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.

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A210494 Biharmonic numbers: numbers m such that ( Hd(m)+Cd(m) )/2 is an integer, where Hd(m) and Cd(m) are the harmonic mean and the contraharmonic (or antiharmonic) mean of the divisors of m.

Original entry on oeis.org

1, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 35, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 119, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263
Offset: 1

Views

Author

Bruno Berselli, Oct 03 2013 - proposed by Umberto Cerruti (Department of Mathematics "Giuseppe Peano", University of Turin, Italy)

Keywords

Comments

Equivalently, numbers m such that ( m*sigma_0(m)+sigma_2(m) ) / (2*sigma_1(m)) = (A038040(m) + A001157(m))/A074400(m) is an integer.
All odd primes belong to the sequence. In fact, if p is an odd prime, (p*sigma_0(p)+sigma_2(p))/(2*sigma_1(p)) = (p+1)/2, therefore p is a biharmonic number.

Links

Crossrefs

Cf. A001599 (harmonic numbers), A020487 (antiharmonic numbers), A038040 (n*sigma_0(n)), A001157 (sigma_2(n)), A074400 (2*sigma_1(n)), A230214 (nonprime terms of A210494).
Cf. A189835.

Programs

  • Haskell
    a210494 n = a210494_list !! (n-1)
a210494_list = filter
   (\x -> (a001157 x + a038040 x) `mod` a074400 x == 0) [1..]
-- Reinhard Zumkeller, Jan 21 2014
  • Magma
    IsInteger := func; [n: n in [1..300] | IsInteger((n*NumberOfDivisors(n)+DivisorSigma(2,n))/(2*SumOfDivisors(n)))];
  • Maple
    with(numtheory); P:=proc(q) local a,k,n;
for n from 1 to q do a:=divisors(n);
if type((n*tau(n)+add(a[k]^2,k=1..nops(a)))/(2*sigma(n)),integer) then print(n); fi; od; end; P(1000); # Paolo P. Lava, Oct 11 2013
  • Mathematica
    B[n_] := (n DivisorSigma[0, n] + DivisorSigma[2, n])/(2 DivisorSigma[1, n]); Select[Range[300], IntegerQ[B[#]] &]
  • PARI
    isok(n) = denominator((n*sigma(n,0) + sigma(n,2))/(2*sigma(n)))==1; \\ Michel Marcus, Jan 14 2016

A246857 Numbers k such that sigma(k + sigma(k)) = 2*sigma(k).

Original entry on oeis.org

2, 3, 5, 11, 23, 29, 41, 53, 83, 89, 113, 131, 173, 179, 191, 233, 239, 251, 281, 293, 329, 359, 413, 419, 431, 443, 491, 509, 593, 623, 641, 653, 659, 683, 719, 743, 761, 809, 869, 911, 953, 979, 1013, 1019, 1031, 1049, 1103, 1223, 1229, 1289, 1409, 1439, 1451
Offset: 1

Views

Author

Jaroslav Krizek, Sep 05 2014

Keywords

Comments

Union of A005384 (Sophie Germain primes) and A246858.
First composite number in sequence is 329 (see A246858).

Examples

			Composite number 329 (with sigma(329) = 384) is in sequence because sigma(329+sigma(329)) = sigma(713) = 768 = 2*384.
Prime 359 (with sigma(359) = 360) is in sequence because sigma(359+sigma(359)) = sigma(719) = 720 = 2*360.

Crossrefs

Cf. A074400, A246456.
Cf. A005384, A246858.

Programs

  • Magma
    [n:n in[1..10000] | SumOfDivisors(n+SumOfDivisors(n)) eq 2*SumOfDivisors(n)]
  • Mathematica
    Select[Range[1500], DivisorSigma[1, # + DivisorSigma[1, #]] == 2 DivisorSigma[1, #] &] (* Michael De Vlieger, Aug 05 2021 *)
  • PARI
    select(n -> sigma(n+sigma(n))==2*sigma(n),[1..1000]) \\ Edward Jiang, Sep 05 2014

A272400 Square array read by antidiagonals upwards in which T(n,k) is the product of the n-th noncomposite number and the sum of the divisors of k, n>=1, k>=1.

