A228947 -id:A228947 - cp's OEIS frontend

A229978 Numbers k such that (2k+1) + phi(2k+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

M. F. Hasler, Oct 05 2013

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)

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

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

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

PARI
```
select(n->(2*n+1+eulerphi(2*n+1)
				
```

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

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cp's OEIS Frontend

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

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Author

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Crossrefs

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Mathematica

PARI

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