A067558 -id:A067558 - cp's OEIS frontend

A201880 Numbers n such that sigma_2(n) - n^2 is prime.

4, 18, 21, 33, 39, 72, 93, 99, 100, 159, 171, 177, 189, 207, 213, 231, 245, 249, 261, 275, 291, 297, 303, 333, 338, 357, 369, 381, 399, 400, 453, 471, 475, 477, 484, 495, 537, 539, 543, 561, 609, 633, 648, 657, 669, 681, 711, 717, 783, 801, 833, 861, 909, 927

Offset: 1

Michel Lagneau, Dec 06 2011

nonn

Numbers n such that sum of the squares of the proper (or aliquot) divisors of n is a prime number.

			a(3)=21 because the aliquot divisors of 21 are 1, 3, 7, the sum of whose squares is 1^2 + 3^2 + 7^2 = 59, prime.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001157, A037020, A067558.

A067558 := proc(n)
    numtheory[sigma][2](n)-n^2 ;
end proc:
isA201880 := proc(n)
    isprime(A067558(n)) ;
end proc:
for n from 1 to 1000 do
    if isA201880(n) then
        printf("%d,",n);
    end if;
end do: # R. J. Mathar, Dec 07 2011

Select[Range[400], PrimeQ[DivisorSigma[2, #]-#^2]&]

is(n)=isprime(sigma(n,2)-n^2) \\ Charles R Greathouse IV, Dec 06 2011

{n: A067558(n) in A000040} - R. J. Mathar, Dec 07 2011

A279814 Numbers n such that the average of the squares of the proper divisors of n is an integer.

Original entry on oeis.org

2, 3, 5, 7, 8, 9, 10, 11, 13, 14, 17, 19, 22, 23, 25, 26, 29, 31, 34, 35, 37, 38, 41, 43, 46, 47, 49, 50, 53, 55, 58, 59, 61, 62, 65, 67, 71, 73, 74, 75, 77, 79, 80, 81, 82, 83, 85, 86, 89, 91, 94, 95, 97, 101, 103, 106, 107, 109, 113, 115, 118, 119, 121, 122, 125, 127, 131, 133, 134, 137, 139, 140, 142, 143, 145, 146, 149

Offset: 1

Ilya Gutkovskiy, Dec 19 2016

Numbers n such that number of proper divisors of n (A032741) divides sum of squares of proper divisors of n (A067558).

All the prime numbers are in this sequence.

			8 is in the sequence because 8 has 3 proper divisors {1,2,4}, 1^2 + 2^2 + 4^2 = 21 and 3 divides 21.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sums of divisors

Cf. A020486, A023884, A032741, A067558.

Select[Range[150], Mod[DivisorSigma[2, #1] - #1^2, DivisorSigma[0, #1] - 1] == 0 &]
Select[Range[200],IntegerQ[Mean[Most[Divisors[#]]^2]]&] (* Harvey P. Dale, Jul 26 2019 *)

is(n)=my(d=divisors(n)); d=apply(k->k^2, d[1..#d-1]); n>1 && sum(i=1,#d,d[i])%#d==0 \\ Charles R Greathouse IV, Dec 19 2016

A291108 Expansion of Sum_{k>=2} k^2x^(2k)/(1 - x^k).

Original entry on oeis.org

0, 0, 0, 4, 0, 13, 0, 20, 9, 29, 0, 65, 0, 53, 34, 84, 0, 130, 0, 145, 58, 125, 0, 273, 25, 173, 90, 265, 0, 399, 0, 340, 130, 293, 74, 614, 0, 365, 178, 609, 0, 735, 0, 625, 340, 533, 0, 1105, 49, 754, 298, 865, 0, 1183, 146, 1113, 370, 845, 0, 1859, 0, 965, 580, 1364, 194, 1743, 0, 1465, 538, 1599, 0, 2550, 0, 1373, 884

Offset: 1

Ilya Gutkovskiy, Aug 17 2017

nonn

Sum of squares of divisors of n except 1 and n^2 (sum of squares of nontrivial divisors of n).

			a(6) = 13 because 6 has 4 divisors {1, 2, 3, 6} among which 2 are nontrivial {2, 3} and 2^2 + 3^2 = 13.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to sums of divisors

Cf. A000290, A001157, A001248, A001358, A005063, A008578, A027751, A048050, A053807, A067558, A070824.

nmax = 75; Rest[CoefficientList[Series[Sum[k^2 x^(2 k)/(1 - x^k), {k, 2, nmax}], {x, 0, nmax}], x]]
Join[{0}, Table[DivisorSigma[2, n] - n^2 - 1, {n, 2, 75}]]

A291108(n) = sumdiv(n,d,if((1==d)||(n==d),0,d^2)); \\ Antti Karttunen, Jan 22 2025

G.f.: Sum_{k>=2} k^2*x^(2*k)/(1 - x^k).
a(n) = A001157(n) -  A000290(n) - 1 for n > 1.
a(n) = A067558(n) - 1 for n > 1.
a(n) = A005063(n) if n is a semiprime (A001358).
a(n) = 0 if n is a prime or 1 (A008578).
a(n) = n if n is a square of prime (A001248).
a(p^k) = (p^(2*k) - p^2)/(p^2 - 1) for p is a prime and k > 0.

