A037020 -id:A037020 - cp's OEIS frontend

A377766 Even numbers whose sum of proper (or aliquot) divisors is a prime.

4, 8, 32, 50, 98, 128, 242, 324, 338, 392, 722, 784, 800, 1058, 1250, 1444, 2304, 2312, 2450, 2704, 2738, 3600, 3872, 5408, 5476, 5618, 6272, 6728, 7442, 7688, 8192, 9248, 11552, 12482, 12800, 14400, 14884, 15488, 15842, 16562, 16900, 16928, 17672, 18050, 19208, 21632, 21904, 22500, 23762, 25088

Offset: 1

Ophir Spector, Nov 06 2024

Even terms of A037020.

Numbers from A088827 (2n^2 or 4n^2) are the only aliquot sum transition from even to odd.

			The aliquot divisors of 32 are 1, 2, 4, 8 and 16, whose sum is 31, a prime, so 32 is a term.

Michel Marcus, Table of n, a(n) for n = 1..3000

Intersection of A005843 and A037020.

Cf. A088827.

Select[2Range[13000],PrimeQ[DivisorSigma[1,#]-#] &] (* Stefano Spezia, Nov 08 2024 *)

is_a377766(n) = !(n%2) && isprime(sigma(n)-n) \\ Hugo Pfoertner, Nov 07 2024

A177891 Numbers n such that sum of proper (or aliquot) divisors of n is a semiprime.

Original entry on oeis.org

6, 9, 14, 15, 16, 18, 20, 22, 25, 33, 36, 38, 45, 46, 51, 52, 62, 68, 70, 72, 75, 80, 86, 87, 91, 93, 95, 99, 104, 105, 110, 116, 117, 118, 119, 130, 136, 141, 142, 143, 144, 145, 148, 154, 158, 159, 160, 162, 165, 166, 169, 183, 195, 196, 200

Offset: 1

Jonathan Vos Post, Dec 14 2010

This is to A037020 as semiprimes A001358 are to primes A000040. The first four values are themselves semiprime.

Contains k^2 if k is in A005383. - Robert Israel, Feb 16 2020

			a(2) = 9 because the aliquot divisors of 9 are 1 and 3, whose sum is 4 = 2*2, semiprime.
a(5) = 16 because the aliquot divisors of 16 are 1, 2, 4, and 8, whose sum is 15 = 3*5, semiprime.

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

Cf. A001065, A001358, A005383, A037020.

filter:= proc(n) uses numtheory;
  bigomega(sigma(n)-n) = 2
end proc:
select(filter, [$1..1000]); # Robert Israel, Feb 16 2020

semiPrimeQ[x_] := Plus @@ Last /@ FactorInteger@ x == 2; fQ[n_] := semiPrimeQ[ DivisorSigma[1, n] - n]; Select[ Range@ 200, fQ]

isok(n) = bigomega(sigma(n)-n) == 2; \\ Michel Marcus, Apr 05 2015

A001065(a(n)) is in A001358.

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

Original entry on oeis.org

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

A277794 Numbers k such that the sum of proper divisors of k is a prime, and the sum of the numbers less than k that do not divide k is also a prime.

Original entry on oeis.org

4, 21, 85, 129, 201, 237, 324, 325, 517, 549, 669, 837, 865, 1081, 1137, 1161, 1165, 1309, 1389, 1677, 1765, 2169, 2233, 2304, 2305, 2469, 2709, 2737, 2761, 3265, 3297, 3745, 3961, 4161, 4285, 4693, 4705, 4741, 4989, 5061, 5221, 5349, 5817, 5949, 6249, 6457, 6517, 6685, 6789, 6813, 6853, 6921

Offset: 1

Ilya Gutkovskiy, Oct 31 2016

Intersection of A037020 and A200981.
Numbers k such that A000005(A001065(k)) = A000005(A024816(k)) = 2 or A000005(A000203(k) - k) = A000005(A000217(k) - A000203(k)) = 2.
All terms are composite (A002808).

			21 is in the sequence because 21 has three proper divisors {1, 3, 7}, and therefore seventeen non-divisors {2, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, so the sum of proper divisors is 1 + 3 + 7 = 11 (which is prime) and the sum of non-divisors is 2 + 4 + 5 + 6 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 = 199 (which is also prime).
22 is not in the sequence because its three proper divisors {1, 2, 11} add up to 14, which is composite.

