A244329 -id:A244329 - cp's OEIS frontend

A244327 a(n) = floor((n*(n+1)/2) / sigma(n)) = floor(A000217(n) / A000203(n)).

1, 1, 1, 1, 2, 1, 3, 2, 3, 3, 5, 2, 6, 4, 5, 4, 8, 4, 9, 5, 7, 7, 11, 5, 10, 8, 9, 7, 14, 6, 15, 8, 11, 11, 13, 7, 18, 12, 13, 9, 20, 9, 21, 11, 13, 15, 23, 9, 21, 13, 18, 14, 26, 12, 21, 13, 20, 19, 29, 10, 30, 20, 19, 16, 25, 15, 33, 18, 25, 17, 35, 13, 36

Offset: 1

Jaroslav Krizek, Jul 08 2014

nonn

RECORD transform of a(n) is A140475 (union of number 1 and primes >= 5).

Sequence of numbers n such that a(n) = a(n+1) = A244666.

			For n = 10; a(10) = floor(A000217(10) / A000203(10)) = floor(55 / 18) = 3.

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Cf. A000203, A000217, A024816, A244328, A244329, A244666.

[Floor((n*(n+1)div 2) div (SumOfDivisors(n))): n in [1..1000]];

a:= n-> floor(n*(n+1)/(2*numtheory[sigma](n))):
seq(a(n), n=1..100);  # Alois P. Heinz, Mar 28 2018

A244327[n_] := Floor[n*(n + 1)/(2*DivisorSigma[1, n])];
Array[A244327, 100] (* Paolo Xausa, Sep 01 2024 *)

a(n) = A244328(n) + A244329(n) for n >= 7.

A244328 a(1) = a(2) = 0; for n >= 3: a(n) = floor((n*(n+1)/2) / antisigma(n)) = floor(A000217(n) / A024816(n)).

Original entry on oeis.org

0, 0, 3, 3, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Jaroslav Krizek, Jul 08 2014

Decimal expansion of 29809/900000. - Stefano Spezia, Sep 02 2024

			For n = 10; a(10) = floor(A000217(10) / A024816(10)) = floor(55 / 37) = 1.

Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A000203, A000217, A024816, A244327, A244329.

[Floor((n*(n+1)div 2) div ((n*(n+1)div 2)-SumOfDivisors(n))): n in [3..1000]];

PadRight[{0, 0, 3, 3, 1, 2}, 100, 1] (* Paolo Xausa, Sep 01 2024 *)

if(n>6,1,[0, 0, 3, 3, 1, 2][n]) \\ Charles R Greathouse IV, May 15 2015

a(n) = 1 for n >= 7.
a(n) = A244327(n) - A244329(n) for n >= 7.
G.f.: x^3*(3 - 2*x^2 + x^3 - x^4)/(1 - x). - Elmo R. Oliveira, Aug 03 2024
E.g.f.: exp(x) - x - 1 + x^2*(x^4 + 60*x^2 + 240*x - 360)/720. - Stefano Spezia, Sep 02 2024

A244666 Numbers n such that floor(antisigma(n) / sigma(n)) = floor(antisigma(n+1) / sigma(n+1)).

Original entry on oeis.org

1, 2, 3, 9, 21, 33, 81, 261, 897, 1334, 1364, 2974, 4364, 14282, 26937, 46593, 64665, 74918, 79833, 92685, 145215, 147454, 161001, 162602, 166934, 289454, 347738, 383594, 422073, 430137, 440013, 443402, 445874, 621027, 649154, 655005, 1174305, 1187361, 1670955

Offset: 1

Jaroslav Krizek, Jul 08 2014

nonn

Also numbers n such that floor((n*(n+1)/2) / sigma(n)) = floor(((n+1)*(n+2)/2) / sigma(n+1)).
Numbers n such that A244327(n) = A244327(n+1).
Numbers n such that A244329(n) = A244329(n+1).

Jaroslav Krizek, Table of n, a(n) for n = 1..39

Cf. A000203, A000217, A024816, A046022, A244327, A244328, A244329.

[n: n in [1..10^6] | Floor((n*(n+1)div 2) div (SumOfDivisors(n))) eq Floor(((n+1)*(n+2)div 2) div (SumOfDivisors(n+1)))]

A244926 Numbers m such that there is an integer k with the property that antisigma(m) = k * sigma(m) + k.

Original entry on oeis.org

1, 2, 247, 2279, 9167, 57479, 200479, 518039, 2119207, 3685439, 9240079, 16384279, 31536647, 101601359, 140558807, 189771287, 299142967, 354032447, 384150199, 486103279, 565468637, 802008239, 853795074, 1107541759, 1328438479, 1494742004, 1580837719, 1768013279

Offset: 1

Jaroslav Krizek, Jul 08 2014

nonn

Numbers m such that A244329(m) = floor(antisigma(m) / sigma(m)) = antisigma(m) mod sigma(m) = A232324(n).
Corresponding values of integers k: 0, 0, 108, 1092, 4488, 28500, 99792, 258300, 1058148, ...
Numbers m such that sigma(m) + 1 divides antisigma(m). - Kevin P. Thompson, Nov 27 2021

			247 is in sequence because 30348 = antisigma(247) = 108 * sigma(247) + 108 = 108*280 + 108.

Kevin P. Thompson, Table of n, a(n) for n = 1..49

Cf. A024816 (antisigma), A000203 (sigma), A244329, A232324.

[n: n in [1..100000] | Floor(((n*(n+1)div 2) - (SumOfDivisors(n))) div (SumOfDivisors(n))) eq ((n*(n+1)div 2) - (SumOfDivisors(n))) mod (SumOfDivisors(n))]

isok(m) = my(s=sigma(m)); denominator((m*(m+1)/2-s)/(s+1)) == 1; \\ Michel Marcus, Jan 21 2022

a(10)-a(28) from Kevin P. Thompson, Nov 27 2021

cp's OEIS Frontend

A244327 a(n) = floor((n*(n+1)/2) / sigma(n)) = floor(A000217(n) / A000203(n)).

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A244328 a(1) = a(2) = 0; for n >= 3: a(n) = floor((n*(n+1)/2) / antisigma(n)) = floor(A000217(n) / A024816(n)).

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A244666 Numbers n such that floor(antisigma(n) / sigma(n)) = floor(antisigma(n+1) / sigma(n+1)).

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A244926 Numbers m such that there is an integer k with the property that antisigma(m) = k * sigma(m) + k.

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