A066218 -id:A066218 - cp's OEIS frontend

A066363 The floor[n^(3/4)]-perfect numbers, where f-perfect numbers for an arithmetical function f is defined in A066218.

2, 4, 8, 10, 16, 98, 236, 244, 268, 284, 292, 316, 332, 638, 2198, 2282, 2338, 2422, 2674, 4653, 12274, 30753, 65018, 225267, 231478, 289605, 376995, 422684, 449756, 453092, 515175, 521925, 522825, 524325, 527025, 527925, 528225, 534075, 538275, 539025, 540975

Offset: 1

Joseph L. Pe, Dec 20 2001

nonn

Amiram Eldar, Table of n, a(n) for n = 1..268
J. Pe, On a Generalization of Perfect Numbers, J. Rec. Math., 31(3) (2002-2003), 168-172.

Cf. A066218.

f[x_] := Floor[ x^(3/4)]; Select[ Range[ 2, 10^4], 2 * f[ # ] == Apply[ Plus, Map[ f, Divisors[ # ] ] ] & ]

a(21)-a(41) from Amiram Eldar, Sep 26 2019

A238905 The tau(sigma)-perfect numbers, where the set of f-perfect numbers for an arithmetical function f is defined in A066218.

Original entry on oeis.org

6, 15, 22, 33, 39, 57, 69, 111, 129, 141, 183, 201, 214, 219, 237, 309, 453, 471, 489, 573, 579, 633, 669, 813, 831, 849, 939, 993, 1101, 1149, 1191, 1263, 1371, 1389, 1461, 1519, 1569, 1623, 1641, 1821, 1839, 1893, 1942, 1983, 2019, 2073, 2199, 2253, 2271

Offset: 1

Paolo P. Lava, Mar 07 2014

			Aliquot divisors of 39 are 1, 3, 13. Then tau(sigma(1)) + tau(sigma(3)) + tau(sigma(13)) = 1 + 3 + 4 = 8 and tau(sigma(39)) = 8.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Paolo P. Lava)

Cf. A000005, A000230, A066218.

with(numtheory); P:=proc(q) local a,b,i,n;
for n from 1 to q do a:=divisors(n); b:=0;
for i from 1 to nops(a)-1 do b:=b+tau(sigma(a[i])); od;
if tau(sigma(n))=b then print(n); fi; od; end: P(10^6);

q[n_] := DivisorSum[n, DivisorSigma[0, DivisorSigma[1, #]] &, # < n &] == DivisorSigma[0, DivisorSigma[1, n]]; Select[Range[2300], q] (* Amiram Eldar, Aug 22 2023 *)

isok(n) = numdiv(sigma(n)) == sumdiv(n, d, (dMichel Marcus, Mar 08 2014

A066226 The sigma(EulerPhi)-perfect numbers, where the set of f-perfect numbers for an arithmetical function f is defined in A066218.

Original entry on oeis.org

2, 88, 328, 5128, 9075, 327688, 1310728, 2066056, 2259976, 188186624, 560889856, 847020032, 1342177288

Offset: 1

Joseph L. Pe, Dec 18 2001

These are all the terms less than 10^5. Problem: Find an expression generating all the terms.

			Let f(n) = sigma(EulerPhi(n)). The proper divisors of 88 are {1, 2, 4, 8, 11, 22, 44}; adding their f-values: 1 + 1 + 3 + 7 + 18 + 18 + 42 = 90 = f(88). Hence 88 is a term of the sequence.

J. Pe, On a Generalization of Perfect Numbers, J. Rec. Math., 31(3) (2002-2003), 168-172.

Cf. A000010, A000203, A066218.

f[x_] := DivisorSigma[1, EulerPhi[x]]; Select[ Range[ 1, 10^5], 2 * f[ # ] == Apply[ Plus, Map[ f, Divisors[ # ] ] ] & ]

a(6)-a(13) from Amiram Eldar, Sep 26 2019

A066228 The EulerPhi(sigma(EulerPhi))-perfect numbers, where the f-perfect numbers for an arithmetical function f are defined in A066218.