Original entry on oeis.org

1, 2, 3, 3, 6, 4, 5, 9, 8, 7, 7, 15, 12, 14, 6, 11, 21, 20, 21, 12, 12, 13, 33, 28, 35, 18, 24, 8, 17, 39, 44, 49, 30, 36, 16, 15, 19, 51, 52, 77, 42, 60, 24, 30, 13, 23, 57, 68, 91, 66, 84, 40, 45, 26, 18, 29, 69, 76, 119, 78, 132, 56, 75, 39, 36, 12, 31, 87, 92, 133, 102, 156, 88, 105, 65, 54, 24, 28
Offset: 1

Views

Author

Omar E. Pol, Apr 28 2016

Keywords

Examples

			The corner of the square array begins:
1,   3,   4,   7,   6,  12,   8,  15,  13,  18...
2,   6,   8,  14,  12,  24,  16,  30,  26,  36...
3,   9,  12,  21,  18,  36,  24,  45,  39,  54...
5,  15,  20,  35,  30,  60,  40,  75,  65,  90...
7,  21,  28,  49,  42,  84,  56, 105,  91, 126...
11, 33,  44,  77,  66, 132,  88, 165, 143, 198...
13, 39,  52,  91,  78, 156, 104, 195, 169, 234...
17, 51,  68, 119, 102, 204, 136, 255, 221, 306...
19, 57,  76, 133, 114, 228, 152, 285, 247, 342...
23, 69,  92, 161, 138, 276, 184, 345, 299, 414...
...

Crossrefs

Rows 1-3: A000203, A074400, A272027.
Columns 1-2: A008578, A112773.
The diagonal 2, 9, 20... is A272211, the main diagonal of A272214.
Cf. A272173.

Programs

  • Mathematica
    Table[If[# == 1, 1, Prime[# - 1]] DivisorSigma[1, k] &@(n - k + 1), {n, 12}, {k, n}] // Flatten (* Michael De Vlieger, Apr 28 2016 *)

Formula

T(n,k) = A008578(n)*A000203(k), n>=1, k>=1.
T(n,k) = A272214(n-1,k), n>=2.

A325299 a(n) = 9 * sigma(n).

Original entry on oeis.org

9, 27, 36, 63, 54, 108, 72, 135, 117, 162, 108, 252, 126, 216, 216, 279, 162, 351, 180, 378, 288, 324, 216, 540, 279, 378, 360, 504, 270, 648, 288, 567, 432, 486, 432, 819, 342, 540, 504, 810, 378, 864, 396, 756, 702, 648, 432, 1116, 513, 837, 648, 882, 486, 1080, 648, 1080, 720, 810, 540, 1512
Offset: 1

Views

Author

Omar E. Pol, Jun 26 2019

Keywords

Comments

9 times the sum of the divisors of n.
a(n) is also the total number of horizontal rhombuses in the terraces of the n-th level of an irregular stepped pyramid (starting from the top) in which the structure of every 40-degree-three-dimensional sector arises after the 40-degree-zig-zag folding of every row of the diagram of the isosceles triangle A237593. The top of the pyramid is a nine-pointed star formed by nine rhombuses (see Links section).

Links

Crossrefs

k times sigma(n), k=1..8: A000203, A074400, A272027, A239050, A274535, A274536, A319527, A319528.
Cf. A008591, A237593.

Programs

  • GAP
    List([1..70],n->9*Sigma(n)); # After Muniru A Asiru
  • Maple
    with(numtheory): seq(9*sigma(n), n=1..64);
  • Mathematica
    9*DivisorSigma[1,Range[70]] (* After Harvey P. Dale *)
  • PARI
    a(n) = 9 * sigma(n);

Formula

a(n) = 9*A000203(n) = 3*A272027(n).
a(n) = A000203(n) + A319528(n) = A074400(n) + A319527(n).
Dirichlet g.f.: 9*zeta(s-1)*zeta(s). - (After Ilya Gutkovskiy)

A246858 Composite numbers k such that sigma(k + sigma(k)) = 2*sigma(k).

