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 1-6 of 6 results.

A270389 Numbers that are equal to the sum of the number of divisors of their k first powers, for some k.

Original entry on oeis.org

1, 2, 5, 64, 203, 505, 524, 649, 818, 1295, 2469, 2869, 4355, 5048, 6083, 10415, 14909, 15021, 22329, 27433, 29189, 29369, 35719, 38023, 44099, 48229, 56372, 85329, 85343, 89270
Offset: 1

Views

Author

Paolo P. Lava, Mar 16 2016

Keywords

Examples

			d(1^1) = 1;
d(2^1) = 2;
d(5^1) + d(5^2) = 2 + 3 = 5;
d(64^1) + d(64^2) + d(64^3) + d(64^4) = 7 + 13 + 19 + 25 = 64;
d(203^1) + d(203^2) + d(203^3)+ d(203^4)+ d(203^5)+ d(203^6)+ d(203^7) = 4 + 9 + 16 + 25 + 36 + 49 + 64 = 203.

Links

Crossrefs

Cf. A000005, A270713.

Programs

  • Maple
    with(numtheory): P:=proc(q) local a,k,n;
for n from 1 to q do a:=0; k:=0;
while a
  • Mathematica
    Select[Range[10^4], Function[n, IntegerQ@ SelectFirst[Range@ 25, Total@ Map[DivisorSigma[0, #] &, n^Range[#]] == n &]]] (* Michael De Vlieger, Mar 17 2016, Version 10 *)
  • PARI
    is(n)=my(e=factor(n)[,2],k,t); while(tCharles R Greathouse IV, Mar 31 2016

Formula

Solutions of the equation n = Sum_{i=1..k}{d(n^k)}.

A283757 Numbers n such that phi(n) = Sum_{j=1..k} d(n^j) for some k, where phi(n) is the Euler totient function of n and d(n) is the number of divisors of n.

Original entry on oeis.org

1, 3, 8, 10, 18, 24, 30, 435, 485, 579, 678, 759, 1052, 1593, 3243, 3857, 3913, 4085, 4445, 4773, 4953, 5685, 6078, 6278, 6322, 6836, 7570, 9823, 10199, 10703, 12474, 12913, 12927, 14180, 14511, 14623, 16958, 17013, 17014, 17174, 17518, 17966, 18238, 19334, 19432
Offset: 1

Views

Author

Paolo P. Lava, Mar 16 2017

Keywords

Comments

Values of k: {1, 1, 1, 1, 1, 1, 1, 4, 9, 9, 4, 5, 8, 9, 8, 9, 9, 9, 9, 9, 9, 9, 8, 9, 9, 16, 9, 12, 12, 12, 4, 32, 12, 9, 12, 32, 12, 13, 12, 12, 12, 12, 12, 12, 9, ...}. - Michael De Vlieger, Mar 17 2017

Examples

			d(1052) + d(1052^2) + d(1052^3) + d(1052^4) + d(1052^5) + d(1052^6) + d(1052^7) + d(1052^8) = 524 = phi(1052).

Crossrefs

Cf. A000005, A000010, A270389, A270713, A275660.

Programs

  • Maple
    with(numtheory): P:=proc(q) local a,k,n; for n from 1 to q do a:=0; k:=0; while a
  • Mathematica
    Select[Range@ 4000, Module[{k = 1, e = EulerPhi@ #, b}, While[Set[b, Sum[DivisorSigma[0, #^j], {j, k}]] < e, k++]; If[b == e, True, False]] &] (* Michael De Vlieger, Mar 17 2017 *)

A283758 Numbers whose sum of divisors is equal to the product of the number of divisors of their k first powers, for some k.

Original entry on oeis.org

5, 22, 23, 102, 110, 382, 497, 510, 517, 527, 719, 1436, 4509, 5039, 6906, 8426, 8786, 9051, 9598, 9741, 9951, 10011, 10505, 10795, 11005, 11431, 11501, 11891, 11995, 12121, 13661, 13777, 13891, 13919, 14101, 14129, 14141, 28780, 31636, 32572, 32756, 33028, 33356
Offset: 1

Views

Author

Paolo P. Lava, Mar 16 2017

Keywords

Comments

Values of k: {2, 2, 3, 2, 2, 3, 3, 2, 3, 3, 5, 3, 3, 6, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 3, 4, 4, 4, 4, 4, 4, 4, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, ...}. - Michael De Vlieger, Mar 17 2017

Examples

			sigma(382) = 576 and d(382) * d(382^2) * d(382^3) = 4 * 9 * 16 = 576;
sigma(9598) = 14400 and d(9598) * d(9598^2) * d(9598^3) * d(9598^4) = 4 * 9 * 16 * 25 = 14400.

Links

Crossrefs

Cf. A000005, A000203, A270389, A270713, A275660, A283757, A283759.

Programs

  • Maple
    with(numtheory): P:=proc(q) local a,k,n; for n from 1 to q do a:=1; k:=0; while a
  • Mathematica
    Select[Range[2, 40000], Module[{k = 1, d = DivisorSigma[1, #], b}, While[Set[b, Product[DivisorSigma[0, #^j], {j, k}]] < d, k++]; If[b == d, True, False]] &] (* Michael De Vlieger, Mar 17 2017 *)

A283759 Numbers whose Euler totient function is equal to the product of the number of divisors of their k first powers, for some k.