Original entry on oeis.org

2, 4, 28, 40, 448, 500380

Offset: 1

Joseph L. Pe, Dec 18 2001

There are no terms between 449 and 10^5. Are there any more terms? Are there infinitely many?

No more terms below 10^9. - Amiram Eldar, Sep 26 2019

			Let f(n) = EulerPhi(sigma(EulerPhi(n))). Proper divisors of 28 = {1, 2, 4, 7, 14}; the sum of their f-values = 1+1+2+4+4 = 12 = f(28); hence 28 belongs to the sequence.

J. Pe, On a Generalization of Perfect Numbers, J. Rec. Math., 31(3) (2002-2003), 168-172.

Cf. A000010, A000203, A066218.

f[x_] := EulerPhi[DivisorSigma[1, EulerPhi[x]]]; Select[ Range[ 1, 10^5], 2 * f[ # ] == Apply[ Plus, Map[ f, Divisors[ # ] ] ] & ]

a(6) from Amiram Eldar, Sep 26 2019

A066238 The floor(n/3)-perfect numbers, where f-perfect numbers for an arithmetical function f are defined in A066218.

Original entry on oeis.org

2, 12, 18, 40, 56, 304, 550, 748, 1504, 3230, 3770, 6976, 29824, 124672, 351351, 382772, 510464, 537248, 698528, 791264, 1081568, 1279136, 2065408, 2279072, 211855016, 561841408, 731378944, 3365232128, 3557004544

Offset: 1

Joseph L. Pe, Dec 19 2001

It appears that there are more floor(n/N)-perfect numbers the larger N is. (Here N = 3.)

			Let f(n) = floor(n/3). Then f(12) = 6 = 3+2+1+0 = f(6)+f(4)+f(3)+f(1); so 12 is a term of the sequence.

J. Pe, On a Generalization of Perfect Numbers, J. Rec. Math., 31(3) (2002-2003), 168-172.

Cf. A066218.

f[x_] := Floor[x/3]; Select[ Range[2, 10^5], 2 * f[ # ] == Apply[ Plus, Map[ f, Divisors[ # ] ] ] & ]

a(14)-a(29) from Amiram Eldar, Sep 26 2019

A066239 The floor(1.001*x)-perfect numbers, where f-perfect numbers for an arithmetical function f are defined in A066218.

Original entry on oeis.org

6, 28, 496, 32445, 130304, 388076, 199272950

Offset: 1

Joseph L. Pe, Dec 19 2001

The floor(n)-perfect numbers are the ordinary perfect numbers. The first three floor[1.001x]-perfect numbers are also ordinary perfect numbers and the first discrepancy comes at the fourth term, 32445 (the fourth perfect number is 8128). Consider other coefficients > 1 but < 1.001. There is some kind of continuity working here. The first discrepancies, if they exist, come at later and later terms as these coefficients are made closer to 1.

			Let f(n) = floor(1.001*n). Then f(6) = 6 = 3+2+1 = f(3)+f(2)+f(1); so 6 is a term of the sequence.

J. Pe, On a Generalization of Perfect Numbers, J. Rec. Math., 31(3) (2002-2003), 168-172.

Cf. A066218.

f[x_] := Floor[1.001*x]; Select[ Range[1, 10^5], 2 * f[ # ] == Apply[ Plus, Map[ f, Divisors[ # ] ] ] & ]

a(5)-a(7) from Amiram Eldar, Sep 26 2019

A066240 The floor(n/2)-perfect numbers, where f-perfect numbers for an arithmetical function f are defined in A066218.

Original entry on oeis.org

18, 20, 70, 104, 464, 1952, 45356, 91388, 130304, 254012, 388076, 437745, 522752, 8382464, 134193152

Offset: 1

Joseph L. Pe, Dec 19 2001

			Let f(n) = floor(n/2). Then f(18) = 9 = 4+3+1+1+0 = f(9)+f(6)+f(3)+f(2)+f(1); so 18 is a term of the sequence.

