A073040 -id:A073040 - cp's OEIS frontend

A048699 Nonprime numbers whose sum of aliquot divisors (A001065) is a perfect square.

1, 9, 12, 15, 24, 26, 56, 75, 76, 90, 95, 119, 122, 124, 140, 143, 147, 153, 176, 194, 215, 243, 287, 332, 363, 386, 407, 477, 495, 507, 511, 524, 527, 536, 551, 575, 688, 738, 791, 794, 815, 867, 871, 892, 924, 935, 963, 992, 1075, 1083, 1159, 1196, 1199, 1295, 1304

Offset: 1

Enoch Haga

The sum of aliquot divisors of prime numbers is 1.

If a^2 is an odd square for which a^2-1 = p + q with p,q primes, then p*q is a term. If m = 2^k-1 is a Mersenne prime then m*(2^k) (twice an even perfect number) is a term. If b = 2^j is a square and b-7 = 3s is a semiprime then 4s is a term. - Metin Sariyar, Apr 02 2020

			a(3)=15; aliquot divisors are 1,3,5; sum of aliquot divisors = 9 and 3^2=9.

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

Cf. A001065, A006532, A020477, A048698, A073040 (includes primes).

a := []; for n from 1 to 2000 do if sigma(n) <> n+1 and issqr(sigma(n)-n) then a := [op(a), n]; fi; od: a;

nn=1400;Select[Complement[Range[nn],Prime[Range[PrimePi[nn]]]],IntegerQ[ Sqrt[DivisorSigma[1,#]-#]]&] (* Harvey P. Dale, Apr 25 2011 *)

isok(k) = !ispseudoprime(k) && issquare(sigma(k) - k); \\ Michel Marcus, May 13 2025

A194948 Numbers k such that sum of aliquot divisors of k, sigma(k) - k, is a cube.

Original entry on oeis.org

1, 2, 3, 5, 7, 10, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49, 53, 56, 59, 61, 67, 69, 71, 73, 76, 79, 83, 89, 97, 101, 103, 107, 109, 113, 122, 127, 131, 133, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233

Offset: 1

Martin Renner, Oct 13 2011

nonn

For prime numbers, the sum of their aliquot divisors is exactly 1 = 1^3.

			a(6) = 10, since the sum of aliquot divisors 1 + 2 + 5 = 8 = 2^3.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..2500 from Harvey P. Dale)

Union of A000040 and A048698.

Cf. A020477, A073040.

for n do s:=numtheory[sigma](n)-n; if root(s,3)=trunc(root(s,3)) then print(n); fi; od:

Select[Range[250],IntegerQ[Power[DivisorSigma[1,#]-#, (3)^-1]]&] (* Harvey P. Dale, Nov 25 2011 *)

A176996 Numbers n such that sum of divisors, sigma(n), and sum of the proper divisors, sigma(n)-n, are both square.

Original entry on oeis.org

1, 3, 119, 527, 935, 3591, 3692, 6887, 12319, 47959, 65151, 97767, 99116, 202895, 237900, 373319, 438311, 699407, 734111, 851927, 957551, 1032156, 1064124, 1437599, 1443959, 2858687, 3509231, 3699311, 4984199, 7237415

Offset: 1

Claudio Meller, Dec 08 2010

nonn

The only prime in this sequence is 3. All prime numbers have the square 1 as the sum of their proper divisors. But since 3 is the only prime of the form n^2 - 1, it is the only prime that satisfies the first condition for inclusion in this sequence.

			119 has divisors 1, 7, 17, 119; it is in the list because 1+7+17+119 = 144 = 12^2 and 1+7+17 = 25 = 5^2.

Donovan Johnson, Table of n, a(n) for n = 1..300
Antonio Roldan Martinez, La suma de sus divisores es cuadrado perfecto

Cf. A006532, which considers all divisors; A048699, which for nonprime numbers considers all divisors except the number itself; A073040, which is the union of A048699 and the prime numbers (A000040).

