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 1-10 of 51 results. Next

A321543 a(n) = Sum_{d|n} (-1)^(d-1)*d^2.

Original entry on oeis.org

1, -3, 10, -19, 26, -30, 50, -83, 91, -78, 122, -190, 170, -150, 260, -339, 290, -273, 362, -494, 500, -366, 530, -830, 651, -510, 820, -950, 842, -780, 962, -1363, 1220, -870, 1300, -1729, 1370, -1086, 1700, -2158, 1682, -1500, 1850, -2318, 2366, -1590, 2210, -3390, 2451, -1953, 2900, -3230, 2810, -2460, 3172
Offset: 1

Views

Author

N. J. A. Sloane, Nov 23 2018

Keywords

Links

Crossrefs

Apart from signs, same as A064027.
Glaisher's zeta_i, i=0..12: A048272, A002129, A321543, A138503, A279395, A321544, A321545, A321546, A321547, A321548, A321549, A321550, A321551.
Cf. A321552 - A321565, A321807 - A321836 for similar sequences.

Programs

  • Maple
    with(numtheory):
a := n -> add( (-1)^(d-1)*d^2, d in divisors(n) ): seq(a(n), n = 1..40);
#  Peter Bala, Jan 11 2021
  • Mathematica
    f[p_, e_] := (p^(2*e + 2) - 1)/(p^2 - 1); f[2, e_] := 2 - (2^(2*e + 2) - 1)/3; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 55] (* Amiram Eldar, Nov 04 2022 *)
  • PARI
    apply( a(n)=sumdiv(n, d, (-1)^(d-1)*d^2), [1..30]) \\ M. F. Hasler, Nov 26 2018

Formula

G.f.: Sum_{k>=1} (-1)^(k-1)*k^2*x^k/(1 - x^k). - Ilya Gutkovskiy, Dec 23 2018
G.f.: Sum_{n >= 1} x^n*(1 - x^n)/(1 + x^n)^3. - Peter Bala, Jan 11 2021
Multiplicative with a(2^e) = 2 - (2^(2*e + 2) - 1)/3, and a(p^e) = (p^(2*e + 2) - 1)/(p^2 - 1) for p > 2. - Amiram Eldar, Nov 04 2022

A321836 a(n) = Sum_{d|n, n/d==1 mod 4} d^12 - Sum_{d|n, n/d==3 mod 4} d^12.

Original entry on oeis.org

1, 4096, 531440, 16777216, 244140626, 2176778240, 13841287200, 68719476736, 282429005041, 1000000004096, 3138428376720, 8916083671040, 23298085122482, 56693912371200, 129746094281440, 281474976710656, 582622237229762, 1156829204647936
Offset: 1

Views

Author

N. J. A. Sloane, Nov 24 2018

Keywords

Links

Crossrefs

Cf. A101455.
Cf. A321543 - A321565, A321807 - A321835 for similar sequences.
Glaisher's E'_i (i=0..12): A002654, A050469, A050470, A050471, A050468, A321829, A321830, A321831, A321832, A321833, A321834, A321835, this sequence.

Programs

  • Mathematica
    s[n_,r_] := DivisorSum[n, #^12 &, Mod[n/#,4]==r &]; a[n_] := s[n,1] - s[n,3]; Array[a, 30] (* Amiram Eldar, Nov 26 2018 *)
s[n_] := If[OddQ[n], (-1)^((n-1)/2), 0]; (* A101455 *)
f[p_, e_] := (p^(12*e+12) - s[p]^(e+1))/(p^12 - s[p]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 04 2023 *)
  • PARI
    apply( a(n)=sumdiv(n, d, if(bittest(n\d,0),(2-n\d%4)*d^12)), [1..30]) \\ M. F. Hasler, Nov 26 2018

