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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.
2
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
1,1
LINKS
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
MATHEMATICA
Select[Range[1000], DivisorSigma[1, #] - 1 - # == EulerPhi[#]/2 + DivisorSigma[0, #] &] (* Paolo Xausa, Apr 17 2024 *)
PROG
(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
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Jason Earls, Aug 02 2001
STATUS
approved