A179871 -id:A179871 - cp's OEIS frontend

A179876 Numbers h such that h and h-1 have same antiharmonic mean of the numbers k < h such that gcd(k, h) = 1.

2, 7, 11, 23, 47, 59, 66, 70, 78, 83, 107, 130, 167, 179, 186, 195, 211, 222, 227, 238, 255, 263, 266, 310, 322, 331, 347, 359, 366, 383, 399, 418, 438, 455, 463, 467, 470, 474, 479, 483, 494, 498, 503

Offset: 1

Jaroslav Krizek, Jul 30 2010, Jul 31 2010

nonn

Corresponding values of numbers h-1 see A179875.
Numbers h such that A175505(h) = A175505(h-1).
Numbers h such that A175506(h) = A175506(h-1).
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).

			For n=3: a(3) = 11; B(11) = A175505(11) / A175506(11) = 7, B(10) = A175505(10) / A175506(10) = 7.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..2047 from R. J. Mathar)

Cf. A179871-A179880, A179882-A179887, A179890, A179891.

antiHMeanGcd := proc(h)
        option remember;
        local a023896,a053818,k ;
        a023896 := 0 ;
        a053818 := 0 ;
        for k from 1 to h do
                if igcd(k,h) = 1 then
                        a023896 := a023896+k ;
                        a053818 := a053818+k^2 ;
                end if;
        end do:
        a053818/a023896 ;
end proc:
n := 1:
for h from 2 do
        if antiHMeanGcd(h) = antiHMeanGcd(h-1) then
                printf("%d %d\n",n,h) ;
                n := n+1 ;
        end if;
end do: # R. J. Mathar, Sep 26 2013

hmax = 1000;
antiHMeanGcd[h_] := antiHMeanGcd[h] = Module[{num = 0, den = 0, k}, For[k = 1, k <= h, k++, If[GCD[k, h] == 1, den += k; num += k^2]]; num/den];
Reap[n = 1; For[h = 2, h <= hmax, h++, If[antiHMeanGcd[h] == antiHMeanGcd[h - 1], Sow[h]; n++]]][[2, 1]] (* Jean-François Alcover, Mar 23 2020, after R. J. Mathar *)

A179879 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 not integer.

Original entry on oeis.org

6, 65, 69, 77, 129, 185, 194, 210, 221, 237, 254, 309, 321, 330, 365, 398, 417, 437, 462, 473, 482, 497, 533, 546, 554, 570, 573, 581, 597, 614, 626, 662, 669, 681, 690, 714, 753, 758, 785, 789, 794, 813, 858, 893, 905, 914, 966, 993, 1037, 1073, 1094, 1101, 1122

Offset: 1

Jaroslav Krizek, Jul 30 2010

nonn

Subsequence of A179875 and A179883.

For corresponding values of numbers h+1 see A179880. - Jaroslav Krizek, Jul 31 2010

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

Cf. A179871, A179872, A179873, A179874, A179875, A179876, A179877, A179878, A179880, A179882, A179883, A179884, A179885, A179886, A179887, A179890, A179891.

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) = A179880(n) - 1. - Jaroslav Krizek, Jul 31 2010

a(36) corrected and more terms added by Amiram Eldar, May 24 2025

A179872 Numbers h such that antiharmonic mean B(h) of the numbers k < h such that gcd(k, h) = 1 is not integer.

Original entry on oeis.org

3, 4, 6, 7, 8, 9, 12, 13, 14, 15, 16, 18, 19, 20, 21, 24, 25, 26, 27, 28, 30, 31, 32, 33, 35, 36, 37, 38, 39, 40, 42, 43, 44, 45, 48, 49, 50, 51, 52, 54, 56, 57, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 84, 86, 87, 88

Offset: 1

Jaroslav Krizek, Jul 30 2010, Jul 31 2010

nonn

Numbers h such that B(h) = A053818(h) / A023896(h) = A175505(h) / A175506(h) is not integer.
Numbers h such that A175506(h) > 1.
Complement of A179871.
Union of A007645 and A179891.

			a(6) = 9 because B(9) = A053818(9) / A023896(9) = 159/27 = 53/9 (not integer).

