A353012 Numbers N such that gcd(N - d, N*d) >= d^2, where d = A000005(N) is the number of divisors of N.

1, 2, 136, 156, 328, 444, 584, 600, 712, 732, 776, 876, 904, 1096, 1164, 1176, 1308, 1544, 1864, 1884, 1928, 2056, 2172, 2248, 2316, 2504, 2601, 2696, 2748, 2824, 2892, 2904, 3208, 3240, 3249, 3272, 3324, 3464, 3592, 3656, 3756, 4044, 4056, 4168, 4188, 4476, 4552, 4616

Offset

1,2

Comments

As d^2 | N-d we have N = k*d^2 + d for some k >= 0 and d > 1. So gcd(k*d^2 + d - d, (N*d^2 + d)*d) = gcd(k*d^2, k*d^3 + d^2) = gcd(k*d^2, d^2) = d^2. So for any N such that d^2 | gcd(N - d, N*d) we have gcd(N - d, N*d) = d^2. - David A. Corneth, Apr 20 2022

Since gcd(N - d, N*d) is never larger than d^2 (if N = n*g, d = f*g with gcd(n,f) = 1, then gcd(N - d, N*d) = g*gcd(n-f,n*f*g) = g*gcd(n-f, f*f*g) <= g*g, since by assumption, no factor of f divides n), so one can also replace "=" by ">=" in the definition.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Formula

For all m in A033950, the sequence contains all numbers m*p^k with k = m/d(m) - 1, and p^k == 1 (mod m), in particular 8*A007519 and 12*A068228 (k = 1, m = 8 and 12), 9*A129805^2, 18*A129805^2 and 24*A215848^2 (k = 2, m = 9, 18 and 24, A^2 = {x^2, x in A}), etc.

Example

N = 1 is in the sequence because d(N) = 1, gcd(1 - 1, 1*1) = 1 = d^2.
N = 2 is in the sequence because d(N) = 2, gcd(2 - 2, 2*2) = 4 = d^2.
N = 136 = 8*17 is in the sequence because d(N) = 4*2 = 8, gcd(8*17 - 8, 8*17*8) = gcd(8*16, 8*8*17) = 8*8 = d^2. Similarly for N = 8*p with any prime p = 8*k + 1.
N = 156 = 2^2*3*13 is in the sequence because d(n) = 3*2*2 = 12, gcd(12*13 - 12, 12*13*12) = gcd(12*12, 12*12*13) = 12*12 = d^2. Similarly for any N = 12*p with prime p = 12*k + 1.
More generally, when N = m*p^k with p^k == 1 (mod m) and m = (k+1)*d(m), then d(N) = d(m)*(k+1) = m and gcd(n - d, n*d) = gcd(m*p^k - m, m*p^k*m) = m*gcd(p^k - 1, p^k*m) = m^2. This holds for m = 8 and 12 with k = 1, for m = 9, 18 and 24 with k = 2, etc: see sequence A033950 for the m-values.

Mathematica

Select[Range[4650], GCD[#1 - #2, #1 #2] == #2^2 & @@ {#, DivisorSigma[0, #]} &] (* Michael De Vlieger, Apr 21 2022 *)

Prog

(PARI) select( {is(n, d=numdiv(n))=gcd(n-d, d^2)==d^2}, [1..10^4])

Crossrefs

Cf. A000005 (number of divisors), A352483 (numerator of (n-d)/(n*d)), A352482 (denominator), A049820 (n - d), A146566 (n*d is divisible by n-d), A033950 (refactorable or tau numbers: d(n) | n, supersequence of this).

Sequence in context: A318965 A097641 A065963 * A226118 A216030 A216083

Adjacent sequences: A353009 A353010 A353011 * A353013 A353014 A353015

Keyword

nonn

Author

M. F. Hasler, Apr 15 2022

Status

approved