A087998 -id:A087998 - cp's OEIS frontend

A088005 Numbers whose abundance is (+-1) times a prime.

3, 9, 10, 18, 20, 25, 36, 49, 50, 81, 100, 104, 121, 136, 196, 225, 289, 324, 338, 392, 464, 484, 576, 650, 676, 784, 800, 841, 900, 961, 968, 1058, 1089, 1156, 1225, 1250, 1352, 1682, 1936, 1952, 2209, 2312, 2450, 2500, 2888, 3136, 3364, 3721, 4225, 4356

Offset: 1

Labos Elemer, Oct 18 2003

nonn

			n=18: sigma[18]=18+9+6+3+2+1=39, 2n=36, abundance=3, a prime;
n=25: sigma[25]=25+5+1=31, 2n=50, abundance=-19.

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

Cf. A000203, A000040, A077374, A087998.

ab[x_] := DivisorSigma[1, x]-2*x Do[If[PrimeQ[s=ab[n]], Print[{n, s}]], {n, 1, 10000}]

isok(n) = isprime(abs(sigma(n)-2*n)); \\ Michel Marcus, Sep 21 2019

abs(sigma(n)-2n) is a prime.

A088006 Abundance values of numbers whose abundance is (+-1) times a prime.

Original entry on oeis.org

-2, -5, -2, 3, 2, -19, 19, -41, -7, -41, 17, 2, -109, -2, 7, -47, -271, 199, -127, 71, 2, -37, 499, 2, -71, 199, 353, -811, 1021, -929, 59, -457, -449, -163, -683, -157, 41, -751, 251, 2, -2161, -19, 401, 467, -61, 967, -631, -3659, -2777, 3391, -4421, 269, -2333, -1201, -4969, 4999, -1103, -2647, -1097, 269

Offset: 1

Labos Elemer, Oct 18 2003

sign

			For n = 4: A088005(4) = 18, sigma(18) = 18 + 9 + 6 + 3 + 2 + 1 = 39, 2*18 = 36, abundance = 39 - 36 = 3 = a(4).
For n = 6: A088005(6) = 25, sigma(25)_= 25 + 5 + 1 = 31, 2*25 = 50, abundance = 31 - 50 = -19 = a(6).

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

Cf. A000040, A000203, A033880, A077374, A087998, A088005.

ab[x_] := DivisorSigma[1, x]-2*x Do[If[PrimeQ[s=ab[n]], Print[s]], {n, 1, 10000}]
Select[Table[DivisorSigma[1,n]-2n,{n,7000}],PrimeQ[Abs[#]]&] (* Harvey P. Dale, Aug 21 2011 *)

list(lim) = {my(ab); for(k = 1, lim, ab = sigma(k) - 2*k; if(isprime(abs(ab)), print1(ab, ", ")));} \\ Amiram Eldar, Feb 16 2025

Solutions x to sigma(k) - 2k = x where abs(x) is a prime number.

a(n) = A033880(A088005(n)). - Amiram Eldar, Feb 16 2025

A217769 Least number k > n such that sigma(k) = 2*(k-n), or 0 if no such k exists.

Original entry on oeis.org

6, 3, 5, 7, 22, 11, 13, 27, 17, 19, 46, 23, 124, 58, 29, 31, 250, 57, 37, 55, 41, 43, 94, 47, 1264, 106, 53, 87, 118, 59, 61, 85, 134, 67, 142, 71, 73, 712, 158, 79, 166, 83, 405, 115, 89, 141, 406, 119, 97, 202, 101, 103, 214, 107, 109, 145, 113, 177, 418, 143

Offset: 0

Jayanta Basu, Mar 28 2013

nonn

a(0) = 6 corresponds to the smallest perfect number.
Is n = 144 the first number for which a(n) = 0? - T. D. Noe, Mar 28 2013
No, a(144) = 95501968. - Giovanni Resta, Mar 28 2013
We can instead compute k - sigma(k)/2 for increasing k, which is computationally much faster. In this case, we stop computing when all n have been found for a range of numbers. - T. D. Noe, Mar 28 2013
Also, the first number whose deficiency is 2n. This is the even bisection of A082730. Hence, the first number in the following sequences: A000396, A191363, A125246, A141548, A125247, A101223, A141549, A141550, A125248, A223608, A223607, A223606. - T. D. Noe, Mar 29 2013
10^12 < a(654) <= 618970019665683124609613824. - Donovan Johnson, Jan 04 2014

			a(4)=22, since 22 is the least number such that sigma(22)=36=2*(22-4).