Formula

G.f.: Sum_{k>=1} k^12*x^k/(1 + x^(2*k)). - Ilya Gutkovskiy, Nov 26 2018
From Amiram Eldar, Nov 04 2023: (Start)
Multiplicative with a(p^e) = (p^(12*e+12) - A101455(p)^(e+1))/(p^12 - A101455(p)).
Sum_{k=1..n} a(k) ~ c * n^13 / 13, where c = beta(13) = 540553*Pi^13/1569592442880 = 0.999999373583..., and beta is the Dirichlet beta function. (End)
a(n) = Sum_{d|n} (n/d)^12*sin(d*Pi/2). - Ridouane Oudra, Sep 27 2024

A321807 a(n) = Sum_{d divides n} (-1)^(d + n/d) * d^10.

Original entry on oeis.org

1, -1025, 59050, -1047553, 9765626, -60526250, 282475250, -1072692225, 3486843451, -10009766650, 25937424602, -61858004650, 137858491850, -289537131250, 576660215300, -1098436836353, 2015993900450, -3574014537275, 6131066257802
Offset: 1

Views

Author

N. J. A. Sloane, Nov 23 2018

Keywords

Links

Crossrefs

Column k=10 of A322083.
Cf. A321543 - A321565, A321808 - A321836 for similar sequences.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (-1)^(# + n/#)*#^10 &]; Array[a, 50] (* Amiram Eldar, Nov 22 2022 *)
  • PARI
    apply( A321807(n)=sumdiv(n, d, (-1)^(d-n\d)*d^10), [1..30]) \\ M. F. Hasler, Nov 26 2018

Formula

G.f.: Sum_{k>=1} (-1)^(k+1)*k^10*x^k/(1 + x^k). - Ilya Gutkovskiy, Dec 22 2018
Multiplicative with a(2^e) = -(511*2^(10*e+1) + 2047)/1023, and a(p^e) = (p^(10*e+10) - 1)/(p^10 - 1) for p > 2. - Amiram Eldar, Nov 22 2022

A322083 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{d|n} (-1)^(n/d+d)*d^k.

Original entry on oeis.org

1, 1, -2, 1, -3, 2, 1, -5, 4, -1, 1, -9, 10, -3, 2, 1, -17, 28, -13, 6, -4, 1, -33, 82, -57, 26, -12, 2, 1, -65, 244, -241, 126, -50, 8, 0, 1, -129, 730, -993, 626, -252, 50, -3, 3, 1, -257, 2188, -4033, 3126, -1394, 344, -45, 13, -4, 1, -513, 6562, -16257, 15626, -8052, 2402, -441, 91, -18, 2
Offset: 1

Views

Author

Ilya Gutkovskiy, Nov 26 2018

Keywords

Comments

For each k, the k-th column sequence (T(n,k))(n>=1) is a multiplicative function of n, equal to (-1)^(n+1)*(Id_k * 1) in the notation of the Bala link. - Peter Bala, Mar 19 2022

Examples

			Square array begins:
   1,   1,   1,    1,     1,     1,  ...
  -2,  -3,  -5,   -9,   -17,   -33,  ...
   2,   4,  10,   28,    82,   244,  ...
  -1,  -3, -13,  -57,  -241,  -993,  ...
   2,   6,  26,  126,   626,  3126,  ...
  -4, -12, -50, -252, -1394, -8052,  ...

Links

Crossrefs

Columns k=0..12 give A228441, A109506, A321558, A321559, A321560, A321561, A321562, A321563, A321564, A321565, A321807, A321808, A321809.
Cf. A109974, A279394, A279396, A285425, A322081, A322082, A322084.

