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

1,2

Comments

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

Links

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

Wikipedia, Contraharmonic mean.

Formula

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

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

Example

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]

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]]]

Prog

(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

Crossrefs

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

Sequence in context: A157917 A242234 A104867 * A216048 A079861 A014008

Adjacent sequences: A179874 A179875 A179876 * A179878 A179879 A179880

Keyword

nonn

Author

Jaroslav Krizek, Jul 30 2010, Jul 31 2010

Extensions

More terms from Michael De Vlieger, Jul 30 2018

Status

approved