A088005 -id:A088005 - cp's OEIS frontend

A125236 Numbers n for which the absolute value of the abundance of n^2 is a prime number.

3, 5, 6, 7, 9, 10, 11, 14, 15, 17, 18, 22, 24, 26, 28, 29, 30, 31, 33, 34, 35, 44, 47, 50, 56, 58, 61, 65, 66, 67, 68, 69, 71, 72, 74, 75, 80, 85, 91, 94, 96, 97, 100, 101, 102, 104, 116, 118, 120, 121, 124, 125, 127, 131, 134, 136, 139, 144, 148, 149, 152, 164, 166, 170

Offset: 1

Jason G. Wurtzel, Nov 25 2006

nonn

			n=5: n^2=25, sigma(25)=25+5+1=31, 2*25=50, abundance=-19.

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

Cf. A088005.

Select[Range[170], PrimeQ[DivisorSigma[1, #^2] - 2*#^2] &] (* Amiram Eldar, Sep 23 2019 *)

for(n=1,170,if(isprime(abs(sigma(n^2)-2*n^2)),print1(n,","))) \\ Klaus Brockhaus, Nov 25 2006

More terms from Klaus Brockhaus, Nov 25 2006

A125237 Numbers n for which the absolute value of the abundance of both n and n^2 is a prime number.

Original entry on oeis.org

3, 9, 10, 18, 50, 100, 104, 121, 136, 289, 464, 576, 650, 900, 5041, 6962, 7225, 10201, 14400, 55225, 65025, 87025, 102152, 147456, 171698, 174050, 179776, 182329, 189225, 201601, 222784, 291848, 312481, 380689, 410881, 469225, 481636, 488601

Offset: 1

Jason G. Wurtzel, Nov 25 2006

nonn

			n=3: The abundance of 3 is -2, the negative of a prime. n^2=9, the abundance of 9 is -5, the negative of a prime as well.

Amiram Eldar, Table of n, a(n) for n = 1..500

Cf. A088005, A125236.

pQ[n_] := PrimeQ[DivisorSigma[1, n] - 2n]; Select[Range[10^4], pQ[#] && pQ[#^2] &] (* Amiram Eldar, Sep 24 2019 *)

{for(n=1, 500000, if(isprime(abs(sigma(n)-2*n)) && isprime(abs(sigma(n^2)-2*n^2)), print1(n, ",")))} \\ Klaus Brockhaus, Nov 25 2006

More terms from Klaus Brockhaus, Nov 25 2006

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

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

A125249 Numbers n for which the absolute values of the abundances of n, n^2 and n^3 are all prime numbers.

Original entry on oeis.org

289, 201601, 222784, 638401, 868562, 910116, 4694048, 4950625, 8994001, 9054081, 19855936, 30085225, 32385152, 47623801, 55100929, 72182016, 78952178, 85099058, 86303522, 91910569, 104040000, 105678400, 111175936, 112530002, 128504896, 133702969, 193043236, 204404209, 216001809, 237961476

Offset: 1

Jason G. Wurtzel, Nov 26 2006

nonn

			n=289; The abundance of 289 is -271, the abundance of 83521 (289^2) is -78301, the abundance of 24137569 (289^3) is -22628971 - all of which are negatives of primes.

Cf. A088005, A125236, A125237.

{for(n=1,1000000,if(isprime(abs(sigma(n)-2*n))&&isprime(abs(sigma(n^2)-2*n^2))&&isprime(abs(sigma(n^3)-2*n^3)),print1(n,",")))}

More terms from Stefan Steinerberger, May 29 2007
a(21)-a(24) from Donovan Johnson, Feb 19 2009
a(25)-a(30) from Jason G. Wurtzel, Sep 24 2014

cp's OEIS Frontend

A125236 Numbers n for which the absolute value of the abundance of n^2 is a prime number.

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A125237 Numbers n for which the absolute value of the abundance of both n and n^2 is a prime number.

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

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

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A125249 Numbers n for which the absolute values of the abundances of n, n^2 and n^3 are all prime numbers.

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PARI

Extensions