A229978 Numbers k such that (2*k+1) + phi(2*k+1) <= sigma(2*k+1)

7, 22, 31, 37, 52, 67, 82, 94, 97, 112, 115, 127, 136, 142, 148, 157, 172, 178, 187, 199, 202, 214, 217, 220, 232, 241, 247, 262, 277, 283, 292, 304, 307, 322, 325, 337, 346, 352, 367, 382, 388, 397, 409, 412, 427, 430, 442, 445, 451, 457, 472, 487, 502, 517, 532, 535, 547, 562, 577, 592, 598, 607, 622, 637, 643, 652, 661, 667, 682, 697, 712, 724, 727, 742, 757, 772, 787, 802, 808, 817

Offset

1,1

Comments

It appears that the equation x + phi(x) = sigma(x) has the unique solution x=2. It is easy to show that this is the only even solution to the equation, but for odd solutions this is less obvious. The present sequence is motivated by the observation that for most odd numbers, the l.h.s. is larger than the r.h.s. (while the opposite is the case for all even numbers). (See also formulas in A228947.)

From Amiram Eldar, Dec 23 2024: (Start)

If k is an odd abundant number (A005231), then (k-1)/2 is a term of this sequence.

The numbers of terms that do not exceed 10^k, for k = 1, 2, ..., are 1, 9, 99, 981, 9879, 98613, 984293, 9850470, 98496984, 985005850, 9850433480, ... . Apparently, the asymptotic density of this sequence exists and equals 0.0985... . (End)

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Mathematica

Select[Range[1000], DivisorSigma[1, 2*#+1] > EulerPhi[2*#+1] + 2*#+1 &] (* Amiram Eldar, Dec 23 2024 *)

Prog

(PARI) select(n->(2*n+1+eulerphi(2*n+1)<sigma(2*n+1)), vector(900, n, n-1))

Crossrefs

Cf. A000010, A000203, A005231, A051612 and references there, A228947.

Sequence in context: A041090 A042639 A287166 * A316840 A063240 A063157

Adjacent sequences: A229975 A229976 A229977 * A229979 A229980 A229981

Keyword

nonn,easy

Author

M. F. Hasler, Oct 05 2013

Status

approved