A066436 -id:A066436 - cp's OEIS frontend

A049226 Composite numbers n such that the sum of divisors of n, sigma(n), divided by the number of divisors, d(n) and sigma(n) minus n are both rational squares.

119, 527, 1196, 3591, 5831, 6887, 12319, 15407, 18575, 33271, 47959, 51119, 56853, 63119, 65151, 116399, 176911, 328151, 373319, 437999, 438311, 520319, 568519, 724687, 734111, 851927, 957551, 1059191, 1140071, 1437599, 1760831, 1813511, 2320919, 3354479, 3383420

Offset: 1

Enoch Haga

nonn

The prime numbers with this property are primes of the form 2*k^2 - 1 (A066436). - Amiram Eldar, Aug 15 2019

The first terms for which the ratio sigma(n)/d(n) is not an integer are 267910912, 1398459816, and 1703794876. - Giovanni Resta, Aug 30 2019

			a(27) = 957551 is a term since the sum of its 16 divisors is sigma(957551) = 1166400 and both 1166400/16 = 72900 = 270^2 and 1166400 - 957551 = 208849 = 457^2 are perfect squares.

Giovanni Resta, Table of n, a(n) for n = 1..1000 (first 300 terms from Amiram Eldar)

Cf. A000005, A000203, A049207, A049227, A049228, A048699, A066436.

[m:m in [1..3400000]|not IsPrime(m) and IsSquare(SumOfDivisors(m)/#Divisors(m)) and IsSquare(SumOfDivisors(m)-m)]; // Marius A. Burtea, Aug 15 2019

Select[Range[10^5], CompositeQ[#] && And @@ IntegerQ /@ Sqrt[{(s = DivisorSigma[1, #]) * DivisorSigma[0, #], s - #}] &] (* Amiram Eldar, Aug 15 2019 *)
cnQ[n_]:=With[{sg=DivisorSigma[1,n]},CompositeQ[n]&&AllTrue[{Sqrt[sg/DivisorSigma[0,n]],Sqrt[sg-n]},IntegerQ]]; Select[Range[ 339*10^4],cnQ] (* Harvey P. Dale, Mar 31 2025 *)

is(n) = my(f = factor(n), s = sigma(f), nd = numdiv(f)); issquare(s/nd) && issquare(s - n) && !isprime(n) \\ David A. Corneth, Aug 15 2019

Name and offset corrected by Amiram Eldar, Aug 15 2019

A143832 Primes of the form 14 n^2-1.

Original entry on oeis.org

13, 223, 349, 503, 1399, 1693, 3583, 6173, 9463, 16183, 18143, 27103, 28349, 33613, 40823, 42349, 48733, 59149, 66653, 70573, 80863, 89599, 101149, 115933, 126349, 129023, 139999, 151423, 157303, 169399, 181943, 185149, 201599, 204973, 218749

Offset: 1

Artur Jasinski, Sep 02 2008

nonn

Primes of the form k n^2-1 k = 2 A066436 these n are A066049 k = 4 only one prime 3 when n = 1 k = 6 A090686 these n are A143826 k = 8 A090684 these n are A143827 k =10 A143828 these n are A143829 k =12 A143830 these n are A143831 k =14 A143832 these n are A143833 k =16 lack of primes

A066436, A066049, A090686, A090684, A143826, A143827, A143828, A143829, A143830, A143831, A143833

p = 14; a = {}; Do[k = p x^2 - 1; If[PrimeQ[k], AppendTo[a, k]], {x, 1, 1000}]; a
Select[14*Range[200]^2-1,PrimeQ] (* Harvey P. Dale, Jul 29 2024 *)

A160697 Record values in A160696.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 10, 11, 13, 15, 17, 18, 21, 22, 24, 25, 28, 34, 36, 38, 39, 41, 42, 43, 45, 46, 49, 50, 52, 56, 59, 62, 63, 64, 69, 73, 76, 80, 81, 85, 87, 91, 92, 95, 98, 102, 108, 109, 112, 113, 115, 118, 125, 126, 127, 132, 134, 137, 140, 141, 143, 153, 154, 155

Offset: 1

Reinhard Zumkeller, May 24 2009

nonn

a(n)=A160696(A160698(n)) and A160696(m)A160698(n);

for n>1: a(n)=A066049(n-1) and A066436(n-1)+1=2*a(n)^2.

A182784 Primes of the form 2*n^4-1.

