A179878 -id:A179878 - cp's OEIS frontend

A179882 a(n) is the corresponding value of contraharmonic mean B(h) of numbers k such that gcd(k, h) = 1 (k < h) for numbers h from A179877(n) and A179878(n).

1, 7, 15, 31, 39, 55, 71, 111, 119, 151, 175, 177, 231, 239, 255, 303, 311, 313, 319, 329, 335, 337, 345, 375, 391, 393, 479, 521, 559, 575, 591, 593, 601, 607, 623, 655, 657, 679, 777, 785, 791, 823, 855, 863, 871, 879, 889, 905, 911, 929, 937, 959, 961, 991

Offset: 1

Jaroslav Krizek, Jul 30 2010, Jul 31 2010

nonn

Subsequence of A179873 and A179874.
It appears that for n >= 3, (4*A005384(n)+1)/3 is a subsequence. - Hilko Koning, Jul 27 2018
This happens for this subsequence of A179877: 10, 22, 46, 58, 82, 106, 166, 178, ... apparently "Semiprimes of form prime - 1" >= 10 (see A077065). - Michel Marcus, Jul 27 2018

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

Cf. A179871, A179872, A179873, A179874, A179875, A179876, A179877, A179878, A179879, A179880, A179883, A179884, A179885, A179886, A179887, A179890, A179891.

{1}~Join~Select[Partition[Table[ContraharmonicMean@ Select[Range[n - 1], GCD[#, n] == 1 &], {n, 2, 1500}], 2, 1], And[IntegerQ@ First@ #, SameQ @@ #] &][[All, 1]] (* Michael De Vlieger, Jul 30 2018 *)

lista(nn) = {vch = vector(nn, k, ch(k)); for (i=1, nn-1, if ((vch[i] == vch[i+1]) && !frac(vch[i]), print1(vch[i], ", ")););} \\ Michel Marcus, Jul 27 2018

a(n) = A175505(A179877(n)) / A175506(A179877(n)).

a(n) = A175505(A179878(n)) / A175506(A179878(n)).

More terms from Michel Marcus, Jul 27 2018

A007645 Generalized cuban primes: primes of the form x^2 + xy + y^2; or primes of the form x^2 + 3*y^2; or primes == 0 or 1 (mod 3).

Original entry on oeis.org

3, 7, 13, 19, 31, 37, 43, 61, 67, 73, 79, 97, 103, 109, 127, 139, 151, 157, 163, 181, 193, 199, 211, 223, 229, 241, 271, 277, 283, 307, 313, 331, 337, 349, 367, 373, 379, 397, 409, 421, 433, 439, 457, 463, 487, 499, 523, 541, 547, 571, 577, 601, 607, 613