Robert Israel, Table of n, a(n) for n = 1..1000
Index entries for sequences related to sums of divisors

Cf. A000005, A000203, A000217, A002808, A037020, A200981.

f:= proc(n) local t; t:= numtheory:-sigma(n) - n; isprime(t) and isprime(n*(n-1)/2 - t) end proc:
select(f, [$1..10^4]); # Robert Israel, Nov 10 2016

Select[Range[7000], DivisorSigma[0, #1 ((#1 + 1)/2) - DivisorSigma[1, #1]] == 2 && DivisorSigma[0, DivisorSigma[1, #1] - #1] == 2 & ]

A290841 a(n) is the least number k such that the sum of the n-th powers of the proper divisors of k is a prime number.

Original entry on oeis.org

4, 4, 8, 4, 115, 33, 119, 4, 8, 18, 35, 15, 21, 177, 565, 4, 21, 501, 155, 275, 175, 72, 63, 21, 161, 207, 50, 100, 415, 393, 493, 453, 1250, 33, 75, 15, 85, 777, 655, 351, 649, 833, 327, 219, 1727, 123, 57, 15, 21, 357, 183, 1113, 50, 87, 57, 135, 831, 291, 341, 196, 175, 249, 2107, 783, 57, 927, 800, 39, 209

Offset: 1

Altug Alkan, Aug 12 2017

nonn

Corresponding primes are 3, 5, 73, 17, 6439469, 1772291, 411162217, 257, ...

a(n) = 4 if and only if 2^n + 1 is a Fermat prime (A019434).

			a(5) = 115 because 1^5 + 5^5 + 23^5 = 6439469 is prime and 115 is the smallest number with this property.

Iain Fox, Table of n, a(n) for n = 1..1000

Cf. A001065, A037020, A252040.

Table[SelectFirst[Range[10^4], PrimeQ[DivisorSigma[n, #] - #^n] &], {n, 69}] (* Michael De Vlieger, Aug 14 2017 *)

a(n) = {my(k=1); while(!isprime(sigma(k,n)-k^n), k++); k;}

A306490 Numbers k such that sigma(k) - k - 2 is prime.

Original entry on oeis.org

8, 9, 15, 16, 18, 27, 32, 33, 35, 36, 45, 50, 51, 64, 65, 75, 77, 87, 91, 95, 98, 119, 123, 125, 135, 143, 144, 147, 153, 161, 162, 175, 177, 185, 195, 200, 207, 209, 213, 215, 217, 221, 231, 247, 259, 261, 273, 285, 287, 297, 303, 315, 321

Offset: 1

Jan Koornstra, Feb 19 2019

Maple and Mathematica programs adapted from A085842.

			The divisors of 8 are {1, 2, 4, 8}. sigma(8) - 8 - 2 = 5, which is prime.

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

Cf. A085842, A037020, A000203.

Filtered([2..330],k->IsPrime(Sigma(k)-k-2)); # Muniru A Asiru, Feb 24 2019

with(numtheory): b := []: for n from 3 to 2000 do t1 := divisors(n); t2 := convert(t1, list); t3 := add(t2[i], i=1..nops(t2)); if isprime(t3-2-n) then b := [op(b), n]; fi; od: b;

f[n_]:=Plus@@Divisors[n]-n-2; lst={}; Do[a=f[n]; If[PrimeQ[a], AppendTo[lst, n]], {n, 7!}]; lst
Select[Range[2, 500], PrimeQ[DivisorSigma[1, #] - # - 2] &] (* Vaclav Kotesovec, Feb 23 2019 *)

isok(n) = isprime(sigma(n) - n - 2); \\ Michel Marcus, Feb 23 2019

A306492 Numbers k such that sigma(k) - 3k is prime.

Original entry on oeis.org

3600, 17424, 22500, 32400, 72900, 291600, 345744, 360000, 476100, 518400, 562500, 656100, 685584, 756900, 1040400, 1382976, 1411344, 1742400, 1904400, 1988100, 2073600, 2250000, 2340900, 2624400, 3027600, 3111696, 4161600, 4284900, 5760000, 6051600, 6170256, 6200100, 6969600

Offset: 1

Jan Koornstra, Feb 19 2019

nonn

			The divisors of 3600 are {1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 30, 36, 40, 45, 48, 50, 60, 72, 75, 80, 90, 100, 120, 144, 150, 180, 200, 225, 240, 300, 360, 400, 450, 600, 720, 900, 1200, 1800, 3600}. sigma(3600) - 3 * 3600 = 12493 - 10800 = 1693, which is prime.

