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.

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

A211780 a(n) = Sum_{d|n, dA000005 is the number of divisors.

Original entry on oeis.org

0, 2, 2, 7, 2, 14, 2, 18, 9, 18, 2, 43, 2, 22, 20, 41, 2, 54, 2, 57, 24, 30, 2, 106, 13, 34, 31, 71, 2, 110, 2, 88, 32, 42, 28, 162, 2, 46, 36, 142, 2, 138, 2, 99, 81, 54, 2, 237, 17, 102, 44, 113, 2, 178, 36, 178, 48, 66, 2, 325, 2, 70, 99, 183, 40, 194, 2
Offset: 1

Views

Author

Jaroslav Krizek, Apr 20 2012

Keywords

Comments

Numbers n such that n divides a(n) are given in A068978.

Examples

			For n = 12: Sum_{d|n, d<n} d * tau(n / d) = 1*6 + 2*4 + 3*3 + 4*2 + 6*2 = 43.

Links

Crossrefs

Cf. A000005, A000203, A007429, A068978, A098198, A211779.

Programs

  • Mathematica
    Table[Sum[d*DivisorSigma[0, n/d], {d, Most[Divisors[n]]}], {n, 100}] (* T. D. Noe, Apr 27 2012 *)
  • PARI
    A211780(n) = sumdiv(n, d, sigma(d))-n; \\ Antti Karttunen, Nov 13 2017
  • Python
    A211780=lambda n:sum(sigma(d) for d in divisors(n, generator=True))-n
from sympy import divisor_sigma as sigma, divisors # M. F. Hasler, Jun 03 2024

Formula

a(n) = A007429(n) - n = A211779(n) + A000203(n) - n .
a(n) = (Sum_{d|n} A000203(d)) - n. - Antti Karttunen, Nov 13 2017
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Pi^4/36 - 1 = 1.705808... . - Amiram Eldar, Jun 06 2024

Extensions

Name edited by M. F. Hasler, Jun 03 2024

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 *)

A276737 a(n) = denominator of Sum_{d|n} tau(d)/d.

Original entry on oeis.org

1, 1, 3, 4, 5, 3, 7, 4, 1, 5, 11, 12, 13, 7, 3, 16, 17, 1, 19, 20, 7, 11, 23, 12, 25, 13, 27, 28, 29, 3, 31, 4, 33, 17, 5, 2, 37, 19, 13, 20, 41, 7, 43, 4, 5, 23, 47, 16, 49, 25, 51, 52, 53, 27, 55, 28, 19, 29, 59, 12, 61, 31, 7, 64, 13, 33, 67, 68, 69, 5, 71
Offset: 1

Views

Author

Jaroslav Krizek, Sep 16 2016

Keywords

Comments

Also denominator of (Sum_{d|n} sigma(d)) / n.

Examples

			For n=6; {d_6} = {1, 2, 3, 6}; {tau(d)_6} = {1, 2, 2, 4}; Sum_{d|6} tau(d)/d = 1/1 + 2/2 + 2/3 + 4/6 = 20/6 = 10/3; a(6) = 3.
For n=6; {d_6} = {1, 2, 3, 6}; {sigma(d)_6} = {1, 3, 4, 12};  (Sum_{d|6} sigma(d))/6 = (1+3+4+12)/6 = 10/3; a(6) = 3.

Links

Crossrefs

Cf. A000005, A007429, A068978, A276736 (numerators).

Programs

  • Magma
    [Denominator(&+[NumberOfDivisors(d)/d: d in Divisors(n)]): n in [1..100]]
  • Mathematica
    Table[Denominator@ Total[DivisorSigma[0, #]/#] &@ Divisors@ n, {n, 71}] (* Michael De Vlieger, Sep 16 2016 *)
  • PARI
    a(n) = denominator(sumdiv(n, d, numdiv(d)/d)); \\ Michel Marcus, Sep 16 2016

Formula

a(A068978(n)) = 1.
For all n, n = (Sum_{d|n} sigma(d)) / (Sum_{d|n} tau(d)/d) = (Sum_{d|n} d*tau(n/d)) / (Sum_{d|n} tau(d)/d) = A007429(n) * a(n) / A276736(n).
Showing 1-6 of 6 results.

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

Content is available under The OEIS End-User License Agreement.