A179877 Numbers h such that h and h+1 have same contraharmonic mean of the numbers k < h such that gcd(k, h) = 1 and simultaneously this mean is an integer (see A179882).

1, 10, 22, 46, 58, 82, 106, 166, 178, 226, 262, 265, 346, 358, 382, 454, 466, 469, 478, 493, 502, 505, 517, 562, 586, 589, 718, 781, 838, 862, 886, 889, 901, 910, 934, 982, 985, 1018, 1165, 1177, 1186, 1234, 1282, 1294, 1306, 1318, 1333, 1357, 1366, 1393

Offset: 1

Author

Jaroslav Krizek, Jul 30 2010, Jul 31 2010

Keywords

• nonn

Comments

For corresponding values of numbers h+1 see A179878. Subsequence of A179875, A179871 and A179883.

Examples

			From _Michael De Vlieger_, Jul 30 2018: (Start)
10 is in the sequence since the reduced residue system of 10 is {1, 3, 7, 9} and that of 11 is {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, the mean of the squares of these 2 systems, divided by the mean of the systems themselves, is 7 in both cases.
6 is not in the sequence, because though the RRS of 6, {1, 5}, and that of 7, {1, 2, 3, 4, 5, 6}, have the same contraharmonic mean of 13/3, it is not integral. (End) [corrected by _Hilko Koning_, Aug 20 2018]
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Wikipedia, Contraharmonic mean.

Crossrefs

Cf. A179871, A179872, A179873, A179874, A179875, A179876, A179878, A179879, A179880, A179882, A179883, A179884, A179885, A179886, A179887, A179890, A179891.

Programs

Mathematica

With[{s = Partition[Table[Mean[#^2]/Mean[#] &@ Select[Range[n - 1], GCD[#, n] == 1 &], {n, 1400}], 2, 1]}, Position[s, _?(And[IntegerQ@ First@ #, SameQ @@ #] &), 1, Heads -> False][[All, 1]]]

PARI

ah(n) = {my(f = factor(n)); if(n == 1, 1, 2*n/3 + (1/3) * prod(i = 1, #f~, 1 - f[i, 1])/eulerphi(f));}
isok(k) = {my(ah1 = ah(k), ah2 = ah(k+1)); ah1 == ah2 && denominator(ah1) == 1;} \\ Amiram Eldar, May 24 2025

Formula

a(n) = (3*A179882(n) - 1)/2. - Hilko Koning, Aug 01 2018

a(n) = A179878(n) - 1. - Amiram Eldar, May 24 2025

Extensions

More terms from Michael De Vlieger, Jul 30 2018