A063534 Numbers k such that C(k) = H(k) + d(k), where C(k) is Chowla's function A048050, H(k) is the half-totient function A023022 and d(k) is the number of divisors function A000005.

6, 8, 15, 21, 33, 39, 51, 57, 69, 87, 93, 111, 123, 129, 141, 159, 177, 183, 201, 213, 219, 237, 249, 267, 291, 303, 309, 321, 327, 339, 381, 393, 411, 417, 447, 453, 471, 489, 501, 519, 537, 543, 573, 579, 591, 597, 633, 669, 681, 687, 699, 717, 723, 753

Offset: 1

Author

Jason Earls, Aug 02 2001

Keywords

• easy
• nonn

Links

• Harry J. Smith, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A048050, A023022, A000005, A001748.

Programs

Mathematica

Select[Range[1000], DivisorSigma[1, #] - 1 - # == EulerPhi[#]/2 + DivisorSigma[0, #] &] (* Paolo Xausa, Apr 17 2024 *)

PARI

C(n)=sigma(n)-n-1;
H(n)=eulerphi(n)/2;
j=[]; for(n=1,1200, if(C(n)==H(n)+numdiv(n),j=concat(j,n))); j

PARI

{ n=0; for (m=1, 10^9, if (sigma(m) - m - 1 == eulerphi(m)/2 + numdiv(m), write("b063534.txt", n++, " ", m); if (n==1000, break)) ) } \\ Harry J. Smith, Aug 25 2009

PARI

is(n) = {my(f = factor(n)); sigma(f) - n - 1 == eulerphi(f) / 2 + numdiv(f);} \\ Amiram Eldar, Apr 15 2024

Formula

Conjecture: a(n) = A001748(n), n <> 2. - R. J. Mathar, Dec 15 2008

The conjecture is false. The least counterexample is a(11546) = 368335 = 5 * 11 * 37 * 181. The next counterexample is 4922335, and there are no more below 10^10. - Amiram Eldar, Apr 15 2024