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 1-9 of 9 results.

A270713 Numbers that are equal to the product of the number of divisors of their first k powers, for some k.

Original entry on oeis.org

1, 2, 225, 4050, 66528, 113400, 120960, 92802153185280, 726046074908612178739200000000000, 3524292573661555639437312000000000000, 2308850758786565168980497090478080000000000, 142039354014714204088514497565910023710398021722450165760000000000000000
Offset: 1

Views

Author

Paolo P. Lava, Mar 22 2016

Keywords

Comments

a(2) = 2 is the only prime term possible, since the product of tau(p^i) is always even, and 2 is the only even prime. - Michael De Vlieger, Mar 27 2016
The corresponding k are: 1, 1, 2, 2, 2, 2, 2, 3, 4, 4, 4, 5, 5, 5. - Michel Marcus, Apr 08 2016; updated by Max Alekseyev, Jun 11 2025

Examples

			d(4050) * d(4050^2) = 30 * 135 = 4050;
d(66528) * d(66528^2) = 96 * 693 = 66528.

Links

Crossrefs

Cf. A000005, A270389.

Programs

  • Maple
    with(numtheory): P:=proc(q) local a,k,n;
for n from 1 to q do a:=tau(n); k:=1;
while a
  • Mathematica
    Select[Insert[Complement[Range@ #, Prime@ Range@ PrimePi@ #] &[2 10^5], 2, 2], Function[k, AnyTrue[Range@ 3, Product[DivisorSigma[0, k^i], {i, #}] == k &]]] (* Michael De Vlieger, Mar 25 2016 *)
  • PARI
    isok(m) = my(k = 1, prd = 1); while (prd < m, prd *= numdiv(m^k); k++); prd == m; \\ Michel Marcus, Apr 08 2016, Jun 12 2025

Extensions

a(8)-a(10) from Hiroaki Yamanouchi, Apr 07 2016
a(11)-a(14) from Max Alekseyev, Jun 10 2025

A275660 Numbers n such that sigma(n) = Sum_{j=1..k} d(n^j) for some k, where sigma(n) is the sum of the divisors of n and d(n) is the number of divisors of n.

Original entry on oeis.org

1, 13, 19, 34, 43, 53, 58, 89, 103, 151, 229, 251, 254, 329, 341, 349, 404, 433, 463, 593, 674, 701, 739, 859, 1033, 1223, 1429, 1483, 1506, 1670, 1709, 1826, 1846, 1886, 1889, 1948, 1951, 2067, 2091, 2143, 2255, 2308, 2431, 2699, 3001, 3079, 3319, 3739, 4003, 4093
Offset: 1

Views

Author

Paolo P. Lava, Aug 04 2016

Keywords

Comments

The primes in this sequence are A124199. - Robert Israel, Feb 20 2024

Examples

			d(53^1) + d(53^2) + d(53^3) + d(53^4) + d(53^5) + d(53^6) + d(53^7) + d(53^8) + d(53^9) = 54 = sigma(53).

Links

Crossrefs

Cf. A000005, A000203, A124199, A270389, A275661.

Programs

  • Maple
    P:= proc(q) local a,k,n;
for n from 1 to q do a:=sigma(n); k:=0;
while a>0 do k:=k+1; a:=a-tau(n^k); od;
if a=0 then print(n); fi; od; end: P(10^9);

A270337 Composite numbers equal to the number of divisors of one of their powers.

Original entry on oeis.org

9, 25, 28, 40, 45, 49, 81, 121, 153, 169, 225, 289, 325, 343, 361, 441, 529, 625, 640, 841, 961, 976, 1089, 1225, 1369, 1521, 1681, 1849, 2133, 2197, 2209, 2401, 2541, 2601, 2809, 3025, 3249, 3481, 3721, 4225, 4489, 4753, 4761, 4851, 5041, 5329, 5929, 6241, 6348, 6561, 6859, 6889
Offset: 1

Views

Author

Paolo P. Lava, Mar 15 2016

Keywords

Comments

Prime numbers are not considered since every prime p satisfies p = d(p^(p-1)), where d() represents the number of divisors.
In general, p^k = d((p^k)^((p^k-1)/k)) for any prime p and for any power k such that (p^k-1)/k is an integer.

Examples

			9 = d(9^4); 28 = d(28^3); 153 = d(153^8); etc.

Links

Crossrefs

Cf. A000005, A073049, A270389.

Programs

  • Maple
    with(numtheory): P:=proc(q) local a,k,n;
for n from 2 to q do if not isprime(n) then a:=tau(n); k:=0;
while a
  • Mathematica
    nn = 2000; Select[Select[Range@ nn, CompositeQ], Function[k, (SelectFirst[k^Range[nn/2], DivisorSigma[0, #] == k &] /. n_ /; MissingQ@ n -> 0) > 0]] (* Michael De Vlieger, Mar 17 2016, Version 10.2 *)

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 *)

A270443 Least m such that d(n^m) > n, where d(n) is the number of divisors of n.

Original entry on oeis.org

2, 3, 2, 5, 2, 7, 3, 5, 3, 11, 2, 13, 3, 3, 4, 17, 3, 19, 3, 4, 4, 23, 3, 13, 5, 9, 4, 29, 3, 31, 7, 5, 5, 5, 3, 37, 6, 6, 4, 41, 3, 43, 4, 5, 6, 47, 3, 25, 5, 7, 5, 53, 4, 7, 4, 7, 7, 59, 3, 61, 7, 5, 11, 8, 4, 67, 6, 8, 4, 71, 4, 73, 8, 6, 6, 8, 4, 79, 4, 21
Offset: 2

Views

Author

Paolo P. Lava, Mar 17 2016

Keywords

Comments

a(p) = p for any prime p.

Examples

			d(4^1) = 3, d(4^2) = 5 then a(4) = 2;
d(9^1) = 3, d(9^2) = 5, d(9^3) = 7, d(9^4) = 9, d(9^5) = 11, then a(9) = 5.

Links

Crossrefs

Cf. A000005, A270337, A270389.

Programs

  • Maple
    with(numtheory): P:=proc(q) local a,k,n;
for n from 2 to q do a:=tau(n); k:=1;
while a
  • Mathematica
    nn = 100; Table[SelectFirst[Range@ nn, DivisorSigma[0, n^#] > n &], {n, 2, nn}] (* Michael De Vlieger, Mar 17 2016, Version 10 *)
  • PARI
    a(n) = {p=1; until (numdiv(n^p) > n, p++); p;} \\ Michel Marcus, Mar 17 2016

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 1-9 of 9 results.

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