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 1-5 of 5 results.

A226564 Numbers k such that sum(d|k, sigma(d)^2/d) is an integer, where d are the divisors of k.

Original entry on oeis.org

1, 205, 3895, 8525, 17050, 71951, 74005, 148010, 359755, 451825, 903650, 1628110, 1632005, 1798775, 2346674, 3597550, 4218285, 8436570, 8993875, 14749955, 17987750, 50471410, 59071771, 92802270, 95335075, 190670150, 280249145, 295358855, 451356495, 481068170
Offset: 1

Views

Author

Paolo P. Lava, Jun 11 2013

Keywords

Examples

			Divisors of 71951 are 1, 11, 31, 211, 341,  2321, 6541, 71951.
sigma(1) = 1, sigma(11) = 12, sigma(31) = 32, sigma(211) = 212, sigma(341) = 384, sigma(2321) = 2544, sigma(6541) = 6784, sigma(71951) = 81408.
(1^2/1 + 12^2/11 + 32^2/31 + 212^2/211 + 384^2/341 + 2544^2/2321 + 6784^2/6541 + 81408^2/71951) = 102625.

Links

Crossrefs

Cf. A068978, A068986, A226563, A226564, A226566

Programs

  • Maple
    with(numtheory); ListA226564:=proc(q) local a,b,k,n;
for n from 1 to q do a:=[op(divisors(n))]; b:=add(sigma(a[k])^2/a[k],k=1..nops(a));
if type(b,integer) then print(n); fi; od; end: ListA226564(10^6);

Extensions

a(12)-a(30) from Giovanni Resta, Jun 11 2013

A226566 Numbers k such that Sum_{d|k} sigma(d)^3/d is an integer, where d are the divisors of k.

Original entry on oeis.org

1, 201, 981, 1962, 3663, 7326, 10791, 12753, 15879, 21582, 25506, 30411, 56898, 60822, 135749, 140283, 172161, 212454, 266727, 280566, 334521, 344322, 360027, 395343, 399267, 407247, 507177, 625878, 669042, 720054, 739674, 790686, 798534, 881919, 1014354, 1221741
Offset: 1

Views

Author

Paolo P. Lava, Jun 11 2013

Keywords

Examples

			Divisors of 981 are 1, 3, 9, 109, 327, 981.
sigma(1) = 1, sigma(3) = 4, sigma(9) = 13, sigma(109) = 110, sigma(327) = 440, sigma(981) = 1430.
(1^3/1 + 4^3/3 + 13^3/9 + 110^3/109 + 440^3/327 + 1430^3/981) = 3253822.

Links

Crossrefs

Cf. A068978, A068986, A226563, A226564, A226565

Programs

  • Maple
    with(numtheory); ListA226566:=proc(q) local a,b,k,n;
for n from 1 to q do a:=[op(divisors(n))]; b:=add(sigma(a[k])^3/a[k],k=1..nops(a));
if type(b,integer) then print(n); fi; od; end: ListA226566(10^6);
  • Mathematica
    aQ[n_] := IntegerQ[DivisorSum[n, DivisorSigma[1, #]^3/# &]]; Select[Range[10^5], aQ] (* Amiram Eldar, Sep 18 2019 *)
  • PARI
    isok(n) = denominator(sumdiv(n, d, sigma(d)^3/d)) == 1; \\ Michel Marcus, Sep 18 2019

A226563 Numbers k such that sum(d|k, sigma(d)^2) is a multiple of k.

Original entry on oeis.org

1, 2, 10, 185, 370, 3145, 6290, 40885, 53465, 63750, 81770, 106930, 241400, 348750, 427720, 828750, 866200, 1207000, 1306875, 1635449, 2613750, 3138200, 3270898, 7149375, 8054345, 8177245, 14298750, 14725400, 15691000, 16108690, 16354490, 16989375, 30368120
Offset: 1

Views

Author

Paolo P. Lava, Jun 11 2013

Keywords

Examples

			Divisors of 3145 are 1, 5, 17, 37, 85, 185, 629, 3145.
sigma(1) = 1, sigma(5) = 6, sigma(17) = 18, sigma(37) = 38, sigma(85) = 108, sigma(185) = 228, sigma(629) = 684, sigma(3145) = 4104.
(1^2 + 6^2 + 18^2 + 38^2 + 108^2 + 228^2 + 684^2 + 4104^2) / 3145 = 5525.