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

Cf. A000203, A037020, A064271.

with(numtheory): b := []: for n from 3 to 1000000 do t1 := divisors(n); t2 := convert(t1, list); t3 := add(t2[i], i=1..nops(t2)); if isprime(t3-3*n) then b := [op(b), n]; fi; od: b;

f[n_]:=Plus@@Divisors[n]-3*n; lst={}; Do[a=f[n]; If[PrimeQ[a], AppendTo[lst, n]], {n, 9!}]; lst
Select[Range[1000000], DivisorSigma[1,#] > 3*# && PrimeQ[DivisorSigma[1,#] - 3*#] &] (* Vaclav Kotesovec, Feb 23 2019 *)

isok(n) = isprime(sigma(n) - 3*n); \\ Michel Marcus, Feb 19 2019

More terms from Michel Marcus, Feb 19 2019

A358199 a(n) is the least integer whose sum of the i-th powers of the proper divisors is a prime for 1 <= i <= n, or -1 if no such number exists.

Original entry on oeis.org

4, 4, 981, 8829, 8829, 122029105, 2282761881

Offset: 1

Jean-Marc Rebert, Nov 02 2022

			4 is a term since the strict divisors of 4 are {1, 2}, 1^1 + 2^1 = 3 and 1^2 + 2^2 = 5 are prime and no number < 4 has this property.

Cf. A001065 (s1), A180852, A357324, A057709, A000203, A001157, A001158.

Subsequence of A037020.

card(n)=my(c=1,s=0);s=sigma(n)-n;while(isprime(s),c++;s=sigma(n,c)-n^c);c--
a(n)=my(x=0);for(k=1,+oo,x=card(k);if(x>=n,return(k)))

from itertools import count
from math import prod
from sympy import isprime, factorint
def A358199(n):
    for m in count(2):
        f = factorint(m).items()
        if all(map(isprime,(prod((p**((e+1)*i)-1)//(p**i-1) for p,e in f) - m**i for i in range(1,n+1)))):
            return m # Chai Wah Wu, Nov 15 2022

A365351 Exponents e such that the aliquot sequence starting with 2^e ends with a prime number at index 2.

Original entry on oeis.org

6, 11, 18, 27, 41, 74, 157, 197, 294, 549, 581

Offset: 1

Jean Luc Garambois, Sep 02 2023

That is, exponents e such that s(s(2^e)) is prime, where s(n) = sigma(n)-n (A001065).
Note that exponents e such that aliquot sequences starting with 2^e end with a prime number at index 1 (exponents e such that s(2^e) is prime) are called "Mersenne exponents" (see A000043).
From Amiram Eldar, Sep 02 2023: (Start)
Numbers k such that 2^k - 1 is a term of A037020.
1206 < a(12) <= 2351 (2351 is a term). (End)

Jean-Luc Garambois, Aliquot sequences starting on integer powers n^i.
Mersenne forum, Results presentation page.

Cf. A000043 (Mersenne exponents), A001065, A037020.

Select[Range[100], PrimeQ[DivisorSigma[1, 2^# - 1] - 2^# + 1] &] (* Amiram Eldar, Sep 02 2023 *)

f(n) = sigma(n) - n; \\ A001065
isok(k) = ispseudoprime(f(f(2^k))); \\ Michel Marcus, Sep 02 2023

def s(n):
    sn = sigma(n) - n
    return sn
e = 1
exponents_list = []
while e<=200:
    m = 2^e
    index = 0
    if is_prime(s(s(m))):
        exponents_list.append(e)
    e+=1
print (exponents_list)

cp's OEIS Frontend

A377766 Even numbers whose sum of proper (or aliquot) divisors is a prime.

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A177891 Numbers n such that sum of proper (or aliquot) divisors of n is a semiprime.

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

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A277794 Numbers k such that the sum of proper divisors of k is a prime, and the sum of the numbers less than k that do not divide k is also a prime.

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A290841 a(n) is the least number k such that the sum of the n-th powers of the proper divisors of k is a prime number.

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Mathematica

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A306490 Numbers k such that sigma(k) - k - 2 is prime.

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GAP

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A306492 Numbers k such that sigma(k) - 3k is prime.

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