Donovan Johnson, Table of n, a(n) for n = 0..653 (first 241 terms from T. D. Noe)
Nichole Davis, Dominic Klyve and Nicole Kraght, On the difference between an integer and the sum of its proper divisors, Involve, Vol. 6 (2013), No. 4, 493-504; DOI: 10.2140/involve.2013.6.493.

Cf. A000203, A000396.

Cf. A087998 (negative n).

Table[Min[Select[Range[2000], DivisorSigma[1, #] == 2*(# - i) &]], {i, 0, 60}]
nn = 144; t = Table[0, {nn}]; k = 0; While[k++; Times @@ t == 0, s = (2*k - DivisorSigma[1, k])/2; If[s >= 0 && s < nn && IntegerQ[s] && t[[s + 1]] == 0, t[[s + 1]] = k]]; t (* T. D. Noe, Mar 28 2013 *)

A124109 Numbers whose abundance (A033880) or deficiency (A033879) is a semiprime number.

Original entry on oeis.org

5, 7, 11, 12, 14, 15, 21, 23, 26, 27, 34, 35, 39, 40, 44, 47, 52, 55, 57, 58, 59, 63, 65, 68, 70, 72, 74, 75, 77, 80, 82, 83, 85, 88, 93, 98, 107, 110, 115, 116, 119, 122, 125, 129, 133, 143, 144, 152, 155, 160, 162, 164, 167, 169, 171, 178, 179, 183, 185, 187, 189

Offset: 1

Jonathan Vos Post, Nov 26 2006

If p is prime, then the only divisors of p are 1 and p, so sigma(p) = p + 1 and abundance(p) = abs(sigma(p) - 2*p) = abs((p+1) - 2*p) = abs(1-p) = p-1. Hence this sequence includes all values of the sequence of the primes which are one more than semiprimes. This is identical to A005385 Safe primes p: (p-1)/2 is also prime [then (p-1)/2 is called a Sophie Germain prime: see A005384] since as Zak Seidov commented, this is identical to primes p such that p-1 is a semiprime]. But the current sequence also contains composites, such as a(4) = 12, a(5) = 14, a(6) = 15 and a(7) = 21. If k = p*q is a semiprime (with p and q distinct primes) then the only divisors of k are 1, p, q and p*q, so sigma(k) = 1 + p + q + p*q and abs(abundance(k)) = abs(1 + p + q + p*q - p*q) = abs(1 + p + q) and these are in the sequence if 1 + p + q is semiprime. Note that numbers can be in the sequence which are neither prime nor semiprime, starting with a(4) = 12 and a(10) = 27.

			a(1) = 5 because abs(sigma(5) - 2*5) = abs(6-10) = abs(-4) = 4 = 2^2 is semiprime.
a(2) = 7 because abs(sigma(7) - 2*7) = abs(8-14) = abs(-6) = 6 = 2 * 3 is semiprime.
a(3) = 11 because abs(sigma(11) - 2*11) = abs(12-22) = abs(-10) = 10 = 2 * 5 is semiprime.
a(4) = 12 because abs(sigma(12) - 2*12) = abs(28-24) = abs(-4) = 4 = 2^2 is semiprime.
a(5) = 14 because abs(sigma(14) - 2*14) = abs(24-28) = abs(+4) = 4 = 2^2 is semiprime.
a(6) = 15 because abs(sigma(15) - 2*15) = abs(24-30) = abs(-6) = 6 = 2 * 3 is semiprime.
a(7) = 21 because abs(sigma(21) - 2*21) = abs(32-42) = abs(-10) = 10 = 2 * 5 is semiprime.
a(8) = 23 because abs(sigma(23) - 2*23) = abs(24-46) = abs(-22) = 22 = 2 * 11 is semiprime.

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

Cf. A000203, A001358, A000040, A005384, A005385, A077374, A087998, A088005, A088006, A125236, A125237.

semiPrimeQ[n_] := Plus @@ Last /@ FactorInteger@n == 2; Select[ Range@ 193, semiPrimeQ@Abs[DivisorSigma[1, # ] - 2# ] &] (* Robert G. Wilson v *)

Abs[sigma(a(n)) - 2*a(n)] is a semiprime, where sigma(k) = sum of divisors of k. {Abs[sigma(a(n)) - 2*a(n)]} is in A001358.

More terms from Robert G. Wilson v, Nov 29 2006

cp's OEIS Frontend

A088005 Numbers whose abundance is (+-1) times a prime.

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A088006 Abundance values of numbers whose abundance is (+-1) times a prime.

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A217769 Least number k > n such that sigma(k) = 2*(k-n), or 0 if no such k exists.

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A124109 Numbers whose abundance (A033880) or deficiency (A033879) is a semiprime number.

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