Programs

  • Mathematica
    Table[Function[k, Sum[(-1)^(n/d+d) d^k, {d, Divisors[n]}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten
Table[Function[k, SeriesCoefficient[Sum[(-1)^(j + 1) j^k x^j/(1 + x^j), {j, 1, n}], {x, 0, n}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten
f[p_, e_, k_] := If[k == 0, e + 1, (p^(k*e + k) - 1)/(p^k - 1)]; f[2, e_, k_] := If[k == 0, e - 3, -((2^(k - 1) - 1)*2^(k*e + 1) + 2^(k + 1) - 1)/(2^k - 1)]; T[1, k_] = 1; T[n_, k_] := Times @@ (f[First[#], Last[#], k] & /@ FactorInteger[n]); Table[T[n - k, k], {n, 1, 11}, {k, n - 1, 0, -1}] // Flatten (* Amiram Eldar, Nov 22 2022 *)
  • PARI
    T(n,k)={sumdiv(n, d, (-1)^(n/d+d)*d^k)}
for(n=1, 10, for(k=0, 8, print1(T(n, k), ", ")); print); \\ Andrew Howroyd, Nov 26 2018

Formula

G.f. of column k: Sum_{j>=1} (-1)^(j+1)*j^k*x^j/(1 + x^j).

A064027 a(n) = (-1)^n*Sum_{d|n} (-1)^d*d^2.

Original entry on oeis.org

1, 3, 10, 19, 26, 30, 50, 83, 91, 78, 122, 190, 170, 150, 260, 339, 290, 273, 362, 494, 500, 366, 530, 830, 651, 510, 820, 950, 842, 780, 962, 1363, 1220, 870, 1300, 1729, 1370, 1086, 1700, 2158, 1682, 1500, 1850, 2318, 2366, 1590, 2210, 3390, 2451, 1953
Offset: 1

Views

Author

Vladeta Jovovic, Sep 11 2001

Keywords

Examples

			L.g.f.: L(x) = x + 3*x^2/2 + 10*x^3/3 + 19*x^4/4 + 26*x^5/5 + 30*x^6/6 + ...
where exp(L(x)) = 1 + x + 2*x^2 + 5*x^3 + 10*x^4 + 18*x^5 + 32*x^6 + 59*x^7 + 106*x^8 + 181*x^9 + ... + A224364(n)*x^n + ... - _Paul D. Hanna_, Apr 04 2013

References

  • G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, AMS Chelsea Publishing, Providence, Rhode Island, 2002, p. 142.

Links

Crossrefs

Cf. A001157, A002129, A008457, A050999, A224364.
Cf. A321543 - A321565, A321807 - A321836 for related sequences.

Programs

  • Magma
    m:=60; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&+[k^2*x^k/(1-(-x)^k): k in [1..m]]) )); // G. C. Greubel, Nov 09 2018
  • Mathematica
    a[n_] := (-1)^n DivisorSum[n, (-1)^# * #^2 &]; Array[a, 50] (* Jean-François Alcover, Dec 23 2015 *)
a[n_] := If[OddQ[n], 1, (1 - 6/(4^(IntegerExponent[n, 2] + 1) - 1))] * DivisorSigma[2, n]; Array[a, 100] (* Amiram Eldar, Sep 21 2023 *)
  • PARI
    {a(n)=if(n<1, 0, (-1)^n*sumdiv(n^1, d, (-1)^d*d^2))} \\ Paul D. Hanna, Apr 04 2013

Formula

Multiplicative with a(2^e) = (4^(e+1)-7)/3, a(p^e) = (p^(2*e+2)-1)/(p^2-1), p > 2.
a(n) = (-1)^n*(A001157(n) - 2*A050999(n)).
Logarithmic derivative of A224364. - Paul D. Hanna, Apr 04 2013
Bisection: a(2*k-1) = A001157(2*k-1), a(2*k) = 4*A001157(k) - A050999(2*k), k >= 1. In the Hardy reference a(n) = sigma^*2(n). - _Wolfdieter Lang, Jan 07 2017
G.f.: Sum_{k>=1} k^2*x^k/(1 - (-x)^k). - Ilya Gutkovskiy, Nov 09 2018
Sum_{k=1..n} a(k) ~ 7 * zeta(3) * n^3 / 24. - Vaclav Kotesovec, Nov 10 2018
Dirichlet g.f.: zeta(s) * zeta(s-2) * (1 - 1/2^(s-1) + 1/2^(2*s-3)). - Amiram Eldar, Sep 21 2023

A321829 a(n) = Sum_{d|n, n/d==1 mod 4} d^5 - Sum_{d|n, n/d==3 mod 4} d^5.

