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Any n which is of the form n = A005940(1+(2*(k^2))) for some k >= 1 [that is, a number that A156552 maps to a twice a square, thus n is odd by necessity], certainly cannot satisfy a(n) = 0, because no twice a square x can satisfy 0 = A106315(x) = x*d(x) mod sigma(x), as then x*d(x) will be even, and sigma(x) will be odd. E.g., 21 = A005940(1+(2*9)) certainly cannot be a solution to a(n) = 0. Numbers belonging that specific excluded set all satisfy A067029(n) = 1. But note that some of the solutions also satisfy it, e.g. 4199 = 13*17*19. - Antti Karttunen, Feb 21 2019
Any n which is of the form n = A005940(1+(2*(k^2))) for some k >= 1 [that is, a number that A156552 maps to a twice a square, thus n is odd by necessity], certainly cannot satisfy a(n) = 0, because no twice a square x can satisfy 0 = A106315(x) = 0, x*d(x) mod sigma(x), as then the expression x*d(x) at the left side of A106315 will be even, and sigma(x) at the right hand side will be odd. E.g., 21 = A005940(1+(2*9)) certainly cannot be a solution for to a(n) = 0 for that reason. Numbers belonging that specific excluded set all satisfy A067029(n) = 1. But note that some of the solutions also satisfy it, e.g. 4199 = 13*17*19 . - Antti Karttunen, Feb 21 2019
Any n which is of the form n = A005940(1+(2*(k^2))) for some k >= 1 [that is, a number that A156552 maps to a twice a square, thus n is odd by necessity], certainly cannot satisfy a(n) = 0, because no twice a square x can satisfy A106315(x) = 0, as then the expression x*d(x) at the left side of A106315 will be even, and sigma(x) at the right hand side will be odd, (which as an obvious fact must be well known). E.g., 21 = A005940(1+(2*9)) cannot be a solution for a(n) = 0 for that reason. Numbers belonging that specific excluded set all satisfy A067029(n) = 1. But note that some of the solutions also satisfy it, e.g. 4199 = 13*17*19 - Antti Karttunen, Feb 21 2019
Any n which is of the form n = A005940(1+(2*(k^2))) for some k >= 1 [that is, a number that A156552 maps to a twice a square, thus n is odd by necessity], certainly cannot satisfy a(n) = 0, because then the expression x*d(x) at the left side of A106315 will be even, and sigma(x) at the right hand side will be odd, (which as an obvious fact must be well known). E.g., 21 = A005940(1+(2*9)) cannot be a solution for a(n) = 0 for that reason. Numbers belonging that specific excluded set all satisfy A067029(n) = 1. But note that some of the solutions also satisfy it, e.g. 4199 = 13*17*19 - Antti Karttunen, Feb 21 2019
Any n which is of the form n = A005940(1+(2*(k^2))) for some k >= 1 [that is, a number that A156552 maps to a twice a square, thus n is odd by necessity], certainly cannot satisfy a(n) = 0, because then the expression x*d(x) at the left side of A106315 will be even, and sigma(x) at the right hand side will be odd. E.g., 21 = A005940(1+(2*9)) cannot be a solution for a(n) = 0 for that reason. Numbers belonging that specific excluded set all satisfy A067029(n) = 1. But note that also some of the solutions also satisfy it, e.g. 4199 = 13*17*19 - Antti Karttunen, Feb 21 2019
Any n which is of the form n = A005940(1+(2*(k^2))) for some k >= 1 [that is, a number that A156552 maps to a twice a square, thus n is odd by necessity], certainly cannot satisfy a(n) = 0, because then the expression x*d(x) at the left side of A106315 will be even, and sigma(x) at the right hand side will be odd. E.g., 21 = A005940(1+(2*9)) cannot be a solution for a(n) = 0 for that reason. Numbers belonging that specific excluded set all have satisfy A067029(n) = 1. But note that also some of the solutions also satisfy it, e.g. 4199 = 13*17*19 - Antti Karttunen, Feb 21 2019
Any n which is of the form n = A005940(1+(2*(k^2))) for some k >= 1 [that is, a number that A156552 maps to a twice a square, thus n is odd by necessity], certainly cannot satisfy a(n) = 0, because then the expression x*d(x) at the left side of A106315 will be even, and sigma(x) at the right hand side will be odd. E.g., 21 = A005940(1+(2*9)) cannot be a solution for a(n) = 0 for that reason. Numbers belonging that specific excluded set all have A067029(n) = 1. - Antti Karttunen, Feb 21 2019