A256151 -id:A256151 - cp's OEIS frontend

A256152 Numbers k such that k is the product of two distinct primes and sigma(k) is a square number.

22, 94, 115, 119, 214, 217, 265, 382, 497, 517, 527, 679, 745, 862, 889, 1174, 1177, 1207, 1219, 1393, 1465, 1501, 1649, 1687, 1915, 1942, 2101, 2159, 2201, 2359, 2899, 2902, 2995, 3007, 3143, 3383, 3401, 3427, 3937, 4039, 4054, 4097, 4315, 4529, 4537, 4702, 4741, 5029, 5065, 5398, 5587

Offset: 1

Antonio Roldán, Mar 16 2015

nonn

This sequence is the intersection of A006881 and A006532.

			199 is in the sequence because 119=7*17 (the product of two distinct primes) and sigma(119)=8*18=144=12^2 (a square number).

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A006881, A006532, A256149, A256150, A256151.

Cf. A020639, A010052.

a256152 n = a256152_list !! (n-1)
256152_list = filter f a006881_list where
   f x = a010052' ((spf + 1) * (x `div` spf + 1)) == 1
         where spf = a020639 x
-- Reinhard Zumkeller, Apr 06 2015

f[n_] := Block[{pf = FactorInteger@ n}, Max @@ Last /@ pf == 1 && Length@ pf == 2]; Select[Range@ 6000, IntegerQ@ Sqrt@ DivisorSigma[1, #] && f@ # &] (* Michael De Vlieger, Mar 17 2015 *)

{for(i=1,10^4,if(omega(i)==2&&issquarefree(i)&&issquare(sigma(i)),print1(i,", ")))}

A256149 Square numbers n such that sigma(n) is a triangular number.

Original entry on oeis.org

1, 36, 441, 5625, 6084, 407044, 8444836, 17388900, 35070084, 40729924, 57790404, 80138304, 537822481, 588159504, 659821969, 918999225, 1820387556, 2179862721, 2599062361, 5110963081, 28816420516, 36144473689, 46082779561, 55145598561, 147225690000, 163405126756

Offset: 1

Antonio Roldán, Mar 16 2015

nonn

This sequence is the intersection of A000290 and A045746.

			441 is in the sequence because 441 = 21^2 is square number, and sigma(441) = 441 + 147 + 63 + 49 + 21 + 9 + 7 + 3 + 1 = 741 = 38*39/2 is triangular number.

Amiram Eldar, Table of n, a(n) for n = 1..72

Cf. A000290, A000217, A045746, A083674, A256150, A256151, A256152.

t = Accumulate[Range@ 10000]; Select[Range[10000]^2, MemberQ[t, DivisorSigma[1, #]] &] (* Michael De Vlieger, Mar 17 2015 *)
Select[Range[500000]^2,OddQ[Sqrt[8DivisorSigma[1,#]+1]]&] (* Harvey P. Dale, Feb 25 2017 *)

{for(i=1,10^6,n=i*i;if(ispolygonal(sigma(n), 3),print1(n,", ")))}

A256150 Oblong numbers n such that sigma(n) is a triangular number.

Original entry on oeis.org

2, 12, 56, 342, 992, 16256, 17822, 169332, 628056, 1189190, 2720850, 11085570, 35599122, 67100672, 1147210770, 1317435912, 1707135806, 7800334080, 11208986256, 13366943840, 17109032402, 17179738112, 46343540900, 58413331032, 83717924940, 204574837700, 274877382656, 445968192672, 589130699852

Offset: 1

Antonio Roldán, Mar 16 2015

nonn

The numbers 12, 56, 992, 16256, 67100672, ... (A139256(n), twice even perfect numbers) are in the sequence because they are oblong (A139256(n) = 2^k*(2^k-1)) and sigma(A139256(n)) = sigma(2^k*(2^k-1)) = sigma(2^k)*sigma(2^k-1) = (2^(k+1)-1)*2^(k+1)/2 is a triangular number.

This sequence is the intersection of A002378 and A045746.

			2 is in the sequence because 2=1*2 is oblong, and sigma(2) = 1+2 = 3 = 2*3/2 is triangular.