Original entry on oeis.org

329, 413, 623, 869, 979, 1819, 2585, 3107, 3173, 3197, 3887, 4235, 4997, 5771, 6149, 6187, 6443, 7409, 8399, 8759, 14429, 15323, 18515, 19019, 21181, 21413, 23989, 26491, 29749, 30355, 31043, 32623, 34009, 34177, 39737, 47321, 47845, 51389, 53311, 56419
Offset: 1

Views

Author

Jaroslav Krizek, Sep 05 2014

Keywords

Comments

Complement of A005384 (Sophie Germain primes) with respect to A246857.

Examples

			Number 329 (with sigma(329) = 384) is in sequence because sigma(329 + sigma(329)) = sigma(713) = 768 = 2*384.

Crossrefs

Cf. A074400, A246456.
Cf. A005384, A246858.

Programs

  • Magma
    [n:n in[1..1000] | SumOfDivisors(n+SumOfDivisors(n)) eq 2*SumOfDivisors(n) and not IsPrime(n)]
  • Mathematica
    Select[Range[57000], And[CompositeQ[#], DivisorSigma[1, # + DivisorSigma[1, #]] == 2 DivisorSigma[1, #]] &] (* Michael De Vlieger, Aug 05 2021 *)
  • PARI
    lista(nn) = {forcomposite(n=2, nn, if (sigma(n+sigma(n)) == 2*sigma(n), print1(n, ", ")););} \\ Michel Marcus, Sep 05 2014

A248881 Numbers n such that lambda(sum of even divisors of 2n) = lambda(sum of odd divisors of 2n) where lambda is the Carmichael function (A002322).

Original entry on oeis.org

1, 3, 5, 6, 9, 11, 13, 17, 18, 19, 25, 26, 27, 29, 36, 37, 38, 41, 43, 45, 49, 50, 53, 54, 59, 61, 63, 65, 67, 68, 72, 73, 74, 75, 81, 82, 83, 85, 86, 87, 89, 90, 95, 97, 98, 99, 100, 101, 103, 107, 109, 113, 117, 121, 122, 125, 126, 130, 131, 134, 137, 139
Offset: 1

Views

Author

Michel Lagneau, Mar 05 2015

Keywords

Comments

Number n such that A002322(A074400(n))= A002322(A000593(n)).
The squares of the form p^2 with p prime are in the sequence because the divisors of 2p^2 are {1,2,p,2p,p^2,2p^2} => sum of even divisors s0 = 2+2p+2p^2 = 2(p^2+p+p^2) and sum of odd divisors s1 = 1+p+p^2 and lambda(s0) = lambda(s1) = lambda(2*s0).
A majority of primes are in the sequence: 3, 5, 11, 13, 17, 19, 29, 37, 41, 43, 53, 59, 61, 67, 73, 83, 89, 97, 101, 103, 107, 109, 113, 131, 137, 139, 149, 157, 163, 173, 179, 181, 193, 197, ... but the primes 7, 23, 31, 47, 71, 79, 127, 151, 167, 191, 223, 239, 263, 367, 383, 431, ... are not in the sequence.

Examples

			18 is in the sequence because A002322(A074400(18))= A002322(78)= 12 and because A002322(A000593(18)) = A002322(13) = 12.

Links

Crossrefs

Cf. A000593, A002322, A074400, A077591.

Programs

  • Mathematica
    lst={};f[x_] := Plus @@ Select[Divisors[x], OddQ[#] &]; g[x_] := Plus @@ Select[Divisors[x], EvenQ[#]&]; Do[If[CarmichaelLambda[f[n]]== CarmichaelLambda[g[n]], AppendTo[lst,n/2]], {n, 1, 500}];lst
  • PARI
    a002322(n) = lcm(znstar(n)[2]);
isok(n) = my(sod = sumdiv(2*n, d, d*(d%2))); my(sed = sigma(2*n) - sod); sod && sed && (a002322(sod) == a002322(sed)); \\ Michel Marcus, Mar 07 2015

A326122 a(n) = 10 * sigma(n).

