A144695 -id:A144695 - cp's OEIS frontend

A327830 Numbers m such that the geometric mean of tau(m) and sigma(m) is an integer.

1, 7, 17, 22, 30, 31, 71, 94, 97, 115, 119, 127, 138, 154, 164, 165, 199, 210, 214, 217, 232, 241, 260, 265, 318, 337, 343, 374, 382, 449, 497, 510, 513, 517, 527, 577, 647, 658, 668, 679, 682, 705, 745, 759, 805, 862, 881, 889, 894, 930, 966, 967, 996, 1102, 1125

Offset: 1

Bernard Schott, Sep 27 2019

nonn

The first 20 terms of this sequence are also the first 20 terms of A144695: m such that sigma(m)/tau(m) is a square. Indeed, if sigma(m)/tau(m) is a square then sigma(m)*tau(m) is also a square, but the converse is false. These counterexamples are in A327831; the first one is a(21) = 232.
The primes p of the form 2*k^2 - 1: 7, 17, 31, 71, ... (A066436) form a subsequence because sigma(p) * tau(p) = (2*k)^2.
Another subsequence consists of the terms m such that sigma(m) and tau(m) are both squares; this occurs when m is the product of two distinct primes p*q, p < q where sigma(m) = (p+1)*(q+1) is a square and tau(m) = 4. The first few terms are 22, 94, 115, 119, 214, ... They are in A256152.

			sigma(30) = 72 and tau(30) = 8, sigma(30)*tau(30) = 576 = 24^2, hence 30 is a term.

Cf. A000005 (tau), A000203 (sigma), A064840 (tau*sigma).
Cf. A011257 (similar, with phi(m) and sigma(m)), A144695 (sigma(m)/tau(m) is a square), A327831 (sigma(m) * tau(m) is a square but sigma(m)/tau(m) is not an integer).
Subsequences: A066436, A256152.

[k:k in [1..1150]| IsSquare(#Divisors(k)*DivisorSigma(1,k))]; // Marius A. Burtea, Sep 27 2019

filter:= s -> issqr(sigma(s)*tau(s)) : select(filter, [$1..2500]);

Select[Range[1000], IntegerQ @ Sqrt[DivisorSigma[0, #] * DivisorSigma[1, #]] &] (* Amiram Eldar, Sep 27 2019 *)

isok(m) = issquare(numdiv(m)*sigma(m)); \\ Michel Marcus, Sep 27 2019

A327831 Numbers m such that sigma(m)*tau(m) is a square but sigma(m)/tau(m) is not an integer.

Original entry on oeis.org

232, 2152, 3240, 3560, 3944, 6516, 17908, 22504, 23716, 26172, 32360, 34344, 36584, 37736, 43300, 45612, 48204, 55080, 55912, 60520, 61480, 69352, 73084, 78184, 79056, 79300, 96552, 104168, 105832, 106088, 125356, 130432, 133864, 140040, 149992, 163764, 168424, 172840, 176360, 183204

Offset: 1

Bernard Schott, Oct 14 2019

nonn

If sigma(m)/tau(m) is a square (m is in A144695) then sigma(m)*tau(m) is also a square (m is in A327830), but the converse is false (see 232 in the Example section). This sequence consists of these counterexamples.

It seems that all terms are even. - Marius A. Burtea, Oct 15 2019

			sigma(232) = 450 and tau(232) = 8, so sigma(232)*tau(232) = 450*8 = 3600 = 60^2 and sigma(232)/tau(232) = 450/8 = 225/4 is not an integer, hence 232 is a term.

Equals A144695 \ A327830.
Similar to A327624 with sigma(m) and phi(m).
Cf. A003601 (sigma(m)/tau(m) is an integer), A023883 (sigma(m)/tau(m) is an integer and m is nonprime).
Cf. A000005 (tau), A000203 (sigma).