Original entry on oeis.org

3, 7, 8, 10, 18, 24, 30, 57, 74, 344, 399, 494, 518, 629, 654, 679, 1154, 2408, 2989, 3048, 3175, 3458, 3789, 4218, 4578, 4890, 5022, 7668, 10602, 13720, 14647, 14701, 14837, 15613, 16133, 17563, 17945, 18335, 19608, 20195, 20358, 21243, 21336, 21423, 22083, 22503
Offset: 1

Views

Author

Paolo P. Lava, Mar 16 2017

Keywords

Comments

Values of k: {1, 2, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 2, 3, 3, 2, 3, 2, 3, 2, 3, 2, 2, 2, 2, 2, 2, 2, 4, 4, 4, 4, 3, 3, 3, 3, 2, 3, 2, 3, 2, 3, 3, 3, 3, 3, 4, 3, 2, 3, 2, 2, 3, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, ...}. - Michael De Vlieger, Mar 17 2017

Examples

			phi(629) = 576 and d(629) * d(629^2) * d(629^3) = 4 * 9 * 16= 576;
phi(14647) = 14400 and d(14647) * d(14647^2) * d(14647^3) * d(14647^4) = 4 * 9 * 16 * 25 = 14400.

Links

Crossrefs

Cf. A000005, A000010, A270389, A270713, A275660, A283757, A283758.

Programs

  • Maple
    with(numtheory): P:=proc(q) local a,k,n; for n from 1 to q do a:=1; k:=0; while a
  • Mathematica
    Select[Range[2, 25000], Module[{k = 1, e = EulerPhi@ #, b}, While[Set[b, Product[DivisorSigma[0, #^j], {j, k}]] < e, k++]; If[b == e, True, False]] &] (* Michael De Vlieger, Mar 17 2017 *)

A290592 Numbers x such that x = Sum_{j=1..k} d(j*x), for some k, where d(x) is the number of divisors of x.

Original entry on oeis.org

1, 2, 9, 10, 26, 34, 46, 76, 121, 128, 136, 140, 174, 194, 226, 230, 232, 240, 268, 278, 296, 325, 362, 370, 432, 434, 438, 575, 598, 610, 620, 637, 694, 708, 718, 734, 735, 756, 796, 808, 842, 854, 860, 866, 898, 922, 925, 986, 1048, 1050, 1072, 1168, 1196, 1228
Offset: 1

Views

Author

Paolo P. Lava, Aug 07 2017

Keywords

Comments

Values of k for the listed items are 1, 1, 2, 2, 4, 5, 6, 7, 14, 9, 9, 7, 11, 19, 21, 13, 14, 8, 19, 25, 17, 18, 31, 19, 13, 21, 23, 29, 27, 28, 22, 31, 53, 26, 54, 55, 23, 17, 45, 38, 62, 36, 29, 63, 65, 66, 42, 40, 47, 21, 41, 44, 36, 65.
All squarefree terms > 1 are even. - Robert Israel, Aug 07 2017

Examples

			For 34 we have that d(34) + d(2*34) + d(3*34) + d(4*34) + d(5*34) = 4 + 6 + 8 + 8 + 8 = 34.

Links

Crossrefs

Cf. A000005, A270713, A270389.

Programs

  • Maple
    with(numtheory): P:=proc(n) local a,k; a:=0; k:=0; while a

A269459 Numbers that are equal to the sum of the number of divisors of their first k arithmetic derivatives, for some k.

Original entry on oeis.org

15, 16, 45, 60, 69, 75, 112, 116, 236, 296, 319, 452, 576, 586, 843, 1047, 1184, 1704, 1902, 2852, 2966, 3068, 3122, 4708, 4805, 5684, 6150, 6712, 7126, 10920, 10950, 13107, 16700, 18698, 27828, 29309, 31142, 31448, 31764, 43152, 48584, 51609, 53822, 62472, 63008
Offset: 1

Views

Author

Paolo P. Lava, Apr 06 2016

Keywords

Examples

			The first eight arithmetic derivatives of 75 are 55, 16, 32, 80, 176, 368, 752, 1520 and d(55) + d(16) + d(32) + d(80) + d(176) + d(368) + d(752) + d(1520) = 4 + 5 + 6 + 10 + 10 + 10 + 10 + 20 = 75.

Crossrefs

Cf. A000005, A003415, A270389, A270713.

Programs

  • Maple
    with(numtheory): P:=proc(q) local a,b,k,n,p; for n from 1 to q do a:=0; k:=1; b:=n;
while a0 then a:=a+tau(b); else break; fi; od;
if n=a then print(n); fi; od; end: P(10^6);
  • PARI
    ad(n) = if (n<1, 0, my(f = factor(n)); n*sum(k=1, #f~, f[k,2]/f[k,1]));
isok(n) = {ss = 0; kn = n; while (ss < n, der = ad(kn); if (der == 0, break); ss += numdiv(der); kn = der); ss == n;} \\ Michel Marcus, Apr 08 2016

Extensions

a(35)-a(45) from Amiram Eldar, Jun 23 2023
Showing 1-6 of 6 results.

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