A185383 -id:A185383 - cp's OEIS frontend

A185373 The numerator of the fraction |n^2/A049417(n)-A064380(n)|.

1, 1, 1, 1, 0, 1, 4, 1, 5, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 7, 5, 1, 2, 1, 2, 9, 3, 1, 1, 1, 4, 11, 11, 25, 2, 1, 1, 9, 2, 1, 11, 1, 4, 3, 7, 1, 2, 1, 2, 1, 13, 1, 3, 1, 13, 49, 17, 1, 0, 1, 1, 49, 16, 25, 1, 1, 17, 19, 35, 1, 14, 1, 2

Offset: 2

Vladimir Shevelev, Feb 17 2011

n^2/A049417(n) is a multiplicative function, whereas A064380 is not. This sequence here measures the (small) differences n^2/A049417(n)-A064380(n) = 1/3, 1/4, 1/5, 1/6, 0, 1/8, 4/15, 1/10, 5/9, 1/12, 1/5 ...

Amiram Eldar, Table of n, a(n) for n = 2..10000

Cf. A064380, A049417, A185383 (denominators)

f[p_, e_] := p^(2^(-1 + Position[Reverse @ IntegerDigits[e, 2], ?(# == 1 &)])); isigma[1] = 1; isigma[n] := Times @@ (Flatten@(f @@@ FactorInteger[n]) + 1);
infCoprimeQ[n1_, n2_] := Module[{g = GCD[n1, n2]}, If[g == 1, True, AllTrue[ FactorInteger[g][[;; , 1]], BitAnd @@ IntegerExponent[{n1, n2}, #] == 0 &]]];
a[n_] := Abs[Numerator[n^2 / isigma[n] - Sum[Boole[infCoprimeQ[j, n]], {j, 1, n-1}]]]; Array[a, 100, 2] (* Amiram Eldar, Mar 20 2025 *)

A185079 a(n) = A064380(n) * A049417(n).

Original entry on oeis.org

3, 8, 15, 24, 36, 48, 60, 80, 90, 120, 140, 168, 192, 216, 255, 288, 330, 360, 390, 448, 504, 528, 600, 624, 672, 720, 760, 840, 936, 960, 1020, 1056, 1134, 1200, 1300, 1368, 1440, 1512, 1620, 1680, 1632, 1848, 1920, 1980, 2088, 2208, 2312, 2400, 2496, 2592, 2730, 2808, 2880, 3024, 3240, 3200, 3330

Offset: 2

Vladimir Shevelev, Feb 18 2011

nonn

Amiram Eldar, Table of n, a(n) for n = 2..10000

Cf. A064380, A049417, A007357, A185078, A185373, A185383.

a(n) = n^2 + o(n^(1+eps)).

Corrected and extended by T. D. Noe, Feb 18 2011

A185078 Numbers k for which A064380(k) = k/2.

Original entry on oeis.org

2, 6, 8, 10, 60, 70, 128, 136, 9822, 18632, 32768, 32896, 36720, 69726, 73662, 73686, 73734, 85962, 86046, 87114, 87198, 87222, 87258, 87294, 87306, 87342, 87366, 87546, 87558, 88014, 88278, 88302, 88338, 88386, 127326, 128046, 128082, 128382, 128406, 128598, 128802

Offset: 1

Vladimir Shevelev, Feb 18 2011

nonn

Note that, if there exist infinitely many infinitary perfect numbers (A007357), then, as k tends to infinity over such numbers, A064380(k)/k = 1/2 + o(k^(-1+eps)). We conjecture that here A064380(k)/k = 1/2 infinitely many times, and thus the sequence contains infinitely many infinitary perfect numbers.

Amiram Eldar, Table of n, a(n) for n = 1..201

Cf. A064380, A007357, A064380, A185373, A185383.

a(7)-a(13) from Amiram Eldar, Sep 13 2019

a(14)-a(41) from Amiram Eldar, Mar 26 2023

A185088 a(n) = |n^2 - A185079(n)|.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 4, 1, 10, 1, 4, 1, 4, 9, 1, 1, 6, 1, 10, 7, 20, 1, 24, 1, 4, 9, 24, 1, 36, 1, 4, 33, 22, 25, 4, 1, 4, 9, 20, 1, 132, 1, 16, 45, 28, 1, 8, 1, 4, 9, 26, 1, 36, 1, 104, 49, 34, 1, 0, 1, 4, 49, 16, 25, 36, 1, 34, 57, 140, 1, 84, 1, 4, 9, 76, 73, 36, 1, 26, 1, 80, 1, 16, 11, 128, 9, 4, 1, 180, 105, 64, 55, 92, 25, 36

Offset: 2

Vladimir Shevelev, Feb 18 2011

nonn

Zeros a(z)=0 occur at z=6, 60, 120, 360, 816,... For these z, A049417(z) | z^2, but there may be other numbers like 90, 180, 540,... satisfying this divisibility criterion which are not places of zeros (the criterion is necessary, not sufficient), see A185288.

Amiram Eldar, Table of n, a(n) for n = 2..10000

Cf. A185079, A050376, A064380, A049417, A007357, A185078, A185373, A185383.

a(A050376(n)) = 1.

A185288 Numbers n for which the terms of the multiplicative sequence {n^2/A049417(n)} are integers.

Original entry on oeis.org

1, 6, 60, 90, 120, 180, 360, 540, 816, 840, 1080, 1740, 1980, 2280, 2520, 3060, 3960, 5712, 6120, 8280, 9540, 11880, 12240, 16920, 18360, 19260, 24480, 25296, 25560, 32760, 36720, 42840, 48960, 54672, 57240, 63700, 73440, 74256, 84360, 85680, 97920, 103320, 115560

Offset: 1

Vladimir Shevelev, Feb 20 2011

nonn

The sequence contains all infinitary perfect numbers (see A007357).

			Let n=120. Its representation over distinct terms of A050376 is 2*3*4*5. Therefore A049417(n)=(2+1)*(3+1)*(4+1)*(5+1)=360. Since 360 is a divisor of 120^2, 120 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A007357, A049417, A064380, A185079, A185088, A185098, A185373, A185383.

fun[p_, e_] := Module[{b = IntegerDigits[e, 2]}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]]; isigma[1] = 1; isigma[n_] := Times @@ fun @@@ FactorInteger[n]; aQ[n_] := Divisible[n^2, isigma[n]]; Select[Range[58000], aQ] (* Amiram Eldar, Jul 21 2019 *)

More terms from Nathaniel Johnston, Mar 16 2011

More terms from Amiram Eldar, Jul 21 2019

cp's OEIS Frontend

A185373 The numerator of the fraction |n^2/A049417(n)-A064380(n)|.

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A185079 a(n) = A064380(n) * A049417(n).

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A185078 Numbers k for which A064380(k) = k/2.

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A185088 a(n) = |n^2 - A185079(n)|.

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A185288 Numbers n for which the terms of the multiplicative sequence {n^2/A049417(n)} are integers.

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