Original entry on oeis.org

10, 30, 40, 70, 60, 120, 80, 150, 130, 180, 120, 280, 140, 240, 240, 310, 180, 390, 200, 420, 320, 360, 240, 600, 310, 420, 400, 560, 300, 720, 320, 630, 480, 540, 480, 910, 380, 600, 560, 900, 420, 960, 440, 840, 780, 720, 480, 1240, 570, 930, 720, 980, 540, 1200, 720, 1200, 800, 900, 600, 1680, 620, 960
Offset: 1

Views

Author

Omar E. Pol, Jul 13 2019

Keywords

Comments

10 times the sum of the divisors of n.
a(n) is also the total number of horizontal rhombuses in the terraces of the n-th level of an irregular stepped pyramid (starting from the top) where the structure of every 36-degree-three-dimensional sector arises after the 36-degree-zig-zag folding of every row of the diagram of the isosceles triangle A237593. The top of the pyramid is a 10-pointed star formed by 10 rhombuses (see Links section).

Links

Crossrefs

k times sigma(n), k=1..9: A000203, A074400, A272027, A239050, A274535, A274536, A319527, A319528, A325299.
Cf. A008592, A033583, A124080, A124252, A142342, A153780, A237593.

Programs

  • GAP
    List([1..70],n->10*Sigma(n)); # After Muniru A Asiru
  • Magma
    [10*DivisorSigma(1, n): n in [1..70]]; // Vincenzo Librandi, Jul 26 2019
  • Maple
    with(numtheory): seq(10*sigma(n), n=1..64);
  • Mathematica
    10*DivisorSigma[1,Range[70]] (* After Harvey P. Dale *)
  • PARI
    a(n) = 10 * sigma(n);

Formula

a(n) = 10*A000203(n) = 5*A074400(n) = 2*A274535(n).
a(n) = A000203(n) + A325299(n) = A074400(n) + A319528(n).
Dirichlet g.f.: 10*zeta(s-1)*zeta(s). - (After Ilya Gutkovskiy)

A348412 Numbers whose even divisors have an integer harmonic mean.

Original entry on oeis.org

2, 6, 12, 30, 56, 84, 168, 270, 280, 540, 616, 840, 992, 1092, 1344, 2856, 2976, 3276, 3780, 4590, 5320, 5940, 7560, 12400, 12420, 14880, 16256, 16380, 18848, 24360, 26784, 36036, 37200, 37240, 41664, 48768, 49140, 55692, 60480, 65520, 86304, 86800, 111720, 128520
Offset: 1

Views

Author

Amiram Eldar, Oct 17 2021

Keywords

Comments

The corresponding harmonic means are 2, 3, 4, 5, 6, 7, 9, 9, 10, 12, 11, 15, 10, 13, 16, 17, 15, ...
Equivalently, even numbers k such that the harmonic mean of the divisors of k/2 is either an integer (A001599) or a half-integer (A348411).

Examples

			6 is a term since its even divisors are 2 and 6, and their harmonic mean, 1/((1/2 + 1/6)/2) = 3, is an integer.

Links

Crossrefs

Cf. A001599, A074400, A348411.
A139256 is a subsequence.

Programs

  • Mathematica
    Select[Range[2, 10^5, 2], IntegerQ[HarmonicMean[Select[Divisors[#], EvenQ]]] &]
  • PARI
    isok(m) = if (! (m%2), my(d=select(x->!(x%2), divisors(m))); denominator(#d/sum(k=1, #d, 1/d[k])) == 1); \\ Michel Marcus, Oct 31 2021
  • Python
    from sympy import gcd, divisor_sigma
A348412_list = [2*n for n in range(1,10**3) if (lambda x, y: 2*gcd(x,y*n)>=x)(divisor_sigma(n),divisor_sigma(n,0))] # Chai Wah Wu, Oct 20 2021

A171642 Non-deficient numbers with odd sigma such that the sum of the even divisors is twice the sum of the odd divisors.