Offset: 1

N. J. A. Sloane, Mira Bernstein and Robert G. Wilson v

Also, odd primes p such that -3 is a square mod p. - N. J. A. Sloane, Dec 25 2017
Equivalently, primes of the form p = (x^3 - y^3)/(x - y). If x=y+1 we get the cuban primes A002407, which is therefore a subsequence.
These are not to be confused with the Eisenstein primes, which are the primes in the ring of integers Z[w], where w = (-1+sqrt(-3))/2. The present sequence gives the rational primes which are also Eisenstein primes. - N. J. A. Sloane, Feb 06 2008
Also primes of the form x^2+3y^2 and, except for 3, x^2+xy+7y^2. See A140633. - T. D. Noe, May 19 2008
Conjecture: this sequence is Union(A002383,A162471). - Daniel Tisdale, Jul 04 2009
Primes p such that antiharmonic mean B(p) of the numbers k < p such that gcd(k, p) = 1 is not integer, where B(p) = A053818(p) / A023896(p) = A175505(p) / A175506(p) = (2p - 1) / 3. Primes p such that A175506(p) > 1. Subsequence of A179872. Union a(n) + A179891 = A179872. Example: a(6) = 37 because B(37) = A053818(37) / A023896(37) = A175505(37) / A175506(37) = 16206 / 666 = 73 / 3 (not integer). Cf. A179871, A179872, A179873, A179874, A179875, A179876, A179877, A179878, A179879, A179880, A179882, A179883, A179884, A179885, A179886, A179887, A179890, A179891, A003627, A034934. - Jaroslav Krizek, Aug 01 2010
Subsequence of Loeschian numbers, cf. A003136 and A024614; A088534(a(n)) > 0. - Reinhard Zumkeller, Oct 30 2011
Primes such that there exist a unique x, y, with 1 < x <= y < p, x + y == 1 (mod p) and x * y == 1 (mod p). - Jon Perry, Feb 02 2014
The prime factors of A002061. - Richard R. Forberg, Dec 10 2014
This sequence gives the primes p which solve s^2 == -3 (mod 4*p) (see Buell, Proposition 4.1., p. 50, for Delta = -3). p = 2 is not a solution. x^2 == -3 (mod 4) has solutions for all odd x. x^2 == -3 (mod  p) has for odd primes p, not 3, the solutions of Legendre(-3|p) = +1 which are p == {1, 7} (mod 12). For p = 3 the representative solution is x = 0. Hence the solution of s^2 == -3 (mod 4*p) are the odd primes p = 3 and p == {1, 7} (mod 12) (or the primes p = 0, 1 (mod 3)). - Wolfdieter Lang, May 22 2021

D. A. Buell, Binary Quadratic Forms. Springer-Verlag, NY, 1989, p. 50.
Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 220-223, 1996.
David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989, p. 7.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Wagon, S. "Eisenstein Primes." Section 9.8 in Mathematica in Action. New York: W. H. Freeman, pp. 319-323, 1991.

T. D. Noe, Table of n, a(n) for n = 1..1000
U. P. Nair, Elementary results on the binary quadratic form a^2+ab+b^2, arXiv:math/0408107 [math.NT], 2004.
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
Eric Weisstein's World of Mathematics, Eisenstein Integer.

Subsequence of A003136.
Subsequences include A002407, A002648, and A201477.
Apart from initial term, same as A045331.
Cf. A001479, A001480 (x and y such that a(n) = x^2 + 3y^2).
Cf. A000040, A003627.
Primes in A003136 and A034017.

a007645 n = a007645_list !! (n-1)
a007645_list = filter ((== 1) . a010051) $ tail a003136_list
-- Reinhard Zumkeller, Jul 11 2013, Oct 30 2011

select(isprime,[3, seq(6*k+1, k=1..1000)]); # Robert Israel, Dec 12 2014

Join[{3},Select[Prime[Range[150]],Mod[#,3]==1&]] (* Harvey P. Dale, Aug 21 2021 *)

forprime(p=2,1e3,if(p%3<2,print1(p", "))) \\ Charles R Greathouse IV, Jun 16 2011

p == 0 or 1 (mod 3).
{3} UNION A002476. - R. J. Mathar, Oct 28 2008
A007645 UNION A003627 = A000040. - Juri-Stepan Gerasimov, Jan 28 2010

Entry revised by N. J. A. Sloane, Jan 29 2013

A179871 Numbers h such that antiharmonic mean B(h) of the numbers k < h such that gcd(k, h) = 1 is an integer.

Original entry on oeis.org

1, 2, 5, 10, 11, 17, 22, 23, 29, 34, 41, 46, 47, 53, 55, 58, 59, 71, 82, 83, 85, 89, 91, 94, 101, 106, 107, 110, 113, 115, 118, 131, 133, 137, 142, 145, 149, 166, 167, 170, 173, 178, 179, 182, 187, 191, 197, 202, 205, 214

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 an integer.
Numbers h such that A175506(h) = 1.
Complement of A179872.
See A179873 (odd positive integers) for corresponding values A175505(a(n)).
Union of A003627 (primes of form 3n-1) and A179887.

			a(9) = 29 because B(29) = A053818(29) / A023896(29) = 7714/406 = 19 (integer).

