A185079 -id:A185079 - cp's OEIS frontend

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

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.

A185099 Indices k for which A185079(k) > k^2.

Original entry on oeis.org

18, 21, 22, 24, 30, 36, 40, 48, 52, 56, 70, 80, 82, 85, 86, 90, 93, 94, 100, 102, 105, 115, 116, 118, 129, 132, 135, 165, 177, 180, 182, 183, 189, 201, 203, 205, 210, 213, 214, 216, 217, 224

Offset: 1

Vladimir Shevelev, Feb 18 2011

nonn

For other k >= 2, except for 6, 60, 120, 360, 816, 6120, 8280, ..., we have A185079(k) < k^2. In particular, all terms of the sequence are composite numbers.

Conjecture: The sequence is infinite.

Amiram Eldar, Table of n, a(n) for n = 1..3000

Cf. A185079.

Limit_{n->oo} a(n)/n^2 = 1.

Corrected by R. J. Mathar, Feb 19 2011

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

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

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A185099 Indices k for which A185079(k) > k^2.

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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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