A175505 -id:A175505 - 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 *)

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)).

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

A175506 Denominators of the antiharmonic means B of numbers k such that gcd(k, n) = 1 for numbers n >= 1 and k < n.

Original entry on oeis.org

1, 1, 3, 2, 1, 3, 3, 4, 9, 1, 1, 6, 3, 3, 3, 8, 1, 9, 3, 2, 3, 1, 1, 12, 5, 3, 27, 6, 1, 3, 3, 16, 3, 1, 3, 18, 3, 3, 3, 4, 1, 3, 3, 2, 9, 1, 1, 24, 21, 5, 3, 6, 1, 27, 1, 12, 3, 1, 1, 6, 3, 3, 9, 32, 3, 3, 3, 2, 3, 3, 1, 36, 3, 3, 15, 6, 3, 3, 3, 8, 81, 1, 1, 6, 1, 3, 3, 4, 1, 9, 1, 2, 3, 1, 3, 48, 3, 21, 9

Offset: 1

Jaroslav Krizek, May 31 2010

nonn

See A175505 - numerators of the antiharmonic means B of numbers k such that gcd(k, n) = 1 for numbers n >= 1 and k < n where B = A053818(n) / A023896(n) = A175505(n) / a(n).

f[n_] := 2Plus @@ (Select[ Range@n, GCD[ #, n] == 1 &]^2)/(n*EulerPhi@n); f[1] = 1; Denominator@ Array[f, 80] (* Robert G. Wilson v, Jul 01 2010 *)

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

More terms from Robert G. Wilson v, Jul 01 2010

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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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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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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A175506 Denominators of the antiharmonic means B of numbers k such that gcd(k, n) = 1 for numbers n >= 1 and k < n.

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

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Magma

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