cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 11-18 of 18 results.

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

Views

Author

Jaroslav Krizek, Jul 30 2010, Jul 31 2010

Keywords

Comments

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

Examples

			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]

Links

Crossrefs

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

Programs

  • Mathematica
    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]]]
  • PARI
    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

Formula

a(n) = (3*A179882(n) - 1)/2. - Hilko Koning, Aug 01 2018
a(n) = A179878(n) - 1. - Amiram Eldar, May 24 2025

Extensions

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

Views

Author

Jaroslav Krizek, Jul 30 2010, Jul 31 2010

Keywords

Comments

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

Links

Crossrefs

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

Programs

Formula

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

Views

Author

Jaroslav Krizek, Jul 30 2010, Jul 31 2010

Keywords

Comments

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

Crossrefs

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

Views

Author

Jaroslav Krizek, Jul 30 2010, Jul 31 2010

Keywords

Comments

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.

Crossrefs

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

Formula

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

Views

Author

Jaroslav Krizek, Jul 30 2010, Jul 31 2010

Keywords

Comments

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.

Crossrefs

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

Formula

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

Extensions

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

Views

Author

Jaroslav Krizek, Jul 30 2010, Jul 31 2010

Keywords

Crossrefs

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

Formula

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

Extensions

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

Views

Author

Jaroslav Krizek, Jul 30 2010, Jul 31 2010

Keywords

Comments

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).

Crossrefs

Cf. A023896, A053818, A175505, A175506.
Cf. A179871, A179872, A179873, A179874, A179875, A179876, A179877, A179878, A179879, A179880, A179882, A179883, A179884, A179885, A179887, A179890, A179891.

Formula

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

Extensions

More terms from Amiram Eldar, May 26 2025

A179896 Sum of the numbers between k := n-th nonprime and 2k (like a jump in a Sieve of Eratosthenes).

Original entry on oeis.org

0, 18, 45, 84, 108, 135, 198, 273, 315, 360, 459, 570, 630, 693, 828, 900, 975, 1053, 1134, 1305, 1488, 1584, 1683, 1785, 1890, 2109, 2223, 2340, 2583, 2838, 2970, 3105, 3384, 3528, 3675, 3825, 3978, 4293, 4455, 4620, 4788, 4959, 5310, 5673, 5859, 6048, 6240, 6435
Offset: 1

Views

Author

Odimar Fabeny, Jul 31 2010

Keywords

Comments

The values 4, 7, 10... (A016777 for n>1) are the values of floor( a(k)/ A018252(k) ) where k runs through the indices where A179879(k) mod A018252(k) != 0. - Odimar Fabeny.
Proof: a(k)/A018252(k) is 3*(A081252(k)-1)/2. This is a non-integer iff A018252(k) is even. Since the n-th even nonprime is 2*n+2, floor(3*(2*n+1)/2) = 3*n+1=a(n). - Robert Israel, Aug 27 2014

Examples

			0(0) = 0, 1(2) = 0, 4(8) = 5,6,7 = 18, 6(12) = 7,8,9,10,11 = 45 and so on.

Links

Crossrefs

Cf. A000040, A179545.

Programs

  • Maple
    ithnonprime := proc(n)local k: option remember: if(n=1)then return 1: else k := procname(n-1)+1: while true do if(not isprime(k))then return k fi: k:=k+1: od: fi: end:
A179896 := proc(n)local k: k:=ithnonprime(n): return 3*k*(k-1)/2: end:
seq(A179896(n),n=1..40); # Nathaniel Johnston, Apr 21 2011
  • Mathematica
    f[n_] := Plus @@ Range[n + 1, 2 n - 1]; f /@ Select[ Range@ 64, ! PrimeQ@# &] (* Robert G. Wilson v, Sep 02 2010 *)

Formula

a(n) = A045943(A141468(n+1)-1). - R. J. Mathar, Sep 01 2010

Extensions

More terms from Odimar Fabeny, Aug 11 2010
Offset adapted to A141468 and to match another 0 - R. J. Mathar, Sep 01 2010
Previous Showing 11-18 of 18 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.