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A138193 Odd composite numbers n for which A137576((n-1)/2)-1 is divisible by phi(n). 6
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois Alcover, Jan 11 2019 *)

CROSSREFS

Cf. A137576, A002326, A006694.

Sequence in context: A075638 A145743 A164384 * A036315 A020154 A079290

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

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Last modified November 13 17:34 EST 2019. Contains 329106 sequences. (Running on oeis4.)