J. Pe, On a Generalization of Perfect Numbers, J. Rec. Math., 31(3) (2002-2003), 168-172.

Cf. A066218.

f[x_] := Floor[x/2]; Select[ Range[ 1, 10^5], 2 * f[ # ] == Apply[ Plus, Map[ f, Divisors[ # ] ] ] & ]

a(7)-a(15) from Amiram Eldar, Sep 26 2019

A066242 The floor((log(x))^2)-perfect numbers, where f-perfect numbers for an arithmetical function f are defined in A066218.

Original entry on oeis.org

2, 18, 20, 32, 45, 63

Offset: 1

Joseph L. Pe, Dec 19 2001

			Let f(n) = floor((log(x))^2). Then f(18) = 8 = 4+3+1+0+0 = f(9)+f(6)+f(3)f(2)+f(1); so 18 is a term of the sequence.

J. Pe, On a Generalization of Perfect Numbers, J. Rec. Math., 31(3) (2002-2003), 168-172.

f[x_ ] := Floor[Log[x]^2]; Select[ Range[2, 10^5], 2 * f[ # ] == Apply[ Plus, Map[ f, Divisors[ # ] ] ] & ]

A066245 Floor(|x*sin(x)|)-perfect numbers, where f-perfect numbers for an arithmetical function f are defined in A066218.

Original entry on oeis.org

3, 6, 10, 34, 50, 91, 222, 364, 1485, 6640, 18579, 775698, 1035507, 1706366, 46388515, 75714802

Offset: 1

Joseph L. Pe, Dec 19 2001

			Let f(n) = floor(|x*sin(x)|). Then f(6) = 1 = 0+1+0 = f(3)+f(2)+f(1); so 6 is a term of the sequence.

J. Pe, On a Generalization of Perfect Numbers, J. Rec. Math., 31(3) (2002-2003), 168-172.

Cf. A066218.

f[x_] := Floor[Abs[x*Sin[x]]]; Select[ Range[2, 10^4], 2 * f[ # ] == Apply[ Plus, Map[ f, Divisors[ # ] ] ] & ]

a(11)-a(16) from Amiram Eldar, Sep 26 2019

A066361 The floor(n^(.9999))-perfect numbers, where f-perfect numbers for an arithmetical function f is defined in A066218.

Original entry on oeis.org

2, 18, 17816, 116624

Offset: 1

Joseph L. Pe, Dec 20 2001

Despite the closeness of f(n) = floor(n^(.9999)) to floor(n) (which generates the usual perfect numbers), the perfect numbers generated by these functions are quite different.

J. Pe, On a Generalization of Perfect Numbers, J. Rec. Math., 31(3) (2002-2003), 168-172.

Cf. A066218.

f[x_] := Floor[x^(.9999)]; Select[ Range[2, 10^6], 2 * f[ # ] == Apply[ Plus, Map[ f, Divisors[ # ] ] ] & ]

cp's OEIS Frontend

A066363 The floor[n^(3/4)]-perfect numbers, where f-perfect numbers for an arithmetical function f is defined in A066218.

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A238905 The tau(sigma)-perfect numbers, where the set of f-perfect numbers for an arithmetical function f is defined in A066218.

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A066226 The sigma(EulerPhi)-perfect numbers, where the set of f-perfect numbers for an arithmetical function f is defined in A066218.

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A066228 The EulerPhi(sigma(EulerPhi))-perfect numbers, where the f-perfect numbers for an arithmetical function f are defined in A066218.

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A066238 The floor(n/3)-perfect numbers, where f-perfect numbers for an arithmetical function f are defined in A066218.

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A066239 The floor(1.001*x)-perfect numbers, where f-perfect numbers for an arithmetical function f are defined in A066218.

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A066240 The floor(n/2)-perfect numbers, where f-perfect numbers for an arithmetical function f are defined in A066218.

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A066242 The floor((log(x))^2)-perfect numbers, where f-perfect numbers for an arithmetical function f are defined in A066218.

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