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 1-7 of 7 results.

A095119 Numbers k such that s(k) >= sigma(k), where s(k) = A095118(k) is the sum of the squares of the divisors of k which are <= sqrt(k) and sigma(k) = A000203(k) is the sum of the divisors of k.

Original entry on oeis.org

1, 840, 900, 1080, 1225, 1260, 1440, 1600, 1680, 1800, 1848, 1890, 1980, 2016, 2100, 2160, 2340, 2400, 2520, 2640, 2700, 2772, 2800, 2880, 2970, 3024, 3080, 3120, 3136, 3150, 3240, 3276, 3300, 3360, 3465, 3528, 3600, 3640, 3696, 3780, 3900, 3960, 3969
Offset: 1

Views

Author

Dean Hickerson, following a suggestion of Leroy Quet, May 28 2004

Keywords

Examples

			840 is in the sequence because s(840) = 3070 >= 2880 = sigma(840).

Links

Crossrefs

Cf. A000203, A095118.

Programs

  • Mathematica
    s[n_]:=Plus@@(Select[Divisors[n], #^2<=n&]^2); Select[Range[4000], s[ # ]>=DivisorSigma[1, # ]&]
  • PARI
    isok(n) = sumdiv(n, d, if (d^2 <= n, d^2)) >= sigma(n); \\ Michel Marcus, Aug 13 2019

A339353 G.f.: Sum_{k>=1} k^2 * x^(k*(k + 1)) / (1 - x^k).

Original entry on oeis.org

0, 1, 1, 1, 1, 5, 1, 5, 1, 5, 1, 14, 1, 5, 10, 5, 1, 14, 1, 21, 10, 5, 1, 30, 1, 5, 10, 21, 1, 39, 1, 21, 10, 5, 26, 30, 1, 5, 10, 46, 1, 50, 1, 21, 35, 5, 1, 66, 1, 30, 10, 21, 1, 50, 26, 70, 10, 5, 1, 91, 1, 5, 59, 21, 26, 50, 1, 21, 10, 79, 1, 130, 1, 5, 35, 21, 50, 50, 1, 110
Offset: 1

Views

Author

Ilya Gutkovskiy, Dec 01 2020

Keywords

Comments

Sum of squares of divisors of n that are smaller than sqrt(n).

Links

Crossrefs

Cf. A001157, A056924, A067558, A070039, A095118, A339354.

Programs

  • Mathematica
    nmax = 80; CoefficientList[Series[Sum[k^2 x^(k (k + 1))/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, #^2 &, # < Sqrt[n] &], {n, 80}]
  • PARI
    a(n) = sumdiv(n, d, if (d^2 < n, d^2)); \\ Michel Marcus, Dec 02 2020

A280375 Expansion of Sum_{k>=1} k^3*x^(k^2)/(1 - x^k).

Original entry on oeis.org

1, 1, 1, 9, 1, 9, 1, 9, 28, 9, 1, 36, 1, 9, 28, 73, 1, 36, 1, 73, 28, 9, 1, 100, 126, 9, 28, 73, 1, 161, 1, 73, 28, 9, 126, 316, 1, 9, 28, 198, 1, 252, 1, 73, 153, 9, 1, 316, 344, 134, 28, 73, 1, 252, 126, 416, 28, 9, 1, 441, 1, 9, 371, 585, 126, 252, 1, 73, 28, 477, 1, 828, 1, 9, 153, 73, 344, 252, 1, 710, 757, 9, 1, 659, 126
Offset: 1

Views

Author

Ilya Gutkovskiy, Jan 01 2017

Keywords

Comments

The sum of the cubes of the divisors of n which are <= sqrt(n).

Examples

			The divisors of 12 which are <= sqrt(12) are {1,2,3}, so a(12) = 1^3 + 2^3 + 3^3 = 36.

Links

Crossrefs

Cf. A001158, A038548, A066839, A095118.

Programs

  • Mathematica
    nmax = 85; Rest[CoefficientList[Series[Sum[k^3 x^k^2/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
(* Second program *)
Table[Total[Select[Divisors@ n, # <= Sqrt@ n &]^3], {n, 85}] (* Michael De Vlieger, Jan 01 2017 *)
  • PARI
    a(n) = my(rn = sqrt(n)); sumdiv(n, d, d^3*(d<=rn)); \\ Michel Marcus, Jan 02 2017

Formula

G.f.: Sum_{k>=1} k^3*x^(k^2)/(1 - x^k).

A347143 Sum of 4th powers of divisors of n that are <= sqrt(n).

Original entry on oeis.org

1, 1, 1, 17, 1, 17, 1, 17, 82, 17, 1, 98, 1, 17, 82, 273, 1, 98, 1, 273, 82, 17, 1, 354, 626, 17, 82, 273, 1, 723, 1, 273, 82, 17, 626, 1650, 1, 17, 82, 898, 1, 1394, 1, 273, 707, 17, 1, 1650, 2402, 642, 82, 273, 1, 1394, 626, 2674, 82, 17, 1, 2275, 1, 17, 2483, 4369, 626
Offset: 1

Views

Author

Ilya Gutkovskiy, Aug 19 2021

Keywords

Crossrefs

Cf. A001159, A038548, A066839, A095118, A279363, A280375, A347142.

