Re^2: Two or three more sequences??

[Richard Mathar]

>     1.  Numbers such that  phi(n) / phi(sigma(n))  =  2
> viz., Iannucci's `year numbers':
>      5,13,37,61,65,73,...,185,...,305,...325,...,365

for(n=1,10^4,eulerphi(n) == 2*eulerphi(sigma(n))&print1(n", "))
5, 13, 37, 61, 65, 73, 119, 157, 185, 193, 277, 305, 313, 365, 397,
421, 457, 481, 541, 613, 661, 673, 733, 757, 785, 793, 877, 949, 965,
997, 1093, 1153, 1201, 1213, 1237, 1321, 1381, 1385, 1453, 1547, 1565,
1615, 1621, 1657, 1753, 1873, 1933, 1985, 1993, 2017, 2041, 2105,
2137, 2257, 2285, 2341, 2405, 2473, 2509, 2557, 2593, 2701, 2705,
2797, 2857, 2917, 3061, 3065, 3217, 3253, 3305, 3313, 3365, 3517,
3601, 3665, 3733, 3785, 3965, 4021, 4057, 4069, 4177, 4261, 4273,
4357, 4385, 4403, 4441, 4453, 4561, 4621, 4745, 4933, 4985, 5077,
5101, 5113, 5161, 5233, 5413, 5437, 5465, 5473, 5491, 5581, 5701,
5765, 5809, 5941, 6005, 6037, 6065, 6073, 6121, 6133, 6185, 6217,
6337, 6361, 6373, 6605, 6637, 6661, 6781, 6905, 6997, 7033, 7057,
7141, 7213, 7259, 7265, 7393, 7417, 7477, 7537, 7753, 7933, 7969,
8053, 8075, 8101, 8105, 8221, 8285, 8317, 8353, 8439, 8461, 8521,
8593, 8677, 8687, 8713, 8749, 8765, 8893, 9013, 9133, 9181, 9241,
9277, 9365, 9529, 9577, 9601, 9661, 9665, 9721, 9817, 9841, 9901,
9965, 9973,


>     2.  Non-squarefree such numbers (see above)

for(n=1,10^4,eulerphi(n)
==2*eulerphi(sigma(n))&vecmax(factor(n)[,2])<2&print1(n", "))

5, 13, 37, 61, 65, 73, 119, 157, 185, 193, 277, 305, 313, 365, 397,
421, 457, 481, 541, 613, 661, 673, 733, 757, 785, 793, 877, 949, 965,
997, 1093, 1153, 1201, 1213, 1237, 1321, 1381, 1385, 1453, 1547, 1565,
1615, 1621, 1657, 1753, 1873, 1933, 1985, 1993, 2017, 2041, 2105,
2137, 2257, 2285, 2341, 2405, 2473, 2509, 2557, 2593, 2701, 2705,
2797, 2857, 2917, 3061, 3065, 3217, 3253, 3305, 3313, 3365, 3517,
3601, 3665, 3733, 3785, 3965, 4021, 4057, 4069, 4177, 4261, 4273,
4357, 4385, 4403, 4441, 4453, 4561, 4621, 4745, 4933, 4985, 5077,
5101, 5113, 5161, 5233, 5413, 5437, 5465, 5473, 5581, 5701, 5765,
5809, 5941, 6005, 6037, 6065, 6073, 6121, 6133, 6185, 6217, 6337,
6361, 6373, 6605, 6637, 6661, 6781, 6905, 6997, 7033, 7057, 7141,
7213, 7259, 7265, 7393, 7417, 7477, 7537, 7753, 7933, 7969, 8053,
8101, 8105, 8221, 8285, 8317, 8353, 8439, 8461, 8521, 8593, 8677,
8687, 8713, 8749, 8765, 8893, 9013, 9133, 9181, 9241, 9277, 9365,
9529, 9577, 9601, 9661, 9665, 9721, 9817, 9841, 9901, 9965, 9973,

>     3.  Numbers such that  phi(n) / phi(sigma(n))  =  1/2
> viz.  2, 6, 8, 9, 16, 28, 70, 78

for(n=1,10^4,2*eulerphi(n) == eulerphi(sigma(n))&print1(n", "))
2, 6, 8, 9, 24, 28, 70, 78, 128, 140, 222, 234, 280, 312, 350, 366,
384, 438, 496, 525, 666, 864, 888, 910, 918, 936, 942, 1036, 1098,
1158, 1232, 1314, 1400, 1464, 1575, 1662, 1708, 1752, 1824, 1836,
1878, 1900, 1938, 2044, 2382, 2480, 2526, 2590, 2664, 2742, 2826,
2886, 3246, 3474, 3640, 3678, 3768, 3876, 3966, 4038, 4270, 4392,
4396, 4398, 4480, 4524, 4542, 4550, 4632, 4758, 4986, 4992, 5110,
5180, 5256, 5262, 5404, 5472, 5634, 5694, 5814, 5982, 6448, 6558,
6648, 6825, 6918, 7146, 7206, 7278, 7422, 7512, 7578, 7756, 7926,
8128, 8226, 8262, 8286, 8360, 8540, 8658, 8718, 8764, 8856, 9528,
9726, 9738, 9801, 9942,


Maximilian