A077102 Smallest number m such that GCD(a+b,a-b) = n, where a = sigma(m) and b = phi(m).

4, 1, 18, 21, 200, 14, 3364, 12, 722, 328, 9801, 42, 25281, 116, 1800, 15, 36992, 810, 4414201, 88, 196, 29161, 541696, 35, 2928200, 1413, 103968, 284, 98942809, 488, 1547536, 364, 19602, 17536, 814088, 370, 49042009, 55297, 1521, 440, 3150464641

Offset: 1

Author

Labos Elemer, Nov 12 2002

Keywords

• nonn

Examples

			For n = 10, a(10) = 328, sigma(328) = 630, phi(328) = 160, sigma(328) + phi(328) = 790, sigma(328) - phi(328) = 470, GCD(790,470) = 10.
For n = odd number, a(n) should be either a square or twice a square and so faster search for large values is possible, like e.g., for n = 97: a(97) = 435979^2 is the smallest solution.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..82

Crossrefs

Cf. A000010, A000203, A051612, A065387, A077099, A077100, A077101.

Programs

Mathematica

f[x_] := Apply[GCD, {DivisorSigma[1, x]+EulerPhi[x], DivisorSigma[1, x]-EulerPhi[x]}]; t=Table[0, {100}]; Do[s=f[n]; If[s<101&&t[[s]]==0, t[[s]]=n], {n, 1, 10^13}]; t

PARI

lista(len) = {my(v = vector(len), c = 0, k = 1, a, b, i); while(c < len, f = factor(k); a = sigma(f); b = eulerphi(f); i = gcd(a+b,a-b); if(i <= len && v[i] == 0, c++; v[i] = k); k++); v;} \\ Amiram Eldar, Nov 14 2024

Formula

a(n) = Min{x; A077099(x) = n}.