Original entry on oeis.org

31, 1249, 2591, 4801, 8191, 13121, 76831, 131071, 388961, 1062881, 1229311, 2672671, 6223391, 7496191, 9759361, 10616831, 12499999, 13530401, 29552671, 31505921, 35701249, 42762751, 48019999, 63281249, 66724351, 77900161, 90424351, 99574271, 149610401

Offset: 1

Vincenzo Librandi, Dec 02 2010

Subsequence of A066436. - R. J. Mathar, Dec 02 2010

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A182783, A182785.

[a: n in [1..350] | IsPrime(a) where a is 2*n^4-1];

Select[Table[2 n^4 - 1, {n, 100}], PrimeQ] (* Vincenzo Librandi, Sep 01 2012 *)

a(n) = 2*A182783(n)^4-1. - R. J. Mathar, Dec 02 2010

A234638 Numbers n for which sigma(sigma(n)) is odd.

Original entry on oeis.org

1, 3, 7, 10, 17, 21, 22, 30, 31, 46, 51, 52, 55, 66, 70, 71, 81, 93, 94, 97, 106, 115, 119, 127, 138, 154, 156, 165, 170, 199, 210, 213, 214, 217, 232, 235, 241, 253, 265, 282, 291, 298, 310, 318, 322, 337, 343, 345, 357, 364, 374, 381, 382, 385

Offset: 1

M. F. Hasler, Dec 28 2013

nonn

See A234641 for numbers n such that n^2 is in this sequence.

The primes in this sequence are 3 and A066436. - Robert Israel, Apr 05 2019

			3 is in the sequence because sigma(sigma(3)) = sigma(4) = 7, which is odd.
7 is in the sequence because sigma(sigma(7)) = sigma(8) = 15, which is odd.
8 is not in the sequence because sigma(sigma(8)) = sigma(15) = 24, which is even.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000203, A028982, A066436, A234641.

filter:= proc(n) uses numtheory; sigma(sigma(n))::odd end proc:
select(filter, [$1..1000]); # Robert Israel, Apr 05 2019

Select[Range[400], OddQ[DivisorSigma[1, DivisorSigma[1, #]]] &] (* Alonso del Arte, Dec 29 2013 *)

PARI
```
is(n)=bittest(sigma(sigma(n)),0)
```
PARI
```
for(n=1,999,is(n)&&print1(n","))
```

A249446 Numbers n such that 2(n^2-1) - 1 and 2(n^2-1) + 1 are primes.

Original entry on oeis.org

2, 4, 10, 11, 34, 41, 46, 49, 56, 59, 76, 85, 95, 125, 160, 181, 185, 196, 200, 206, 220, 245, 266, 280, 295, 301, 304, 346, 365, 379, 386, 391, 440, 470, 505, 556, 571, 595, 659, 679, 689, 731, 784, 815, 820, 854, 869, 896, 944, 959, 994, 1001, 1004, 1015, 1025, 1154, 1250, 1345, 1376

Offset: 1

Juri-Stepan Gerasimov, Oct 29 2014

nonn

Subsequence of A066049. - Michel Marcus, Oct 29 2014

n such that 2*n^2 - 2 is in A014574. - Robert Israel, Nov 18 2014

			2 is in this sequence because 2*(2^2-1) - 1 = 5 and 2*(2^2-1) + 1 = 7 are both prime.

Colin Barker, Table of n, a(n) for n = 1..1600

Cf. A001097, A014574, A066049, A066436, A201712.

[ n: n in [1..1400] | IsPrime(2*(n^2-1)-1) and IsPrime(2*(n^2-1)+1) ];

select(n -> isprime(2*n^2-3) and isprime(2*n^2-1), [$1 .. 10000]); # Robert Israel, Nov 18 2014

Select[Range[0, 1500], PrimeQ[2 #^2 - 3] && PrimeQ[2 #^2 - 1] &] (* Vincenzo Librandi, Oct 29 2014 *)

isok(n) = isprime(2*(n^2-1) - 1) && isprime(2*(n^2-1) + 1); \\ Michel Marcus, Oct 31 2014

A164041 Primes of the form 2p^2 + 4p + 1, where p is also prime.