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

Cf. A179872, A179873, 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[Range[215], IntegerQ[B[#]] &] (* Ivan Neretin, May 22 2015 *)

isok(k) = {my(f = factor(k)); if(k == 1, 1, denominator(2*k/3 + (1/3) * prod(i = 1, #f~, 1 - f[i, 1])/eulerphi(f)) == 1);} \\ Amiram Eldar, May 24 2025

A179891 Composites 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

4, 6, 8, 9, 12, 14, 15, 16, 18, 20, 21, 24, 25, 26, 27, 28, 30, 32, 33, 35, 36, 38, 39, 40, 42, 44, 45, 48, 49, 50, 51, 52, 54, 56, 57, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 84, 86, 87, 88, 90, 92, 93, 95, 96, 98, 99, 100, 102, 104, 105, 108, 111

Offset: 1

Jaroslav Krizek, Jul 30 2010

nonn

Composites h such that B(h) = A053818(h) / A023896(h) = A175505(h) / A175506(h) is not integer.
Composites h such that A175506(h) > 1.
Subsequence of A179872.
A179872 is the union of this sequence and A007645.

			a(6) = 14 because B(14) = A053818(14) / A023896(14) = 406/42 = 29/3 (not integer).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1749 from G. C. Greubel)

Cf. A007645, A023896, A053818, A175505, A175506.

Cf. A179871, A179872, A179873, A179874, A179875, A179876, A179877, A179878, A179879, A179880, A179882, A179883, A179884, A179885, A179886, A179887, A179890.

f[n_] := 2 Plus @@ (Select[ Range@n, GCD[ #, n] == 1 &]^2)/(n*EulerPhi@n); Select[ Range@ 111, ! PrimeQ@# && ! IntegerQ@f@# &] (* Robert G. Wilson v, Aug 02 2010 *)

isok(k) = if(isprime(k), 0, 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 Robert G. Wilson v, Aug 02 2010

A179880 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

7, 66, 70, 78, 130, 186, 195, 211, 222, 238, 255, 310, 322, 331, 366, 399, 418, 438, 463, 474, 483, 498, 534, 547, 555, 571, 574, 582, 598, 615, 627, 663, 670, 682, 691, 715, 754, 759, 786, 790, 795, 814, 859, 894, 906, 915, 967, 994, 1038, 1074, 1095, 1102, 1123

Offset: 1

Jaroslav Krizek, Jul 30 2010, Jul 31 2010

nonn

Subsequence of A179876 and A179883.

For corresponding values of numbers h-1 see A179879. - 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, A179879, 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) = {if( k == 1, 0, my(ah1 = ah(k), ah2 = ah(k-1)); ah1 == ah2 && denominator(ah1) > 1);} \\ Amiram Eldar, May 24 2025

a(n) = A179879(n) + 1. - Jaroslav Krizek, Jul 31 2010

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

A179887 Nonprimes q such that antiharmonic mean B(q) of the numbers k < q such that gcd(k, q) = 1 is an integer, where B(q) = A053818(q) / A023896(q) = A175505(q) / A175506(q).

Original entry on oeis.org

1, 10, 22, 34, 46, 55, 58, 82, 85, 91, 94, 106, 110, 115, 118, 133, 142, 145, 166, 170, 178, 182, 187, 202, 205, 214, 217, 226, 230, 235, 247, 253, 259, 262, 265, 266, 274, 290, 295, 298, 301, 319, 334, 346, 355, 358, 374, 382, 391, 394, 403, 410, 415, 427

Offset: 1

Jaroslav Krizek, Jul 30 2010, Jul 31 2010

nonn

Nonprimes q such that A175506(q) = 1.
Subsequence of A179871.
A179871 is the union of this sequence and A003627.
Corresponding values of B(q) in A179890.

			a(6) = 55 because B(55) = A053818(55) / A023896(55) = 40700 / 1100 = 37 (integer).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1653 from G. C. Greubel)

Cf. A003627, A023896, A053818, A175505, A175506.