Intersection[Select[Range[10^5], IntegerQ[Sqrt[-# + Plus@@Divisors[#]]] &], Select[Range[10^5], IntegerQ[Sqrt[Plus@@Divisors[#]]] &]] (* Alonso del Arte, Dec 08 2010 *)
t = {}; Do[If[And @@ IntegerQ /@ Sqrt[{x = DivisorSigma[1, n], x - n}], AppendTo[t, n]], {n, 10^6}]; t (* Jayanta Basu, Jul 27 2013 *)
sdQ[n_]:=Module[{d=DivisorSigma[1,n]},AllTrue[{Sqrt[d],Sqrt[d-n]}, IntegerQ]]; Select[Range[73*10^5],sdQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Apr 17 2018 *)

is_A176996 = lambda n: is_square(sigma(n)) and is_square(sigma(n)-n) # D. S. McNeil, Dec 09 2010

Intersection of A006532 and A073040.

A201879 Numbers n such that sigma_2(n) - n^2 is a square.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 30, 31, 37, 41, 43, 47, 53, 59, 61, 67, 70, 71, 73, 79, 83, 89, 97, 101, 102, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257

Offset: 1

Michel Lagneau, Dec 06 2011

nonn

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

All primes are in this sequence. Nonprimes in the sequence are 1, 30, 70, 102, 282, 286, 646, 730, 920, 1242, ... - Charles R Greathouse IV, Dec 06 2011

			a(12)=30 because the aliquot divisors of 30 are  1, 2, 3, 5, 6, 10, 15, the sum of whose squares is 1^2 + 2^2 + 3^2 + 5^2 + 6^2 + 10^2 + 15^2 = 400 = 20^2.

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

Cf. A000290, A001065, A001157, A073040.

A067558 := proc(n)
    numtheory[sigma][2](n)-n^2 ;
end proc:
isA201879 := proc(n)
    issqr(A067558(n)) ;
end proc:
for n from 1 to 300 do
    if isA201879(n) then
        printf("%d,",n);
    end if;
end do: # R. J. Mathar, Dec 07 2011

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

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

{n: A067558(n) in A000290}. - R. J. Mathar, Dec 07 2011

A279459 Numbers n such that sum of the proper divisors of n is the square of the sum of the digits of n.

Original entry on oeis.org

24, 153, 176, 794, 3071, 3431, 4607, 9671, 15599, 17711, 18167, 19511, 45671, 50927, 56471, 62807, 74639, 119279, 127559, 154199, 165791, 174719, 175871, 695399, 699359

Offset: 1

Ilya Gutkovskiy, Dec 12 2016

Subsequence of A073040.
Numbers n such that A001065(n) = A118881(n) or A000203(n) - n = (A007953(n))^2.
Every term in the sequence is composite (since the only proper divisor of a prime is 1). The sum of the proper divisors of a k-digit composite number n must exceed sqrt(n) >= sqrt(10^(k-1)), but the square of the sum of the digits of a k-digit number cannot exceed (9k)^2 = 81k^2. Since sqrt(10^(k-1)) > 81k^2 for all integers k > 8, every term in the sequence must be less than the smallest 9-digit number, 10^8. An exhaustive search through 10^8 shows that a(25)=699359 is the last term. - Jon E. Schoenfield, Dec 13 2016

			24 is in the sequence because 24 has 7 proper divisors {1,2,3,4,6,8,12}, 1 + 2 + 3 + 4 + 6 + 8 + 12 = 36 and (2 + 4)^2 = 36;
153 is in the sequence because 153 has 5 proper divisors {1,3,9,17,51}, 1 + 3 + 9 + 17 + 51 = 81 and (1 + 5 + 3)^2 = 81;
176 is in the sequence because 176 has 9 proper divisors {1,2,4,8,11,16,22,44,88}, 1 + 2 + 4 + 8 + 11 + 16 + 22 + 44 + 88 = 196 and (1 + 7 + 6)^2 = 196, etc.

Eric Weisstein's World of Mathematics, Digit Sum
Index entries for sequences related to sums of divisors

Cf. A073040, A001065, A118881, A145746.

Select[Range[1000000], DivisorSigma[1, #1] - #1 == Total[IntegerDigits[#1]]^2  &]

is(n) = sigma(n)-n==sumdigits(n)^2 \\ Felix Fröhlich, Dec 13 2016

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A048699 Nonprime numbers whose sum of aliquot divisors (A001065) is a perfect square.

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A194948 Numbers k such that sum of aliquot divisors of k, sigma(k) - k, is a cube.

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A176996 Numbers n such that sum of divisors, sigma(n), and sum of the proper divisors, sigma(n)-n, are both square.

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A201879 Numbers n such that sigma_2(n) - n^2 is a square.

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A279459 Numbers n such that sum of the proper divisors of n is the square of the sum of the digits of n.

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