A179882 -id:A179882 - 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

A179873 Corresponding values of antiharmonic means B(h) to numbers h 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, 1, 3, 7, 7, 11, 15, 15, 19, 23, 27, 31, 31, 35, 37, 39, 39, 47, 55, 55, 57, 59, 61, 63, 67, 71, 71, 73, 75, 77, 79, 87, 89, 91, 95, 97, 99, 111, 111, 113, 115, 119, 119, 121, 125, 127, 131, 135, 137, 143, 145, 151, 151, 153, 155, 157, 159, 165, 167, 169, 171

Offset: 1

Jaroslav Krizek, Jul 30 2010, Jul 31 2010

nonn

Conjecture: nondecreasing sequence of odd numbers.

			a(6) = A175505(A179871(6)) = A175505(17) = 11 = B(17).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..2500 from Ivan Neretin)

Cf. A179871, A179872, A179874, A179875, A179876, A179877, A179878, A179879, A179880, A179882, A179883, A179884, A179885, A179886, A179887, A179890, A179891.

B[n_] := Plus @@ ((ks = Select[Range[n], GCD[n, #] == 1 &])^2)/Plus @@ ks; Select[B /@ Range[215], IntegerQ] (* Ivan Neretin, May 22 2015 *)

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));}
list(lim) = {my(m); for(k = 1, lim, m = ah(k); if(denominator(m) == 1, print1(m, ", ")));} \\ Amiram Eldar, May 25 2025

a(n) = A053818(A179871(n)) / A023896(A179871(n)) = A175505(A179871(n)).

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

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

A317510 Numbers (4p+1)/3 where p is a Sophie Germain prime p > 3.

Original entry on oeis.org

7, 15, 31, 39, 55, 71, 111, 119, 151, 175, 231, 239, 255, 311, 319, 335, 375, 391, 479, 559, 575, 591, 655, 679, 791, 855, 871, 879, 911, 959, 991, 1015, 1079, 1215, 1271, 1351, 1359, 1375, 1399, 1471, 1631, 1639, 1719, 1879, 1919, 1935, 1975, 1999, 2015, 2079, 2111, 2135, 2311, 2415, 2519, 2535, 2575, 2631

Offset: 1

Hilko Koning, Jul 30 2018

nonn

It appears that this is a subsequence of A179882.
Define a set of consecutive positive odd numbers {1,......, (A077065(n)-1)} with n >= 3 and skip the number A077065(n)/2. Then the contraharmonic mean of that set gives the sequence. For example: ContraharmonicMean[{1, 3, 7, 9}] = 7, ContraharmonicMean[{1, 3, 5, 7, 9, 13, 15, 17, 19, 21}] = 15, ContraharmonicMean[{1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 25, 27, 29, 31, 33, 35, 37,39, 41, 43, 45}] = 31. - Hilko Koning, Aug 28 2018
Let p be a Sophie Germain prime and define h = 2p + 1 a safe prime. Then the contraharmonic mean of the totatives of h is given by: CHM(h) =  (Sum_{1 <= k < h, gcd(k, h) = 1} k^2) / (Sum_{1 <= k < h, gcd(k, h) = 1} k). Since h is prime, all integers k = 1, 2, ... , h - 1 are coprime to h. Then, CHM(h) = ((h - 1) * h * (2h-1) / 6) / ((h - 1) * h  / 2). Thus CHM = (2h-1) / 3 = (4p+1) / 3. These values are integers precisely when p == 2 mod 3, which holds for all Sophie Germain primes, p >= 5. The resulting values for the sequence A317510, which is therefore a subsequence of A179882. - Hilko Koning, Jun 17 2025

Muniru A Asiru, Table of n, a(n) for n = 1..5000

Cf. A005384, A179882.

Subsequence of A004767, and of A004771.

a:=[];; for p in [3..2000] do if IsPrime(p) and IsPrime(2*p+1) then Add(a,(4*p+1)/3); fi; od; a; # Muniru A Asiru, Aug 28 2018

lst = {}; Do[If[PrimeQ[p] && PrimeQ[2 p + 1], AppendTo[lst, (4 p + 1)/3]], {p, 5, 2*10^3}]; lst
4 (Select[Prime@Range[3, 300], PrimeQ[2 # + 1] &] + 1)/3 - 1 (* Robert G. Wilson v, Jul 30 2018 *)

lista(nn) = {forprime (p=5, nn, if (isprime(2*p+1), print1((4*p+1)/3, ", ")););} \\ Michel Marcus, Aug 27 2018

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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A179873 Corresponding values of antiharmonic means B(h) to numbers h 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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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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A179886 Corresponding values of antiharmonic mean B(h) of the numbers k < h such that gcd(k, h) = 1 for numbers h from A179884.

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