A357918 Odd numbers that can be written as phi(k) + d(k) for more than one k, where phi(k) = A000010(k) is Euler's totient function and d(k) = A000005(k) is the number of divisors of k.

2061, 4131, 36981, 78765, 14054589, 889978059, 110543990589

Offset: 1

Author

J. M. Bergot and Robert Israel, Oct 19 2022

Keywords

• nonn
• more

Comments

For phi(k) + d(k) to be odd, k must be a square.

Examples

			a(1) = 2061 = phi(57^2) + d(57^2) = phi(64^2) + d(64^2) = phi(84^2) + d(84^2).
a(2) = 4131 = phi(98^2) + d(98^2) = phi(114^2) + d(114^2).
a(3) = 36981 = phi(237^2) + d(237^2) = phi(342^2) + d(342^2).
a(4) = 78765 = phi(486^2) + d(486^2) = phi(492^2) + d(492^2).
a(5) = 14054589 = phi(4593^2) + d(4593^2) = phi(7320^2) + d(7320^2).
a(6) = 889978059 = phi(29833^2) + d(29833^2) = phi(45668^2) + d(45668^2).
a(7) = 110543990589 = phi(337993^2) + d(337993^2) = phi(423891^2) + d(423891^2).
		

Crossrefs

Cf. A000005, A000010, A061468, A357916

Programs

Maple

N:= 10^12: vmax:= evalf(N/(exp(gamma)*log(log(N))+3/log(log(N)))):
Q:= [seq(numtheory:-phi(k^2)+numtheory:-tau(k^2),k=1..sqrt(N))]:
QN := select(`<`,Q,vmax):
QS:= sort(QN):
K:= select(t -> QS[t+1]=QS[t], [$1..nops(QS)-1]):
convert(QS[K],set);