Original entry on oeis.org

1, 32, 242, 1024, 3126, 7744, 16806, 32768, 58807, 100032, 161050, 247808, 371294, 537792, 756492, 1048576, 1419858, 1881824, 2476098, 3201024, 4067052, 5153600, 6436342, 7929856, 9768751, 11881408, 14290100, 17209344, 20511150, 24207744, 28629150, 33554432, 38974100, 45435456, 52535556
Offset: 1

Views

Author

N. J. A. Sloane, Nov 24 2018

Keywords

Links

Crossrefs

Cf. A101455, A175570.
Cf. A321543 - A321565, A321807 - A321836 for similar sequences.
Glaisher's E'_i (i=0..12): A002654, A050469, A050470, A050471, A050468, this sequence, A321830, A321831, A321832, A321833, A321834, A321835, A321836.

Programs

  • Mathematica
    s[n_,r_] := DivisorSum[n, # ^5 &, Mod[n/#,4]==r &]; a[n_] := s[n,1] - s[n,3]; Array[a, 30] (* Amiram Eldar, Nov 26 2018 *)
s[n_] := If[OddQ[n], (-1)^((n-1)/2), 0]; (* A101455 *)
f[p_, e_] := (p^(5*e+5) - s[p]^(e+1))/(p^5 - s[p]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 04 2023 *)
  • PARI
    apply( A321829(n)=factorback(apply(f->f[1]^(5*f[2]+5)\/(f[1]^5+f[1]%4-2),Col(factor(n)))), [1..40]) \\ M. F. Hasler, Nov 26 2018

Formula

G.f.: Sum_{k>=1} k^5*x^k/(1 + x^(2*k)). - Ilya Gutkovskiy, Nov 26 2018
Multiplicative with a(p^e) = round(p^(5e+5)/(p^5 + p%4 - 2)), where p%4 is the remainder of p modulo 4. (Following R. Israel in A321833.) - M. F. Hasler, Nov 26 2018
Sum_{k=1..n} a(k) ~ c * n^6 / 6, where c = A175570. - Amiram Eldar, Nov 04 2023
a(n) = Sum_{d|n} (n/d)^5*sin(d*Pi/2). - Ridouane Oudra, Sep 27 2024

A321830 a(n) = Sum_{d|n, n/d==1 mod 4} d^6 - Sum_{d|n, n/d==3 mod 4} d^6.

Original entry on oeis.org

1, 64, 728, 4096, 15626, 46592, 117648, 262144, 530713, 1000064, 1771560, 2981888, 4826810, 7529472, 11375728, 16777216, 24137570, 33965632, 47045880, 64004096, 85647744, 113379840, 148035888, 190840832, 244156251, 308915840, 386889776
Offset: 1

Views

Author

N. J. A. Sloane, Nov 24 2018

Keywords

Links

Crossrefs

Cf. A101455, A258814.
Cf. A321543 - A321565, A321807 - A321836 for similar sequences.
Glaisher's E'_i (i=0..12): A002654, A050469, A050470, A050471, A050468, A321829, this sequence, A321831, A321832, A321833, A321834, A321835, A321836.

