A349763 Numbers k such that d(k) = A000005(k), sigma(k) = A000203(k) and phi(k) = A000010(k) are all deficient numbers (A005100).

1, 2, 3, 4, 8, 16, 48, 64, 121, 128, 192, 256, 512, 529, 1024, 2116, 2209, 2809, 3072, 3481, 4096, 6889, 8192, 8836, 11449, 12288, 13924, 14641, 16384, 17161, 18769, 22201, 27556, 27889, 29282, 29929, 32041, 32768, 36481, 45796, 51529, 54289, 57121, 63001, 65536

Offset: 1

Author

Amiram Eldar, Nov 29 2021

Keywords

• nonn

Comments

Sándor (2005) proved that this sequence is infinite by showing that any number of the form 2^(p-1), where p is a sufficiently large prime, is a term. d(2^(p-1)) = p and phi(2^(p-1)) = 2^(p-2) are deficient for all primes, while sigma(2^(p-1)) = 2^p - 1 is deficient for a sufficiently large prime, a result of a theorem by Bojanić (1954): lim_{p prime -> oo} sigma(2^p - 1)/(2^p - 1) = 1.

Examples

			2 is a term since d(2) = 2, sigma(2) = 3 and phi(2) = 1 are all deficient numbers.
		

References

• R. Bojanić, Asymptotic evaluations of the sum of divisors of certain numbers (in Serbo-Croatian), Bull. Soc. Math.-Phys. R. P. Macédoine, Vol. 5 (1954), pp. 5-15.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• József Sándor, Selected Chapters of Geometry, Analysis and Number Theory, 2005, pp. 132-134.

Crossrefs

Cf. A000005, A000010, A000203, A005100, A061286.

Subsequence of A349759.

Programs

Mathematica

defQ[n_] := DivisorSigma[1, n] < 2*n; q[n_] := And @@ defQ /@ Join[DivisorSigma[{0, 1}, n], {EulerPhi[n]}]; Select[Range[10^5], q]

PARI

isdef(k) = sigma(k) < 2*k;
isok(k) = my(f=factor(k)); isdef(numdiv(f)) && isdef(sigma(f)) && isdef(eulerphi(k)); \\ Michel Marcus, Dec 01 2021