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

Cf. A179871, A179873, A179874, A179875, A179876, A179877, A179878, A179879, A179880, A179882, A179883, A179884, A179885, A179886, A179887, A179890, A179891.

isok(k) = {my(f = factor(k)); if(k == 1, 0, denominator(2*k/3 + (1/3) * prod(i = 1, #f~, 1 - f[i, 1])/eulerphi(f)) > 1);} \\ Amiram Eldar, May 25 2025

More terms from Amiram Eldar, May 25 2025

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).

Original entry on oeis.org

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

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

A179886 Corresponding values of antiharmonic mean B(h) of the numbers k < h such that gcd(k, h) = 1 for numbers h from A179884.

Original entry on oeis.org

1, 1, 7, 7, 15, 15, 31, 31, 39, 39, 55, 55, 71, 71, 111, 111, 119, 119, 151, 151, 175, 175, 177, 177, 231, 231, 239, 239, 255, 255, 303, 303, 311, 311, 313, 313, 319, 319, 329, 329, 335, 335, 337, 337, 345, 345, 375, 375, 391, 391, 393, 393, 479, 479, 521, 521

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).

Cf. A023896, A053818, A175505, A175506.

Cf. A179871, A179872, A179873, A179874, A179875, A179876, A179877, A179878, A179879, A179880, A179882, A179883, A179884, A179885, A179887, A179890, A179891.

a(2*n-1) = a(2*n) = A179882(n). - Amiram Eldar, May 26 2025

More terms from Amiram Eldar, May 26 2025

A034934 Numbers k such that (3*k + 1)/2 is prime.

Original entry on oeis.org

1, 3, 7, 11, 15, 19, 27, 31, 35, 39, 47, 55, 59, 67, 71, 75, 87, 91, 99, 111, 115, 119, 127, 131, 151, 155, 159, 167, 171, 175, 179, 187, 195, 207, 211, 231, 235, 239, 255, 259, 267, 279, 287, 295, 299, 307, 311, 319, 327, 335, 339, 347, 371, 375, 379, 391

Offset: 1

Jud McCranie

nonn

Related to hyperperfect numbers of a certain form.

The formula by Jaroslav Krizek is explained as follows: If p = (3n+1)/2 is prime, then it is an integer, and p must be of the form p = 3m-1, i.e., p = A003627(k). On the other hand, if p = A003627(k), then all k < p are coprime to p, so we have B(p) = (Sum_{kM. F. Hasler, Nov 29 2010

			a(6) = 19 because for A003627(6) = 29, B(29) = A053818(29)/A023896(29) = 7714/406 = 19. Cf. A179871-A179891, A003627, A007645. - _Jaroslav Krizek_, Aug 01 2010

Amiram Eldar, Table of n, a(n) for n = 1..10000
Judson S. McCranie, A study of hyperperfect numbers, J. Int. Seq., Vol. 3 (2000), Article 00.1.3.

Cf. A003627, A007645, A038536, A045309, A175505, A179871-A179891.

[ n: n in [1..400 by 2] | IsPrime((3*n+1) div 2) ];

Select[Range[500], PrimeQ[(3# + 1)/2] &] (* Harvey P. Dale, Jan 15 2011 *)

is(n)=isprime((3*n+1)/2) \\ Charles R Greathouse IV, Feb 20 2017

a(n) = A175505(A003627(n)). - Jaroslav Krizek, Aug 01 2010

Corrected by Vincenzo Librandi, Mar 24 2010

cp's OEIS Frontend

A179876 Numbers h 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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A179879 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 not integer.

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A179872 Numbers h such that antiharmonic mean B(h) of the numbers k < h such that gcd(k, h) = 1 is not integer.

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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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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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