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.

Showing 1-7 of 7 results.

A074866 Non-balanced numbers in A015763.

Original entry on oeis.org

46, 134, 138, 161, 184, 230, 299, 322, 402, 414, 483, 552, 598, 623, 644, 670, 690, 805, 874, 897, 966, 1173, 1196, 1208, 1242, 1246, 1288, 1495, 1608, 1610, 1702, 1794, 1869, 1909, 1932, 1990, 1992, 2010, 2024, 2070, 2185, 2202, 2346, 2415, 2576
Offset: 1

Views

Author

Labos Elemer, Dec 05 2002

Keywords

Links

Crossrefs

Cf. A000010, A000203, A001160, A015763, A020492, A078539.
Cf. A074868, A077801, A077803, A078544, A078549, A078550.

Programs

  • Mathematica
    q[n_] := Sign[Mod[DivisorSigma[{1, 5}, n], EulerPhi[n]]] == {1, 0}; Select[Range[3000], q] (* Amiram Eldar, Apr 11 2024 *)
  • PARI
    is(n) = {my(f = factor(n), phi = eulerphi(f)); (sigma(f) % phi) && !(sigma(f, 5) % phi);} \\ Amiram Eldar, Apr 11 2024

Formula

sigma_5(a(n)) mod phi(a(n)) = 0; sigma(a(n)) mod phi(a(n)) <> 0.

A074868 Non-balanced numbers in A015765.

Original entry on oeis.org

295, 590, 767, 885, 1038, 1416, 1534, 1589, 1770, 2065, 2301, 2422, 3178, 3186, 3245, 3304, 3448, 3540, 4130, 4602, 4767, 5192, 5230, 5448, 5516, 5605, 6195, 6291, 6356, 6490, 6574, 6860, 7266, 7945, 7965, 8236, 8260, 8437, 8968, 9145, 9204, 9342
Offset: 1

Views

Author

Labos Elemer, Dec 05 2002

Keywords

Links

Crossrefs

Cf. A000010, A000203, A013955, A015765, A020492, A078539.
Cf. A074866, A077801, A077803, A078544, A078549, A078550.

Programs

  • Mathematica
    q[n_] := Sign[Mod[DivisorSigma[{1, 7}, n], EulerPhi[n]]] == {1, 0}; Select[Range[10000], q] (* Amiram Eldar, Apr 11 2024 *)
  • PARI
    is(n) = {my(f = factor(n), phi = eulerphi(f)); (sigma(f) % phi) && !(sigma(f, 7) % phi);} \\ Amiram Eldar, Apr 11 2024

Formula

sigma_7(a(n)) mod phi(a(n)) = 0; sigma(a(n)) mod phi(a(n)) <> 0.

A077801 Non-balanced numbers in A015767.

Original entry on oeis.org

38, 54, 87, 95, 114, 126, 135, 147, 174, 182, 209, 215, 216, 222, 258, 266, 285, 294, 297, 315, 342, 378, 430, 447, 455, 456, 494, 518, 540, 546, 551, 609, 627, 632, 635, 645, 654, 665, 702, 762, 783, 798, 836, 894, 899, 945, 957, 1015, 1022, 1032, 1064
Offset: 1

Views

Author

Labos Elemer, Dec 05 2002

Keywords

Links

Crossrefs

Cf. A000010, A000203, A013957, A015767, A020492, A078539.
Cf. A074866, A074868, A077803, A078544, A078549, A078550.

Programs

  • Mathematica
    q[n_] := Sign[Mod[DivisorSigma[{1, 9}, n], EulerPhi[n]]] == {1, 0}; Select[Range[1000], q] (* Amiram Eldar, Apr 11 2024 *)
  • PARI
    is(n) = {my(f = factor(n), phi = eulerphi(f)); (sigma(f) % phi) && !(sigma(f, 9) % phi);} \\ Amiram Eldar, Apr 11 2024

Formula

sigma_9(a(n)) mod phi(a(n)) = 0; sigma(a(n)) mod phi(a(n)) <> 0.

A077803 Non-balanced numbers in A015769.

Original entry on oeis.org

235, 329, 470, 658, 695, 705, 799, 807, 940, 987, 1316, 1390, 1410, 1529, 1598, 1614, 1645, 1786, 1880, 1969, 1974, 2085, 2115, 2397, 2632, 2734, 2820, 3055, 3058, 3290, 3478, 3938, 3948, 4136, 4170, 4230, 4465, 4587, 4794, 4935, 5264, 5358, 5593, 5640
Offset: 1

Views

Author

Labos Elemer, Dec 05 2002

Keywords

Links

Crossrefs

Cf. A000010, A000203, A013959, A015769, A020492, A078539.
Cf. A074866, A074868, A077801, A078544, A078549, A078550.