Programs

  • Mathematica
    s[n_,r_] := DivisorSum[n, #^6 &, Mod[n/#,4]==r &]; a[n_] := s[n,1] - s[n,3]; Array[a, 30] (* Amiram Eldar, Nov 26 2018 *)
s[n_] := If[OddQ[n], (-1)^((n-1)/2), 0]; (* A101455 *)
f[p_, e_] := (p^(6*e+6) - s[p]^(e+1))/(p^6 - s[p]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 04 2023 *)
  • PARI
    apply( A321830(n)=factorback(apply(f->f[1]^(6*f[2]+6)\/(f[1]^6+f[1]%4-2),Col(factor(n)))), [1..30]) \\ M. F. Hasler, Nov 26 2018

Formula

G.f.: Sum_{k>=1} k^6*x^k/(1 + x^(2*k)). - Ilya Gutkovskiy, Nov 26 2018
Multiplicative with a(p^e) = round(p^(6e+6)/(p^6 + p%4 - 2)), where p%4 is the remainder of p modulo 4. (Following R. Israel in A321833.) - M. F. Hasler, Nov 26 2018
Sum_{k=1..n} a(k) ~ c * n^7 / 7, where c = 61*Pi^7/184320 (A258814). - Amiram Eldar, Nov 04 2023
a(n) = Sum_{d|n} (n/d)^6*sin(d*Pi/2). - Ridouane Oudra, Sep 27 2024

A321831 a(n) = Sum_{d|n, n/d==1 mod 4} d^7 - Sum_{d|n, n/d==3 mod 4} d^7.

Original entry on oeis.org

1, 128, 2186, 16384, 78126, 279808, 823542, 2097152, 4780783, 10000128, 19487170, 35815424, 62748518, 105413376, 170783436, 268435456, 410338674, 611940224, 893871738, 1280016384, 1800262812, 2494357760, 3404825446, 4584374272, 6103593751
Offset: 1

Views

Author

N. J. A. Sloane, Nov 24 2018

Keywords

Links

Crossrefs

Cf. A101455, A258815.
Cf. A321543 - A321565, A321807 - A321836 for similar sequences.
Glaisher's E'_i (i=0..12): A002654, A050469, A050470, A050471, A050468, A321829, A321830, this sequence, A321832, A321833, A321834, A321835, A321836.

Programs

  • Mathematica
    s[n_,r_] := DivisorSum[n, #^7 &, Mod[n/#,4]==r &]; a[n_] := s[n,1] - s[n,3]; Array[a, 30] (* Amiram Eldar, Nov 26 2018 *)
s[n_] := If[OddQ[n], (-1)^((n-1)/2), 0]; (* A101455 *)
f[p_, e_] := (p^(7*e+7) - s[p]^(e+1))/(p^7 - s[p]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 04 2023 *)
  • PARI
    apply( A321831(n)=factorback(apply(f->f[1]^(7*f[2]+7)\/(f[1]^7+f[1]%4-2),Col(factor(n)))), [1..30]) \\ M. F. Hasler, Nov 26 2018

Formula

G.f.: Sum_{k>=1} k^7*x^k/(1 + x^(2*k)). - Ilya Gutkovskiy, Nov 26 2018
Multiplicative with a(p^e) = round(p^(7e+7)/(p^7 + p%4 - 2)), where p%4 is the remainder of p modulo 4. (Following R. Israel in A321833.) - M. F. Hasler, Nov 26 2018
Sum_{k=1..n} a(k) ~ c * n^8 / 8, where c = A258815. - Amiram Eldar, Nov 04 2023
a(n) = Sum_{d|n} (n/d)^7*sin(d*Pi/2). - Ridouane Oudra, Sep 27 2024

A321832 a(n) = Sum_{d|n, n/d==1 (mod 4)} d^8 - Sum_{d|n, n/d==3 (mod 4)} d^8.

Original entry on oeis.org

1, 256, 6560, 65536, 390626, 1679360, 5764800, 16777216, 43040161, 100000256, 214358880, 429916160, 815730722, 1475788800, 2562506560, 4294967296, 6975757442, 11018281216, 16983563040, 25600065536, 37817088000, 54875873280, 78310985280, 110058536960, 152588281251, 208827064832
Offset: 1

Views

Author

N. J. A. Sloane, Nov 24 2018

Keywords

Links

Crossrefs

Cf. A101455, A258816.
Cf. A321543 - A321565, A321807 - A321836 for similar sequences.
Glaisher's E'_i (i=0..12): A002654, A050469, A050470, A050471, A050468, A321829, A321830, A321831, this sequence, A321833, A321834, A321835, A321836.

