A340144 Numerators of the sequence whose Dirichlet convolution with itself yields sequence A247074(x) = phi(x)/(Product_{primes p dividing x} gcd(p-1, x-1)).

1, 1, 1, 7, 1, 3, 1, 25, 11, 7, 1, 19, 1, 11, 3, 363, 1, 31, 1, 43, 5, 19, 1, 63, 19, 23, 61, 3, 1, 19, 1, 1335, 9, 31, 11, 189, 1, 35, 11, 139, 1, 29, 1, 115, 7, 43, 1, 867, 27, 127, 15, 11, 1, 163, 19, 279, 17, 55, 1, 93, 1, 59, 51, 9923, 5, -15, 1, 187, 21, 3, 1, 615, 1, 71, 55, 19, 29, 59, 1, 1875, 1363, 79, 1, 203

Offset: 1

Author

Antti Karttunen, Dec 29 2020

Keywords

• sign
• frac

Examples

			For n = 561 = 3*11*17, its divisors d are: 1, 3, 11, 17, 33, 51, 187, 561.
For this sequence, the corresponding terms a(d) are: 1, 1, 1, 1, 9, 15, 79, -99.
For A046644, the corresponding terms are:            1, 2, 2, 2, 4,  4,  4,   8.
Convolving these ratios as Sum_{d|561} r(d)*r(n/d) = 2*((1/1)*(-99/8) + (1/2)*(79/4) + (1/2)*(15/4) + (1/2)*(9/4)) yields 1 as expected, because 561 is Carmichael number (A002997) and A247074 obtains value 1 on all of them.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..8191
• Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Crossrefs

Cf. A046644 (denominators).

Cf. A247074.

Cf. also A340141, A340145, A340146.

Programs

PARI

up_to = 65537;
A247074(n) = { my(f=factor(n)); eulerphi(f)/prod(i=1, #f~, gcd(f[i, 1]-1, n-1)); }; \\ From A247074
DirSqrt(v) = {my(n=#v, u=vector(n)); u[1]=1; for(n=2, n, u[n]=(v[n]/v[1] - sumdiv(n, d, if(d>1&&dA247074(n)));
A340144(n) = numerator(v340144rat[n]);