Programs

  • Mathematica
    q[n_] := Sign[Mod[DivisorSigma[{1, 11}, n], EulerPhi[n]]] == {1, 0}; Select[Range[6000], q] (* Amiram Eldar, Apr 11 2024 *)
  • PARI
    is(n) = {my(f = factor(n), phi = eulerphi(f)); (sigma(f) % phi) && !(sigma(f, 11) % phi);} \\ Amiram Eldar, Apr 11 2024

Formula

sigma_11(a(n)) mod phi(a(n)) = 0; sigma(a(n)) mod phi(a(n)) <> 0.

A078549 Non-balanced numbers in A015771.

Original entry on oeis.org

749, 1498, 2247, 2568, 2889, 2996, 3745, 3959, 4494, 5778, 5992, 6741, 7490, 7918, 8876, 8988, 9416, 9737, 9994, 11235, 11556, 11877, 11984, 12733, 13482, 14231, 14445, 14980, 16264, 17976, 18404, 19474, 20223, 20804, 22363, 22470, 23112
Offset: 1

Views

Author

Labos Elemer, Dec 05 2002

Keywords

Links

Crossrefs

Cf. A000010, A000203, A013961, A015771, A020492, A078539.
Cf. A074866, A074868, A077801, A077803, A078544, A078550.

Programs

  • Mathematica
    q[n_] := Sign[Mod[DivisorSigma[{1, 13}, n], EulerPhi[n]]] == {1, 0}; Select[Range[24000], q] (* Amiram Eldar, Apr 11 2024 *)
  • PARI
    is(n) = {my(f = factor(n), phi = eulerphi(f)); (sigma(f) % phi) && !(sigma(f, 13) % phi);} \\ Amiram Eldar, Apr 11 2024

Formula

sigma_13(a(n)) mod phi(a(n)) = 0; sigma(a(n)) mod phi(a(n)) <> 0.

A078550 Non-balanced numbers in A015774.

Original entry on oeis.org

38, 46, 54, 87, 95, 114, 126, 134, 135, 138, 147, 161, 174, 182, 184, 209, 215, 216, 222, 230, 258, 285, 294, 297, 299, 315, 322, 398, 402, 414, 430, 437, 455, 456, 483, 540, 546, 551, 552, 598, 609, 623, 627, 632, 635, 644, 645, 670, 690, 762, 783, 805
Offset: 1

Views

Author

Labos Elemer, Dec 05 2002

Keywords

Links

Crossrefs

Cf. A000010, A000203, A013963, A015774, A020492, A078539.
Cf. A074866, A074868, A077801, A077803, A078544, A078549.

Programs

  • Mathematica
    q[n_] := Sign[Mod[DivisorSigma[{1, 15}, n], EulerPhi[n]]] == {1, 0}; Select[Range[1000], q] (* Amiram Eldar, Apr 11 2024 *)
  • PARI
    is(n) = {my(f = factor(n), phi = eulerphi(f)); (sigma(f) % phi) && !(sigma(f, 15) % phi);} \\ Amiram Eldar, Apr 11 2024

Formula

sigma_15(a(n)) mod phi(a(n)) = 0; sigma(a(n)) mod phi(a(n)) <> 0.

Extensions

Name corrected by Amiram Eldar, Apr 11 2024

A071188 Largest prime factor of number of divisors of n; a(1)=1.

Original entry on oeis.org

1, 2, 2, 3, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 2, 5, 2, 3, 2, 3, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 3, 3, 2, 2, 5, 3, 3, 2, 3, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 3, 7, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 3, 3, 2, 2, 2, 5, 5, 2, 2, 3, 2, 2, 2, 2, 2, 3
Offset: 1

Views

Author

Reinhard Zumkeller, May 15 2002

Keywords

Comments

From Robert Israel, Dec 04 2016: (Start)
a(n)=2 if and only if every member of the prime signature of n is of the form 2^k-1.
a(m*k) = max(a(m),a(k)) if m and k are coprime. (End)

Links

Crossrefs

Cf. A000005, A006530, A071187, A071190, A078542, A078543, A078544, A124010.
Cf. A327839, A354181.

Programs

  • Haskell
    a071188 = a006530 . a000005  -- Reinhard Zumkeller, Sep 04 2013
  • Maple
    f:= n -> max(1, numtheory:-factorset(numtheory:-tau(n))):
map(f, [$1..100]); # Robert Israel, Dec 04 2016
  • Mathematica
    Max[Transpose[FactorInteger[#]][[1]]]&/@DivisorSigma[0,Range[100]] (* Harvey P. Dale, Aug 28 2013 *)
  • PARI
    a(n) = if(n == 1, 1, vecmax(factor(numdiv(n))[, 1])); \\ Michel Marcus, Dec 05 2016

Formula

a(n) = A006530(A000005(n)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*d(1) + Sum_{k>=2} prime(k)*(d(k) - d(k-1)) = 2.4365518864..., where d(1) = A327839, and for k >= 2, d(k) is the asymptotic density of numbers whose number of divisors is a prime(k)-smooth number, i.e., d(k) = Product_{p prime} ((1 - 1/p) * Sum_{i, A006530(i) <= prime(k)} 1/p^(i-1)) (see A354181 for an example). - Amiram Eldar, Jan 15 2024
Showing 1-7 of 7 results.

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