolo Xausa, Table of n, a(n) for n = 1..1000

Row sums of A378638.

Cf. A000010, A319048, A362469.

f:= proc(n) local d;
  if n::odd then return 3 fi;
  3+add(convert(numtheory:-invphi(d),`+`), d = select(type,numtheory:-divisors(n),even))
end proc:
map(f, [$1..100]); # Robert Israel, Dec 05 2024

With[{nmax = 100}, Table[If[OddQ[n], 3, Total[PositionIndex[Divisible[n, #[[;; Max[n^2, 6]]]]][True]]], {n, nmax}] & [EulerPhi[Range[nmax^2]]]]

a(n) = vecsum(select(x->!(n % eulerphi(x)), [1..max(n^2, 6)])); \\ Michel Marcus, Dec 05 2024

a(2*k+1) = 3, for k >= 0.

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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