Cf. A000217, A002378, A139256, A045746, A256149, A256151, A256152.

Select[2 Accumulate[Range@10000], MemberQ[Accumulate[Range@10000], DivisorSigma[1, #]] &] (* Michael De Vlieger, Mar 17 2015 *)

{for (i=1,i=10^6,n=i*(i+1);if(ispolygonal(sigma(n), 3),print(n)))}

A287473 Triangular numbers k such that phi(k) is a square number, where phi(k) is the Euler totient function (A000010).

Original entry on oeis.org

1, 10, 136, 630, 2016, 7875, 9180, 18915, 32896, 37128, 46056, 58311, 66430, 103740, 131841, 198135, 225456, 301476, 323610, 332520, 408156, 499500, 738720, 786885, 839160, 862641, 922761, 924120, 1065070, 1079715, 1183491, 1385280, 1851850, 1906128, 1925703

Offset: 1

Amiram Eldar, May 25 2017

nonn

The indices of these triangular numbers are: 1, 4, 16, 35, 63, 125, 135, 194, 256, 272, 303, 341, 364, 455, 513, 629, 671, 776, 804, 815, 903, 999, 1215, 1254, 1295, 1313, 1358, 1359, 1459, 1469, 1538, 1664, 1924, 1952, 1962, ... and their phi values are the squares of: 1, 2, 8, 12, 24, 60, 48, 96, 128, 96, 120, 180, 144, 144, 288, 288, 240, 288, 264, 288, 336, 360, 432, 600, 432, 720, 720, 480, 648, 672, 864, 576, 720, 720, 1080, ...

Similar to A115910, since A115910(n)^2 are squares whose phi is a triangular number.

			136=16*17/2 is triangular, phi(136)=64=8^2 is a square, thus 136 is in the sequence.

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

Cf. A000010, A000217, A000290, A039770, A115910, A256151.

Intersection of A000217 and A039770.

Select[Accumulate[Range[1000]],IntegerQ[Sqrt[EulerPhi[#]]]&]

isok(n) = ispolygonal(n, 3) && issquare(eulerphi(n)); \\ Michel Marcus, May 25 2017

A339599 Quadratically perfect numbers: numbers k such that (sigma(k) - 2k)^2 = sigma(k).

Original entry on oeis.org

1, 3, 66, 491536

Offset: 1

Tomohiro Yamada, Dec 09 2020

Paul Lescot observed that if m is a perfect number and 2m - 1 is prime, then m * (2m - 1) belongs to this sequence.
Moreover, Lescot proved that an even integer k belongs to this sequences if and only if k = m * (2m - 1) for an even perfect number m = 2^(p-1) * (2^p - 1) and 2m - 1 is prime. Among all primes p in A000043 (Mersenne exponents) up to 23209, only p = 2 and 5, which give 66 and 491536 respectively, satisfy this condition.
An even term k in this sequence belongs to A256151 since k = m * (2m - 1) and sigma(k) is square.
Lescot also confirms that there exists no odd integer between 5 and 10^6 in this sequence and conjectures that there exists no further odd term in this sequence.
There exists no further term in this sequence in n <= 2^32.
Question: are there only four terms in this sequence?

			sigma(1) = 1 = (sigma(1) - 2)^2;
sigma(3) = 4 = (sigma(3) - 6)^2;
sigma(66) = 144 = 12^2 = (sigma(66) - 132)^2.

Paul Lescot, An arithmetical question related to perfect numbers, The Mathematical Gazette 104 (2020), 20-26, a preprint available from Archive ouverte HAL.

Cf. A000396, A256151, A033880.

PARI
```
is(k) = {sigma(k) == (sigma(k)-2*k)^2}
```

cp's OEIS Frontend

A256152 Numbers k such that k is the product of two distinct primes and sigma(k) is a square number.

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A256149 Square numbers n such that sigma(n) is a triangular number.

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A256150 Oblong numbers n such that sigma(n) is a triangular number.

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A287473 Triangular numbers k such that phi(k) is a square number, where phi(k) is the Euler totient function (A000010).

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A339599 Quadratically perfect numbers: numbers k such that (sigma(k) - 2k)^2 = sigma(k).

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