Original entry on oeis.org

17, 31, 71, 127, 647, 1151, 2887, 3527, 7687, 12799, 19207, 20807, 23327, 34847, 39199, 49927, 53791, 73727, 79999, 103967, 117127, 145799, 172871, 194687, 220447, 279751, 294911, 323207, 336199, 387199, 394271, 419527, 438047, 587527, 649799, 724807

Offset: 1

Vincenzo Librandi, Aug 08 2009

A subsequence of the primes of the form 2k^2+4k+1 = 2*(k+1)^2-1, A066436. - R. J. Mathar, Aug 10 2009

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

[a: p in PrimesUpTo(700)|IsPrime(a) where a is 2*p^2+4*p+1 ] // Vincenzo Librandi, Sep 01 2012

lst={}; Do[p=Prime@n; a=2*p^2+4*p+1; If[PrimeQ@a,AppendTo[lst,a]],{n,7!}]; lst (* Vladimir Joseph Stephan Orlovsky, Aug 12 2009 *)
Select[Table[2p^2+4p+1,{p,Prime[Range[200]]}],PrimeQ] (* Harvey P. Dale, Aug 25 2019 *)

lista(nn) = {forprime(p=2, nn, if(isprime(q=2*p^2+4*p+1), print1(q, ", ")));} \\ Altug Alkan, Mar 29 2018

a(n) = 2*(A164042(n))^2 + 4*A164042(n) + 1.

a(29) corrected by R. J. Mathar, Aug 11 2009

Edited by N. J. A. Sloane, Aug 11 2009

A245042 Primes of the form (k^2+4)/5.

Original entry on oeis.org

17, 73, 89, 193, 337, 521, 953, 1009, 1249, 1657, 2377, 2833, 3329, 3433, 4441, 4561, 5849, 6553, 7297, 8081, 8737, 9769, 11617, 12401, 12601, 13417, 15569, 16937, 17881, 18121, 20353, 21649, 27529, 28729, 29033, 30577, 33457, 35449, 36809, 46273, 49801

Offset: 1

Chai Wah Wu, Jul 10 2014

nonn

Also equal to primes p such that 5*p-4 is a perfect square.

Chai Wah Wu, Table of n, a(n) for n = 1..1000

Cf. A154418.

Cf. A154616, A002327, A066436.

Select[(Range[500]^2+4)/5,PrimeQ] (* Harvey P. Dale, Jul 13 2014 *)

import sympy
L = (k**2 + 4 for k in range(10**3))
[n//5 for n in L if n % 5 == 0 and sympy.ntheory.isprime(n//5)]

A245045 Primes of the form (k^2+2)/6.

Original entry on oeis.org

3, 11, 17, 43, 67, 113, 131, 193, 241, 353, 523, 641, 683, 1291, 1601, 1667, 1873, 2017, 2243, 2731, 3083, 3361, 3851, 4483, 4817, 4931, 5281, 5521, 7211, 8363, 8513, 8971, 9283, 9923, 10753, 11971, 13633, 16433, 17713, 18371, 18593, 19267, 21841, 22571

Offset: 1

Chai Wah Wu, Jul 10 2014

nonn

			When k=4, (k^2+2)/6 = 3 is prime, so 4 is a member of the sequence. since putting k = 0, 1, 2, or 3 does not give a prime, so 4 is the first term.

Chai Wah Wu, Table of n, a(n) for n = 1..1500

Cf. A154616, A002327, A066436. First 5 terms equal to A078116. First 4 terms equal to A127996.

import sympy
[(k**2+2)/6 for k in range(10**6) if sympy.ntheory.isprime((k**2+2)/6) & ((k**2+2)/6).is_integer()]

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

Original entry on oeis.org

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

cp's OEIS Frontend

A049226 Composite numbers n such that the sum of divisors of n, sigma(n), divided by the number of divisors, d(n) and sigma(n) minus n are both rational squares.

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A143832 Primes of the form 14 n^2-1.

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Mathematica

A160697 Record values in A160696.

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A182784 Primes of the form 2*n^4-1.

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Magma

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A234638 Numbers n for which sigma(sigma(n)) is odd.

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Maple

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PARI

A249446 Numbers n such that 2*(n^2-1) - 1 and 2*(n^2-1) + 1 are primes.

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Magma

Maple

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A164041 Primes of the form 2*p^2 + 4*p + 1, where p is also prime.

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Magma

Mathematica

PARI

Formula

Extensions

A245042 Primes of the form (k^2+4)/5.

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Author

Keywords

A249446 Numbers n such that 2(n^2-1) - 1 and 2(n^2-1) + 1 are primes.

A164041 Primes of the form 2p^2 + 4p + 1, where p is also prime.