Programs

  • Mathematica
    Table[DivisorSum[n, #^4 &, # <= Sqrt[n] &], {n, 1, 65}]
nmax = 65; CoefficientList[Series[Sum[k^4 x^(k^2)/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
  • PARI
    A347143(n) = { my(s=0); fordiv(n,d,if((d^2)>n,return(s)); s += (d^4)); (s); }; \\ Antti Karttunen, Aug 19 2021

Formula

G.f.: Sum_{k>=1} k^4 * x^(k^2) / (1 - x^k).

A347173 Sum of squares of odd divisors of n that are <= sqrt(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 10, 1, 1, 10, 1, 1, 10, 1, 1, 10, 1, 1, 10, 1, 1, 10, 26, 1, 10, 1, 1, 35, 1, 1, 10, 1, 26, 10, 1, 1, 10, 26, 1, 10, 1, 1, 35, 1, 1, 10, 50, 26, 10, 1, 1, 10, 26, 50, 10, 1, 1, 35, 1, 1, 59, 1, 26, 10, 1, 1, 10, 75, 1, 10, 1, 1, 35, 1, 50, 10, 1, 26
Offset: 1

Views

Author

Ilya Gutkovskiy, Aug 21 2021

Keywords

Examples

			a(18) = 10 as the odd divisors of 18 are the divisors of 9 which are 1, 3 and 9. Of those, 1 and 3 are <= sqrt(18) so we find the squares of 1 and 3 then add them i.e., a(18) = 1^2 + 3^2 = 10. - _David A. Corneth_, Feb 24 2024

Links

Crossrefs

Cf. A001157, A050999, A069288, A069289, A095118, A347161, A347174, A347175.

Programs

  • Mathematica
    Table[DivisorSum[n, #^2 &, # <= Sqrt[n] && OddQ[#] &], {n, 1, 80}]
nmax = 80; CoefficientList[Series[Sum[(2 k - 1)^2 x^((2 k - 1)^2)/(1 - x^(2 k - 1)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
  • PARI
    a(n) = sum(k=0, sqrtint(n), if ((k%2) && !(n%k), k^2)); \\ Michel Marcus, Aug 22 2021
  • PARI
    a(n) = {
	my(s = sqrtint(n), res);
	n>>=valuation(n, 2);
	d = divisors(n);
	for(i = 1, #d,
		if(d[i] <= s,
			res += d[i]^2
		,
			return(res)
		)
	); res
} \\ David A. Corneth, Feb 24 2024

Formula

G.f.: Sum_{k>=1} (2*k - 1)^2 * x^((2*k - 1)^2) / (1 - x^(2*k - 1)).

A372625 Expansion of Sum_{k>=1} k^2 * x^(k^2) / (1 + x^k).

Original entry on oeis.org

1, -1, 1, 3, 1, -5, 1, 3, 10, -5, 1, -6, 1, -5, 10, 19, 1, -14, 1, -13, 10, -5, 1, 10, 26, -5, 10, -13, 1, -39, 1, 19, 10, -5, 26, 14, 1, -5, 10, -6, 1, -50, 1, -13, 35, -5, 1, 46, 50, -30, 10, -13, 1, -50, 26, -30, 10, -5, 1, -11, 1, -5, 59, 83, 26, -50, 1, -13, 10, -79
Offset: 1

Views

Author

Ilya Gutkovskiy, May 07 2024

Keywords

Crossrefs

Cf. A064027, A066839, A078306, A095118, A321543, A321558, A348608.

Programs

  • Mathematica
    nmax = 70; CoefficientList[Series[Sum[k^2 x^(k^2)/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, (-1)^(# + n/#) #^2 &, # <= Sqrt[n] &], {n, 1, 70}]

Formula

a(n) = Sum_{d|n, d <= sqrt(n)} (-1)^(d + n/d) * d^2.

A373031 Expansion of Sum_{k>=1} (-1)^(k+1) * k^2 * x^(k^2) / (1 - x^k).

Original entry on oeis.org

1, 1, 1, -3, 1, -3, 1, -3, 10, -3, 1, 6, 1, -3, 10, -19, 1, 6, 1, -19, 10, -3, 1, -10, 26, -3, 10, -19, 1, 31, 1, -19, 10, -3, 26, -46, 1, -3, 10, 6, 1, -30, 1, -19, 35, -3, 1, -46, 50, 22, 10, -19, 1, -30, 26, 30, 10, -3, 1, -21, 1, -3, 59, -83, 26, -30, 1, -19, 10, 71
Offset: 1

Views

Author

Ilya Gutkovskiy, May 20 2024

Keywords

Crossrefs

Cf. A038548, A064027, A066839, A095118, A321543, A333781, A333782, A344300, A372625, A373032.

Programs

  • Mathematica
    nmax = 70; CoefficientList[Series[Sum[(-1)^(k + 1) k^2 x^(k^2)/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

Formula

a(n) = Sum_{d|n, d <= sqrt(n)} (-1)^(d+1) * d^2.
Showing 1-7 of 7 results.

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