Programs

  • Mathematica
    s[n_,r_] := DivisorSum[n, #^8 &, Mod[n/#,4]==r &]; a[n_] := s[n,1] - s[n,3]; Array[a, 30] (* Amiram Eldar, Nov 26 2018 *)
s[n_] := If[OddQ[n], (-1)^((n-1)/2), 0]; (* A101455 *)
f[p_, e_] := (p^(8*e+8) - s[p]^(e+1))/(p^8 - s[p]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 04 2023 *)
  • PARI
    apply( A321832(n)=factorback(apply(f->f[1]^(8*f[2]+8)\/(f[1]^8+f[1]%4-2),Col(factor(n)))), [1..50]) \\ M. F. Hasler, Nov 26 2018

Formula

G.f.: Sum_{k>=1} k^8*x^k/(1 + x^(2*k)). - Ilya Gutkovskiy, Nov 26 2018
Multiplicative with a(p^e) = round(p^(8e+8)/(p^8 + (p mod 4) - 2)). (Following R. Israel in A321833.) - M. F. Hasler, Nov 26 2018
Sum_{k=1..n} a(k) ~ c * n^9 / 9, where c = 277*Pi^9/8257536 (A258816). - Amiram Eldar, Nov 04 2023
a(n) = Sum_{d|n} (n/d)^8*sin(d*Pi/2). - Ridouane Oudra, Sep 27 2024

A321810 Sum of 6th powers of odd divisors of n.

Original entry on oeis.org

1, 1, 730, 1, 15626, 730, 117650, 1, 532171, 15626, 1771562, 730, 4826810, 117650, 11406980, 1, 24137570, 532171, 47045882, 15626, 85884500, 1771562, 148035890, 730, 244156251, 4826810, 387952660, 117650, 594823322, 11406980, 887503682
Offset: 1

Views

Author

N. J. A. Sloane, Nov 24 2018

Keywords

Links

Crossrefs

Column k=6 of A285425.
Cf. A050999, A051000, A051001, A051002, A321811 - A321816 (analog for 2nd .. 12th powers).
Cf. A321543 - A321565, A321807 - A321836 for related sequences.
Cf. A000265, A013665, A013954.

Programs

  • Mathematica
    f[2, e_] := 1; f[p_, e_] := (p^(6*e + 6) - 1)/(p^6 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 02 2022 *)
  • PARI
    apply( A321810(n)=sigma(n>>valuation(n,2),6), [1..30]) \\ M. F. Hasler, Nov 26 2018
  • Python
    from sympy import divisor_sigma
def A321810(n): return int(divisor_sigma(n>>(~n&n-1).bit_length(),6)) # Chai Wah Wu, Jul 16 2022

Formula

a(n) = A013954(A000265(n)) = sigma_6(odd part of n); in particular, a(2^k) = 1 for all k >= 0. - M. F. Hasler, Nov 26 2018
G.f.: Sum_{k>=1} (2*k - 1)^6*x^(2*k-1)/(1 - x^(2*k-1)). - Ilya Gutkovskiy, Dec 22 2018
From Amiram Eldar, Nov 02 2022: (Start)
Multiplicative with a(2^e) = 1 and a(p^e) = (p^(6*e+6)-1)/(p^6-1) for p > 2.
Sum_{k=1..n} a(k) ~ c * n^7, where c = zeta(7)/14 = 0.0720249... . (End)
a(n) + a(n/2)*2^6 = A013954(n) where a(.)=0 for non-integer arguments. - R. J. Mathar, Aug 15 2023
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