Links

Crossrefs

Cf. A068978, A068986, A226564-A226566

Programs

  • Maple
    with(numtheory); ListA226563:=proc(q) local a,b,k,n;
for n from 1 to q do a:=[op(divisors(n))]; b:=add(sigma(a[k])^2,k=1..nops(a));
if type(b/n,integer) then print(n); fi; od; end: ListA226563(10^6);

Extensions

a(14)-a(33) from Giovanni Resta, Jun 11 2013

A318492 a(n) is the denominator of Sum_{d|n} Sum_{j|d} 1/j.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 2, 11, 12, 13, 14, 15, 16, 17, 9, 19, 20, 1, 22, 23, 24, 25, 26, 27, 28, 29, 6, 31, 32, 33, 34, 7, 18, 37, 38, 13, 40, 41, 2, 43, 44, 45, 46, 47, 16, 49, 5, 51, 52, 53, 27, 5, 8, 19, 58, 59, 60, 61, 62, 21, 64, 65, 66, 67, 4, 69, 14, 71, 36, 73, 74, 75
Offset: 1

Views

Author

Ilya Gutkovskiy, Aug 27 2018

Keywords

Examples

			1, 5/2, 7/3, 17/4, 11/5, 35/6, 15/7, 49/8, 34/9, 11/2, 23/11, 119/12, 27/13, 75/14, 77/15, 129/16, ...

Crossrefs

Cf. A000005, A000203, A006171, A007429, A017665, A017666, A060640, A068986 (positions of 1's), A318491 (numerators).

Programs

  • Mathematica
    Denominator[Table[Sum[DivisorSigma[-1, d], {d, Divisors[n]}], {n, 75}]]
Denominator[Table[Sum[DivisorSigma[1, d]/d, {d, Divisors[n]}], {n, 75}]]
Denominator[Table[Sum[d DivisorSigma[0, d], {d, Divisors[n]}]/n, {n, 75}]]
nmax = 75; Rest[Denominator[CoefficientList[Series[Sum[DivisorSigma[1, k] x^k/(k (1 - x^k)), {k, 1, nmax}], {x, 0, nmax}], x]]]
nmax = 75; Rest[Denominator[CoefficientList[Series[-Log[Product[(1 - x^k)^DivisorSigma[0, k], {k, 1, nmax}]], {x, 0, nmax}], x]]]
  • PARI
    a(n) = denominator(sumdiv(n, d, sumdiv(d, j, 1/j))); \\ Michel Marcus, Aug 28 2018

Formula

Denominators of coefficients in expansion of Sum_{k>=1} sigma(k)*x^k/(k*(1 - x^k)), where sigma(k) = sum of divisors of k (A000203).
Denominators of coefficients in expansion of -log(Product_{k>=1} (1 - x^k)^tau(k)), where tau(k) = number of divisors of k (A000005).
a(n) = denominator of Sum_{d|n} sigma(d)/d.
a(n) = denominator of (1/n)*Sum_{d|n} d*tau(d).

A226565 Numbers k such that Sum_{d|k} sigma(d)^3 is a multiple of k.

Original entry on oeis.org

1, 2, 14, 32, 39, 42, 78, 96, 105, 117, 126, 133, 189, 195, 210, 224, 234, 266, 288, 378, 390, 399, 465, 480, 546, 585, 672, 793, 798, 930, 975, 1170, 1197, 1248, 1365, 1470, 1586, 1638, 1862, 1950, 1995, 2016, 2379, 2394, 2646, 2730, 3255, 3360, 3393, 3591
Offset: 1

Views

Author

Paolo P. Lava, Jun 11 2013

Keywords

Examples

			Divisors of 189 are 1, 3, 7, 9, 21, 27, 63, 189, sigma(1) = 1, sigma(3) = 4, sigma(7) = 8, sigma(9) = 13, sigma(21) = 32, sigma(27) = 40, sigma(63) = 104, sigma(189) = 320. (1^3 + 4^3 + 8^3 + 13^3 + 32^3 + 40^3 + 104^3 + 320^3) / 189 = 179854.

Links

Crossrefs

Cf. A068978, A068986, A226563, A226564, A226566, A344044.

Programs

  • Maple
    with(numtheory); ListA226565:=proc(q) local a,b,k,n;
for n from 1 to q do a:=[op(divisors(n))]; b:=add(sigma(a[k])^3/n,k=1..nops(a));
if type(b,integer) then print(n); fi; od; end: ListA226565 (10^6);
  • Mathematica
    Select[Range[4000],Divisible[Total[DivisorSigma[1,#]^3&/@Divisors[#]],#]&] (* Harvey P. Dale, Sep 17 2019 *)
s[n_] := DivisorSum[n, DivisorSigma[1, #]^3 &]; Select[Range[3600], Divisible[s[#], #] &] (* Amiram Eldar, Jul 01 2022 *)
Showing 1-5 of 5 results.

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