A179886 -id:A179886 - cp's OEIS frontend

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

Jaroslav Krizek, Jul 30 2010, Jul 31 2010

nonn

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

			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]

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

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

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]]]

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

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

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

More terms from Michael De Vlieger, Jul 30 2018

A179878 Numbers h such that h and h-1 have same antiharmonic mean of the numbers k < h such that gcd(k, h) = 1 and simultaneously this mean is an integer (see A179882).

Original entry on oeis.org

2, 11, 23, 47, 59, 83, 107, 167, 179, 227, 263, 266, 347, 359, 383, 455, 467, 470, 479, 494, 503, 506, 518, 563, 587, 590, 719, 782, 839, 863, 887, 890, 902, 911, 935, 983, 986, 1019, 1166, 1178, 1187, 1235, 1283, 1295, 1307, 1319, 1334, 1358, 1367, 1394

Offset: 1

Jaroslav Krizek, Jul 30 2010, Jul 31 2010

nonn

For corresponding values of numbers h-1 see A179877. Subsequence of A179876, A179871 and A179883.

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

Cf. A179871, A179872, A179873, A179874, A179875, A179876, A179877, A179879, A179880, A179882, A179883, A179884, A179885, A179886, A179887, A179890, A179891.

With[{s = Partition[Table[Mean[#^2]/Mean[#] &@ Select[Range[n - 1], GCD[#, n] == 1 &], {n, 1400}], 2, 1]}, 1 + Position[s, ?(And[IntegerQ@ First@ #, SameQ @@ #] &), 1, Heads -> False][[All, 1]]] (* _Michael De Vlieger, Jul 30 2018 *)

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) = {if(k == 1, 0, my(ah1 = ah(k), ah2 = ah(k-1)); ah1 == ah2 && denominator(ah1) == 1);} \\ Amiram Eldar, May 24 2025

a(n) = A179877(n) + 1. - Amiram Eldar, May 24 2025

A179874 Possible values of A179873(m) in increasing order, where A179873(m) = corresponding values of antiharmonic means to numbers from A179871 (numbers h such that antiharmonic mean of the numbers k < h such that gcd(k, h) = 1 is an integer).

Original entry on oeis.org

1, 3, 7, 11, 15, 19, 23, 27, 31, 35, 37, 39, 47, 55, 57, 59, 61, 63, 67, 71, 73, 75, 77, 79, 87, 89, 91, 95, 97, 99, 111, 113, 115, 119, 121, 125, 127, 131, 135, 137, 143, 145, 151, 151, 153, 155, 157, 159, 165, 167

Offset: 1

Jaroslav Krizek, Jul 30 2010, Jul 31 2010

nonn

Conjecture: a(n) = sequence of odd numbers.

Cf. A179871, A179872, A179873, A179875, A179876, A179877, A179878, A179879, A179880, A179882, A179883, A179884, A179885, A179886, A179887, A179890, A179891.

A179883 List of twin numbers (h, h+1) such that h and h+1 have same antiharmonic mean of the numbers k < h such that gcd(k, h) = 1.

Original entry on oeis.org

1, 2, 6, 7, 10, 11, 22, 23, 46, 47, 58, 59, 65, 66, 69, 70, 77, 78, 82, 83, 106, 107, 129, 130, 166, 167, 178, 179, 185, 186, 194, 195, 210, 211, 221, 222, 226, 227, 237, 238, 254, 255, 262, 263, 265, 266, 309, 310, 321, 322, 330, 331, 346, 347, 358, 359, 365

Offset: 1

Jaroslav Krizek, Jul 30 2010, Jul 31 2010

nonn

Antiharmonic mean B(h) of numbers k such that gcd(k, h) = 1 for numbers h >= 1 and k < h = A053818(n) / A023896(n) = A175505(h) / A175506(h). Union of A179875 and A179876.

Cf. A179871, A179872, A179873, A179874, A179875, A179876, A179877, A179878, A179879, A179880, A179882, A179884, A179885, A179886, A179887, A179890, A179891.

a(2*n-1) = A179875(n), a(2*n) = A179876(n) = A179875(n)+1. - Amiram Eldar, May 24 2025

A179884 List of twin numbers (h, h+1) such that h and h+1 have same antiharmonic mean of the numbers k < h such that gcd(k, h) = 1 and simultaneously this mean is an integer.

Original entry on oeis.org

1, 2, 10, 11, 22, 23, 46, 47, 58, 59, 82, 83, 106, 107, 166, 167, 178, 179, 226, 227, 262, 263, 265, 266, 346, 347, 358, 359, 382, 383, 454, 455, 466, 467, 469, 470, 478, 479, 493, 494, 502, 503, 505, 506, 517, 518, 562, 563, 586, 587, 589, 590, 718, 719, 781, 782

Offset: 1

Jaroslav Krizek, Jul 30 2010, Jul 31 2010

nonn

Subsequence of A179883 and A179871.
Antiharmonic mean B(h) of numbers k such that gcd(k, h) = 1 for numbers h >= 1 and k < h = A053818(n) / A023896(n) = A175505(h) / A175506(h).
Corresponding values of antiharmonic mean B(a(n)) are in A179886.

Cf. A179871, A179872, A179873, A179874, A179875, A179876, A179877, A179878, A179879, A179880, A179882, A179883, A179885, A179886, A179887, A179890, A179891.

a(2*n-1) = A179877(n), a(2*n) = A179878(n) = A179877(n)+1. - Amiram Eldar, May 24 2025

More terms from Amiram Eldar, May 25 2025

A179885 Antiharmonic mean B(h) of numbers k such that gcd(k, h) = 1 for numbers h >= 1 and k < h = A053818(n) / A023896(n) = A175505(h) / A175506(h).

Original entry on oeis.org

6, 7, 65, 66, 69, 70, 77, 78, 129, 130, 185, 186, 194, 195, 210, 211, 221, 222, 237, 238, 254, 255, 309, 310, 321, 322, 330, 331, 365, 366, 398, 399, 417, 418, 437, 438, 462, 463, 473, 474, 482, 483, 497, 498, 533, 534, 546, 547, 554, 555, 570, 571, 573, 574, 581

Offset: 1

Jaroslav Krizek, Jul 30 2010, Jul 31 2010

nonn

Subsequence of A179883 and A179872.

Cf. A179871, A179873, A179874, A179875, A179876, A179877, A179878, A179879, A179880, A179882, A179884, A179886, A179887, A179890, A179891.

a(2*n-1) = A179879(n), a(2*n) = A179880(n) = A179879(n) + 1. - Amiram Eldar, May 26 2025

More terms from Amiram Eldar, May 26 2025

cp's OEIS Frontend

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).

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A179878 Numbers h such that h and h-1 have same antiharmonic mean of the numbers k < h such that gcd(k, h) = 1 and simultaneously this mean is an integer (see A179882).

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A179874 Possible values of A179873(m) in increasing order, where A179873(m) = corresponding values of antiharmonic means to numbers from A179871 (numbers h such that antiharmonic mean of the numbers k < h such that gcd(k, h) = 1 is an integer).

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A179883 List of twin numbers (h, h+1) such that h and h+1 have same antiharmonic mean of the numbers k < h such that gcd(k, h) = 1.

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A179884 List of twin numbers (h, h+1) such that h and h+1 have same antiharmonic mean of the numbers k < h such that gcd(k, h) = 1 and simultaneously this mean is an integer.

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A179885 Antiharmonic mean B(h) of numbers k such that gcd(k, h) = 1 for numbers h >= 1 and k < h = A053818(n) / A023896(n) = A175505(h) / A175506(h).

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