A294355 E.g.f.: exp( Sum_{n>=1} phi(n) * sigma(n^n) * x^n/n ).

1, 1, 8, 182, 6962, 419242, 36634192, 4488812596, 722684103164, 148732979255324, 38081859158891744, 11886289198519022344, 4436616366171778495192, 1951865016371020657730488, 999444614538375532106921408, 589291073127565867170680837168, 396337742242710677353781867268368, 301596359950397359016053774993717264, 257797318556710190428189878485023244416

Offset

0,3

Comments

Note that exp( Sum_{n>=1} phi(K) * sigma(K^n) * x^n/n ) is an integer series for any fixed positive integer K ; though similar, the generating function for this sequence does not yield an integer series.

Links

Paul D. Hanna, Table of n, a(n) for n = 0..200

Example

E.g.f.: A(x) = 1 + x + 8*x^2/2! + 182*x^3/3! + 6962*x^4/4! + 419242*x^5/5! + 36634192*x^6/6! + 4488812596*x^7/7! + 722684103164*x^8/8! + 148732979255324*x^9/9! + 38081859158891744*x^10/10! +...
such that
log(A(x)) = x + phi(2)*sigma(2^2)*x^2/2 + phi(3)*sigma(3^3)*x^3/3 + phi(4)*sigma(4^4)*x^4/4 + phi(5)*sigma(5^5)*x^5/5 + phi(6)*sigma(6^6)*x^6/6 + phi(7)*sigma(7^7)*x^7/7 +...+ phi(n)*sigma(n^n)*x^n/n +...
Explicitly,
log(A(x)) = x + 7*x^2/2 + 80*x^3/3 + 1022*x^4/4 + 15624*x^5/5 + 277622*x^6/6 + 5764800*x^7/7 + 134217724*x^8/8 + 3486784398*x^9/9 + 99951169828*x^10/10 +...

Prog

(PARI) {a(n) = my(A); A = exp(sum(m=1, n+1, eulerphi(m)*sigma(m^m)*x^m/m) +x*O(x^n)); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A000010 (phi), A000203 (sigma).

Sequence in context: A332118 A261825 A203359 * A180607 A024286 A231795

Adjacent sequences: A294352 A294353 A294354 * A294356 A294357 A294358

Keyword

nonn

Author

Paul D. Hanna, Oct 29 2017

Status

approved