A301482 Composite numbers whose sum of aliquot parts divide the sum of the squares of their aliquot parts.

Original entry on oeis.org

8, 22, 27, 32, 77, 125, 128, 243, 343, 494, 512, 611, 660, 1073, 1281, 1331, 1425, 2033, 2048, 2187, 2197, 2332, 3125, 4172, 4565, 4913, 5293, 6031, 6859, 8192, 9983, 12167, 13969, 15818, 15947, 16807, 17485, 19683, 23489, 23840, 24389, 25241, 25389, 29791, 32768

Offset: 1

Paolo P. Lava, Mar 22 2018

Semiprimes in the sequence: 22, 77, 611, 1073, 2033, 5293, 6031, 9983, 13969, 15947, 23489, 25241, 40301, 49901, 50249, 51101, 56759, 65017, 71677, 85079, 97217, 98099, 99101, .... - Robert Israel, Mar 29 2018

2^k is a term for all odd k > 1. - Michael S. Branicky, Aug 22 2021

			Aliquot parts of 77 are 1, 7, 11. Then (1^2 + 7^2 + 11^2)/(1 + 7 + 11) = 171/19 = 9.

Amiram Eldar, Table of n, a(n) for n = 1..1009 (terms below 10^9, terms 1..100 from Paolo P. Lava)

Cf. A001065, A001157, A020487, A067558.

Contains A056824.

with(numtheory): P:=proc(n)
if not isprime(n) and frac((add(p^2,p=divisors(n))-n^2)/(sigma(n)-n))=0
then n; fi; end: seq(P(i),i=2..35*10^3);

aQ[n_] := CompositeQ[n] && Divisible[DivisorSigma[2, n] - n^2, DivisorSigma[1, n] - n]; Select[Range[33000], aQ] (* Amiram Eldar, Aug 17 2019 *)

isok(n) = (n!=1) && !isprime(n) && (((sigma(n,2) - n^2) % (sigma(n) - n)) == 0); \\ Michel Marcus, Mar 23 2018

from sympy import divisors
def ok(n):
    divs = divisors(n)[:-1]
    return len(divs) > 1 and sum(d**2 for d in divs)%sum(divs) == 0
print(list(filter(ok, range(4, 32769)))) # Michael S. Branicky, Aug 22 2021

A374539 The sum of the squares of the infinitary divisors of n.

Original entry on oeis.org

1, 5, 10, 17, 26, 50, 50, 85, 82, 130, 122, 170, 170, 250, 260, 257, 290, 410, 362, 442, 500, 610, 530, 850, 626, 850, 820, 850, 842, 1300, 962, 1285, 1220, 1450, 1300, 1394, 1370, 1810, 1700, 2210, 1682, 2500, 1850, 2074, 2132, 2650, 2210, 2570, 2402, 3130, 2900

Offset: 1

Amiram Eldar, Jul 11 2024

Also the sum of the infinitary divisors of n^2.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A049417, A050376, A077609.

Similar sequences: A001157, A034676, A050999, A067558, A076577, A351265, A351307, A351647.

f[p_, e_] := p^(2^Position[Reverse@IntegerDigits[e, 2], ?(# == 1 &)]); a[1] = 1; a[n] := Times @@ (Flatten@(f @@@ FactorInteger[n]) + 1); Array[a, 100]

a(n) = {my(f = factor(n), b); prod(i = 1, #f~, b = binary(2 * f[i, 2]); prod(k=1, #b, if(b[k], 1+f[i, 1]^(2^(#b-k)), 1)));}

from math import prod
from sympy import factorint
def A374539(n): return prod(p**(1<Chai Wah Wu, Jul 11 2024

a(n) = A049417(n^2).
a(n) <= A001157(n), with equality if and only if n is in A036537.
Multiplicative with a(p^e) = Product{k>=1, e_k=1} (p^(2^(k+1)) + 1), where e = Sum_{k} e_k * 2^k is the binary representation of e, i.e., e_k is bit k of e.
Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = Product_{P} (1 + 1/(P^2*(P+1))) = 1.14142906130350119631..., and P are numbers of the form p^(2^k) where p is prime and k >= 0 (A050376).

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A201880 Numbers n such that sigma_2(n) - n^2 is prime.

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A279814 Numbers n such that the average of the squares of the proper divisors of n is an integer.

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A291108 Expansion of Sum_{k>=2} k^2*x^(2*k)/(1 - x^k).

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A301482 Composite numbers whose sum of aliquot parts divide the sum of the squares of their aliquot parts.

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A374539 The sum of the squares of the infinitary divisors of n.

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A291108 Expansion of Sum_{k>=2} k^2x^(2k)/(1 - x^k).