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

f[n_] := 2 Plus @@ (Select[ Range@n, GCD[ #, n] == 1 &]^2)/(n*EulerPhi@n); Select[ Range@ 433, ! PrimeQ@# && IntegerQ@ f@# &] (* Robert G. Wilson v, Aug 02 2010 *)

isok(k) = if(isprime(k), 0, my(f = factor(k)); if(k == 1, 1, denominator(2*k/3 + (1/3) * prod(i = 1, #f~, 1 - f[i, 1])/eulerphi(f)) == 1)); \\ Amiram Eldar, May 26 2025

More terms from Robert G. Wilson v, Aug 02 2010

A179890 Values of antiharmonic mean B(q) of the numbers k < q such that gcd(k, q) = 1 is an integer for nonprimes q from A179887, where B(q) = A053818(q) / A023896(q) = A175505(q) / A175506(q).

Original entry on oeis.org

1, 7, 15, 23, 31, 37, 39, 55, 57, 61, 63, 71, 73, 77, 79, 89, 95, 97, 111, 113, 119, 121, 125, 135, 137, 143, 145, 151, 153, 157, 165, 169, 173, 175, 177, 177, 183, 193, 197, 199, 201, 213, 223, 231, 237, 239, 249, 255, 261, 263, 269, 273, 277, 285, 289, 297, 301, 303, 303

Offset: 1

Jaroslav Krizek, Jul 30 2010, Jul 31 2010

nonn

			a(6) = 37 because for A179887(6) = 55 holds: B(55) = A053818(55)/A023896(55) = 40700/1100 = 37.

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

Cf. A023896, A053818, A175505, A175506.

Cf. A179871, A179872, A179873, A179874, A179875, A179876, A179877, A179878, A179879, A179880, A179882, A179883, A179884, A179885, A179886, A179887, A179890, A179891.

list(lim) = print1(1, ", "); for(k = 2, lim, if(!isprime(k), my(f = factor(k), b = 2*k/3 + (1/3) * prod(i = 1, #f~, 1 - f[i, 1])/eulerphi(f)); if(denominator(b) == 1, print1(b, ", ")))); \\ Amiram Eldar, May 26 2025

Incorrect formula removed by Amiram Eldar, May 26 2025

A179875 Numbers h 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, 6, 10, 22, 46, 58, 65, 69, 77, 82, 106, 129, 166, 178, 185, 194, 210, 221, 226, 237, 254, 262, 265, 309, 321, 330, 346, 358, 365, 382, 398, 417, 437, 454, 462, 466, 469, 473, 478, 482, 493, 497, 502

Offset: 1

Jaroslav Krizek, Jul 30 2010, Jul 31 2010

nonn

Corresponding values of numbers h+1 see A179876.
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).
Conjecture: also numbers k such that mu(k) = 1 and mu(k+1) = -1, where mu is the Möbius function (tested on the first 10^4 terms). - Amiram Eldar, Mar 06 2021

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

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

Cf. A179871, A179872, A179873, A179874, A179876, A179877, A179878, A179879, A179880, A179882, A179883, A179884, A179885, A179886, A179887, A179890, A179891.

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

cp's OEIS Frontend

A179882 a(n) is the corresponding value of contraharmonic mean B(h) of numbers k such that gcd(k, h) = 1 (k < h) for numbers h from A179877(n) and A179878(n).

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A007645 Generalized cuban primes: primes of the form x^2 + xy + y^2; or primes of the form x^2 + 3*y^2; or primes == 0 or 1 (mod 3).

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A179871 Numbers h such that antiharmonic mean B(h) of the numbers k < h such that gcd(k, h) = 1 is an integer.

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A179891 Composites 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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A179880 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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A179887 Nonprimes q such that antiharmonic mean B(q) of the numbers k < q such that gcd(k, q) = 1 is an integer, where B(q) = A053818(q) / A023896(q) = A175505(q) / A175506(q).

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A179890 Values of antiharmonic mean B(q) of the numbers k < q such that gcd(k, q) = 1 is an integer for nonprimes q from A179887, where B(q) = A053818(q) / A023896(q) = A175505(q) / A175506(q).

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