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 1-10 of 10 results.

A078544 Non-balanced numbers in A015761.

Original entry on oeis.org

38, 54, 87, 95, 114, 126, 135, 147, 174, 182, 209, 215, 216, 222, 258, 285, 294, 297, 315, 430, 455, 456, 540, 546, 551, 609, 627, 632, 635, 645, 762, 783, 836, 899, 957, 1015, 1032, 1102, 1107, 1118, 1140, 1188, 1211, 1218, 1270, 1290, 1330, 1343, 1349
Offset: 1

Views

Author

Labos Elemer, Dec 05 2002

Keywords

Links

Crossrefs

Cf. A015761, A020492, A000005, A000010, A078539.

Programs

  • Mathematica
    Select[Range[1350], Divisible[DivisorSigma[3, #],(phi = EulerPhi[#])] && !Divisible[DivisorSigma[1, #], phi] &] (* Amiram Eldar, Jun 23 2019 *)
  • PARI
    isok(k) = my(phi=eulerphi(k)); ((sigma(k, 3) % phi) == 0) && (sigma(k) % phi); \\ Michel Marcus, Jun 23 2019

Formula

Mod[sigma[3, a(n)], phi[a(n)]]=0; Mod[sigma[1, a(n)], phi[a(n)]]><0.

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

A078540 Least non-balanced x (i.e., not in A020492) such that sigma(p(n),x)/phi(x) is an integer, where p(n) = n-th prime.

Original entry on oeis.org

22, 38, 46, 295, 235, 749, 3687, 6128, 1415, 4254, 10451, 8351, 334, 4511, 3398, 1286, 148870, 11015, 35519, 10239, 14072, 76088, 5991, 718, 11654, 30761, 7431, 20993, 700654, 22169, 5095, 4198, 27415, 26744, 14318, 48368, 180878, 16991, 173123, 4166, 5033, 7246
Offset: 1

Views

Author

Labos Elemer, Dec 02 2002

Keywords

Examples

			n=6: prime(6)=13, cases of sigma(13,x)/phi(x) is an integer are listed in A015771: 1, 2, 3, 6, 12, etc.; the first term which is non-balanced, i.e., not in A020492, is a(6) = 749 = A020492(31); a(29) = 700854 and a(45) = 510759 are remarkably large.

Links

Crossrefs

Cf. A000005, A000010, A000040, A015759-A015774, A020492, A078538, A078539.

Programs

  • Mathematica
    Table[fl=1; Do[s1=DivisorSigma[1, n]/EulerPhi[n]; sk=DivisorSigma[Prime[k], n]/EulerPhi[n]; If[ !IntegerQ[s1]&&IntegerQ[sk]&&Equal[fl, 1], Print[{n, Prime[k]}]; fl=0], {n, 1, 1000000}], {k, 1, 100}]
  • PARI
    lista(nmax) = {my(ps = primes(nmax), pmax = ps[#ps], v = vector(pmax), c = 0, k = 2, f, e, p); while(c < nmax, f = factor(k); e = eulerphi(f); if(sigma(f) % e > 0, for(i = 1, nmax, p = ps[i]; if(!(sigma(f, p) % e) && v[p] == 0, c++; v[p] = k))); k++); for(i = 1, pmax, if(v[i] > 0, print1(v[i], ", ")));} \\ Amiram Eldar, Aug 29 2024

Formula

a(n) = min{x; A000203(x) mod A000005(x) = 0 but sigma(A000040(n), x) mod phi(x) is not 0}.

Extensions

a(18) corrected and more terms added by Amiram Eldar, Aug 29 2024

A078542 Unbalanced composite numbers.

Original entry on oeis.org

4, 8, 9, 10, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 32, 33, 34, 36, 38, 39, 40, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 72, 74, 75, 76, 77, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 106, 108
Offset: 1

Views

Author

Labos Elemer, Dec 04 2002

Keywords

Examples

			46 = 2*23 and sigma(46)/phi(46) = 72/22 is not an integer, so 46 is in the sequence.

Links

Crossrefs

Cf. A000010, A000203, A020492, A078538, A078539, A078540.

Programs

  • Mathematica
    Do[s=DivisorSigma[1, n]/EulerPhi[n]; If[ !IntegerQ[s]&&!PrimeQ[n], Print[n]], {n, 1, 256}]
Select[Range[150],CompositeQ[#]&&!IntegerQ[DivisorSigma[1,#]/ EulerPhi[ #]]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jan 19 2020 *)
  • PARI
    lista(nn) = forcomposite(n=1, nn, if (denominator(sigma(n)/eulerphi(n)) != 1, print1(n, ", "))); \\ Michel Marcus, Jul 11 2018

A078543 Balanced refactorable numbers.

Original entry on oeis.org

1, 2, 12, 56, 248, 12192, 28896, 60960, 61344, 66528, 86304, 94944, 129504, 133920, 140448, 182880, 201924, 207264, 242316, 282720, 408672, 416640, 426720, 429408, 604128, 664608, 671760, 776736, 792480, 854496, 862752, 906528
Offset: 1

Views

Author

Labos Elemer, Dec 04 2002

Keywords

Examples

			n=56: tau(56)=8, sigma(56)=120, phi(56)=24, q1=120/24=5 for balancedness, q2=56/8=7 for refactorability.

Links

Crossrefs

Intersection of A033950 and A020492.
Cf. A078538, A078539, A078540.

Programs

  • Mathematica
    Do[s=DivisorSigma[1, n]/EulerPhi[n]; If[ !IntegerQ[s]&&!PrimeQ[n], Print[n]], {n, 1, 256}]
Showing 1-10 of 10 results.

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

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