Original entry on oeis.org

18, 162, 450, 882, 1458, 2178, 2450, 3042, 4050, 5202, 6050, 6498, 7938, 8450, 9522, 11250, 13122, 15138, 17298, 19602, 22050, 24642, 27378, 30258, 33282, 36450, 39762, 43218, 46818, 50562, 54450, 58482, 61250, 62658, 66978, 71442, 76050, 80802, 85698
Offset: 1

Views

Author

Peter Luschny, Dec 14 2009

Keywords

Comments

Numbers which are non-deficient (2n <= sigma(n)) [A023196] such that sigma(n) [A000203] is odd and the sum of the even divisors [A074400] is twice the sum of the odd divisors [A000593].
The sequence of terms which are not of the form 72*k^2 + 72*k + 18 starts: 2450, 6050, 8450, 61250, 120050, 151250, 211250, 296450.

Examples

			Divisors(18) = {1, 2, 3, 6, 9, 18}, sigma(18) = 39, and 2 + 6 + 18 = 2*(1 + 3 + 9).

Links

Crossrefs

Cf. A000203, A023196, A074400, A000593.
Cf. A171641, A083207, A023196, A077591, A137933.

Programs

  • Maple
    with(numtheory): A171642 := proc(n) local k,s,a;
s := sigma(n); a := add(k,k=select(x->type(x,odd),divisors(n)));
if 3*a = s and 2*n <= s and type(s,odd) then n else NULL fi end:
  • Python
    from sympy import divisors
A171642 = []
for n in range(1, 10**5):
    d = divisors(n)
    s = sum(d)
    if s % 2 and 2*n <= s and s == 3*sum([x for x in d if x % 2]):
        A171642.append(n)
# Chai Wah Wu, Aug 20 2014

A235671 Triangle read by rows in which row n lists the proper divisors of n in increasing order, 2n, and the proper divisors of n in decreasing order.

Original entry on oeis.org

2, 1, 4, 1, 1, 6, 1, 1, 2, 8, 2, 1, 1, 10, 1, 1, 2, 3, 12, 3, 2, 1, 1, 14, 1, 1, 2, 4, 16, 4, 2, 1, 1, 3, 18, 3, 1, 1, 2, 5, 20, 5, 2, 1, 1, 22, 1, 1, 2, 3, 4, 6, 24, 6, 4, 3, 2, 1, 1, 26, 1, 1, 2, 7, 28, 7, 2, 1, 1, 3, 5, 30, 5, 3, 1, 1, 2, 4, 8, 32, 8, 4, 2, 1
Offset: 1

Views

Author

Omar E. Pol, Jan 24 2014

Keywords

Comments

Numerators of a sequence related to the symmetric structure of sigma, which arises from the structure of A237593. The structure in the first two octants is transformed in a structure in the 6th and 7th octants, which is similar to an isosceles triangle.
Denominators are in A007395.
Row sums give A074400.
Row lengths is A114003 (see the Jovovic's formula in A114003).

Examples

			The irregular triangle begins:
2;
1, 4, 1;
1, 6, 1;
1, 2, 8, 2, 1;
1, 10, 1;
1, 2, 3, 12, 3, 2, 1;
1, 14, 1;
1, 2, 4, 16, 4, 2, 1;
1, 3, 18, 3, 1;
1, 2, 5, 20, 5, 2, 1;
1, 22, 1;
1, 2, 3, 4, 6, 24, 6, 4, 3, 2, 1;
...
Also:
1;
1/2, 2, 1/2;
1/2, 3, 1/2;
1/2, 1, 4, 1, 1/2;
1/2, 5, 1/2;
1/2, 1, 3/2, 6, 3/2, 1, 1/2;
1/2, 7, 1/2;
1/2, 1, 2, 8, 2, 1, 1/2;
1/2, 3/2, 9, 3/2, 1/2;
1/2, 1, 5/2, 10, 5/2, 1, 1/2;
1/2, 11, 1/2;
1/2, 1, 3/2, 2, 3, 12, 3, 2, 3/2, 1, 1/2;
...

Links

Crossrefs

Cf. A000005, A001065, A027750, A056538, A074400, A000203, A114002, A114003, A236104, A237591, A237593, A237270, A233772, A233773.

Programs

  • Mathematica
    pd[n_]:=Module[{d=Most[Divisors[n]]},Flatten[Join[{d,{2n},Reverse[d]}]]]; Flatten[Array[pd,20]] (* Harvey P. Dale, Dec 22 2014 *)
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