[k:k in [1..200000] | not IsIntegral(a/b) and IsSquare(a*b) where a is DivisorSigma(1,k) where b is #Divisors(k)]; // Marius A. Burtea, Oct 15 2019

filter:= u -> sigma(u)/tau(u) <> floor(sigma(u)/tau(u)) and issqr(sigma(u)*tau(u)) : select(filter, [$1..100000]);

sQ[n_] := IntegerQ@Sqrt[n]; aQ[n_] := sQ[(d = DivisorSigma[0, n]) * (s = DivisorSigma[1, n])] && !sQ[s/d]; Select[Range[2*10^5], aQ] (* Amiram Eldar, Oct 15 2019 *)

isok(m) = my(s=sigma(m), t=numdiv(m)); issquare(s*t) && (s % t); \\ Michel Marcus, Oct 15 2019

A341939 Numbers m such that phi(m)/tau(m) is a square of an integer where phi is the Euler totient function (A000010) and tau is the number of divisors function (A000005).

Original entry on oeis.org

1, 3, 8, 10, 18, 19, 24, 30, 34, 45, 52, 57, 73, 74, 85, 102, 125, 135, 140, 152, 153, 156, 163, 182, 185, 190, 202, 219, 222, 252, 255, 333, 342, 360, 375, 394, 416, 420, 436, 451, 455, 456, 459, 476, 489, 505, 514, 546, 555, 570, 584, 606, 625, 629, 640, 646, 679, 680, 730

Offset: 1

Bernard Schott, Feb 24 2021

nonn

The first 11 terms of this sequence are also the first 11 terms of A341938: m such that phi(m)*tau(m) is a square, then, a(12) = 57 while A341938(12) = 54. Indeed, if phi(m)/tau(m) is a perfect square then phi(m)*tau(m) is also a square, but the converse is false. These counterexamples are in A341940, the first one is 54 (last example).
Some subsequences (see examples):
-> The seven terms that satisfy also tau(m) = phi(m) form the subsequence A020488 with phi(m)/tau(m) = 1^2.
-> Primes p of the form 2*k^2 + 1 (A090698) form another subsequence because tau(p) = 2 and phi(p) = p-1 = 2*k^2, so phi(p)/tau(p) = k^2.
-> Cubes p^3 where p is a prime of the form k^2+1 (A002496) form another subset because if p = 2, phi(8)/tau(8)=1, and if p odd, phi(p^3)/tau(p^3) = (k*p/2)^2 with k even.

			phi(30) = 8, tau(30) = 8 so phi(30)/tau(30) = 1^2, and 30 is a term.
phi(45) = 24, tau(45) = 6, so phi(45)/tau(45) = 4 = 2^2, and 85 is a term.
phi(125) = 100, tau(125) = 4, so phi(125)/tau(125) = 25 = 5^2, and 125 is a term.
phi(54) = 18, tau(54) = 8, and phi(54)/tau(54) = 18/8 = 9/4 = (3/2)^2 and 54 is not a term while phi(54)*tau(54) = 12^2.

Intersection of A020491 and A341938.
Similar for: A144695 (sigma(n)/tau(n) perfect square), A293391 (sigma(n)/phi(n) perfect square).
Subsequences: A090698, A020488.
Cf. A069237, A289585, A341940.
Cf. A000005 (phi), A000010(tau).

with(numtheory): filter:= q -> phi(q)/tau(q) = floor(phi(q)/tau(q)) and issqr(phi(q)/tau(q)) : select(filter, [$1..750]);

Select[Range[1000], IntegerQ @ Sqrt[EulerPhi[#]/DivisorSigma[0, #]] &] (* Amiram Eldar, Feb 24 2021 *)

isok(m) = my(x=eulerphi(m)/numdiv(m)); (denominator(x)==1) && issquare(x); \\ Michel Marcus, Feb 24 2021

A152218 Numbers k such that sigma_2(k)*sigma_1(k)/sigma_0(k) is a perfect square.

