A138193 Odd composite numbers n for which A137576((n-1)/2)-1 is divisible by phi(n).

9, 15, 25, 27, 33, 39, 49, 55, 57, 63, 81, 87, 95, 111, 119, 121, 125, 135, 143, 153, 159, 161, 169, 175, 177, 183, 201, 207, 209, 225, 243, 249, 287, 289, 295, 297, 303, 319, 321, 329, 335, 343, 351, 361, 369, 375, 391, 393, 407, 415, 417, 423, 447, 489, 497

Offset

1,1

Comments

If p is an odd prime then A137576((p-1)/2)=p. Therefore the composite numbers n may be considered as quasiprimes. In particular, if (m,n)=1 we have a natural generalization of the little Fermat theorem: m^(A137576((n-1)/ 2)-1)=1 mod n.

Links

Ray Chandler, Table of n, a(n) for n=1..1239

Example

a(1)=9: A137576(4)=13 and 13-1 is divisible by phi(9)=6.

Mathematica

A137576[n_] := Module[{t}, (t = MultiplicativeOrder[2, 2 n + 1])* DivisorSum[2 n + 1, EulerPhi[#]/MultiplicativeOrder[2, #] &] - t + 1];
okQ[n_] := OddQ[n] && CompositeQ[n] && Divisible[A137576[(n - 1)/2] - 1, EulerPhi[n]];
Reap[For[k = 1, k < 500, k += 2, If[okQ[k], Print[k]; Sow[k]]]][[2, 1]] (* Jean-François Alcover, Jan 11 2019 *)

Crossrefs

Cf. A137576, A002326, A006694.

Sequence in context: A075638 A145743 A164384 * A330947 A337237 A036315

Adjacent sequences: A138190 A138191 A138192 * A138194 A138195 A138196

Keyword

nonn

Author

Vladimir Shevelev, May 04 2008

Extensions

Extended by Ray Chandler, May 08 2008

Status

approved