Original entry on oeis.org

1, 4, 529, 2116, 2583, 3249, 3346, 6150, 10332, 12474, 12792, 12996, 28224, 38240, 59245, 85905, 91035, 103607, 142560, 176382, 212949, 236980, 249744, 343620, 360096, 364140, 379050, 414428, 450840, 751530, 787710, 788424, 851796, 1059474, 1132096, 1366407

Offset: 1

Ctibor O. Zizka, Nov 29 2008

nonn

Amiram Eldar, Table of n, a(n) for n = 1..500 (terms 1..200 from Donovan Johnson)

Cf. A000005, A000290, A000203, A001157, A140480, A144695.

fQ[n_] := IntegerQ[ Sqrt[ DivisorSigma[2, n] DivisorSigma[1, n]/DivisorSigma[0, n]]]; k = 1; lst = {}; While[k < 1132096, If[ fQ@k, AppendTo[lst, k]; Print@k]; k++ ]; lst (* Robert G. Wilson v, Sep 10 2010 *)
Select[Range[137*10^4],IntegerQ[Sqrt[(DivisorSigma[2,#]DivisorSigma[ 1,#])/ DivisorSigma[ 0,#]]]&] (* Harvey P. Dale, Jun 18 2018 *)

isok(k) = {my(f = factor(k)); issquare(sigma(f, 2) * sigma(f) / numdiv(f));} \\ Amiram Eldar, Feb 01 2025

{k: A001157(k)*A000203(k)/A000005(k) in A000290}.

Correct definition recovered by Jack Brennen
12 more terms from R. J. Mathar, Aug 25 2010
More terms from Robert G. Wilson v, Sep 10 2010

A272442 Numbers n such that the average of the positive divisors of n^2 is a square.

Original entry on oeis.org

1, 313, 4077, 42217, 75848, 1276101, 2468936, 8105257, 13213921, 23740424, 76136648, 172118709, 249948601, 296474857, 309232296, 772776968, 1236329761, 1254551521, 2288805793, 2372141576, 2410951561, 2536945441, 3202075016, 10065852072

Offset: 1

Altug Alkan, Apr 29 2016

nonn

The values of n are 1, 313, 3^3*151, 7*37*163, 2^3*19*499, 3^3*151*313, 2^3*19*37*439, 37*439*499, ...
Corresponding n^2 values are 1, 97969, 16621929, 1782275089, ...
The next two primes in the sequence (after 313 and 2288805793) are 86136917171761 and 23666840016182721134955337. - Giovanni Resta, Apr 26 2017

			313 is a term because the divisors of 313^2 are 1, 313 and 313^2. Average of 1, 313 and 313^2 is 32761 that is 181^2.

Cf. A000290, A003601, A144695.

Select[Range[10^5], IntegerQ@ Sqrt@ Mean@ Divisors[#^2] &] (* Michael De Vlieger, Apr 29 2016 *)

lista(nn) = for(n=1, nn, if(sigma(n^2) % numdiv(n^2) == 0 && issquare(sigma(n^2)/numdiv(n^2)), print1(n, ", ")));

is(n)=my(f=factor(n),s,d); f[,2]*=2; s=sigma(f); d=numdiv(f); s%d==0 && issquare(s/d) \\ Charles R Greathouse IV, Apr 29 2016

a(10)-a(16) from Charles R Greathouse IV, Apr 29 2016

a(17)-a(24) from Giovanni Resta, Apr 26 2017

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A327830 Numbers m such that the geometric mean of tau(m) and sigma(m) is an integer.

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A327831 Numbers m such that sigma(m)*tau(m) is a square but sigma(m)/tau(m) is not an integer.

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A341939 Numbers m such that phi(m)/tau(m) is a square of an integer where phi is the Euler totient function (A000010) and tau is the number of divisors function (A000005).

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A152218 Numbers k such that sigma_2(k)*sigma_1(k)/sigma_0(k) is a perfect square.

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A272442 Numbers n such that the average of the positive divisors of n^2 is a square.

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