Conjecture concerning the primes for which 8 is a primitive root and the smallest k such that k*n is a Niven (or Harshad) number.

[Davide Rotondo]

Conjecture.
Consider the primes for which 8 is the primitive root:

3, 5, 11, 29, 53, 59, 83, 101, 107, 131, 149, 173, 179, 197, 227, 269, 293, 317, 347, 389, 419, 443, 461, 467, 491, 509, 557, 563, 587, 653, 659, 677, 701, 773, 797, 821, 827, 941, 947, 1019, 1061, 1091, 1109, 1187, 1229, 1259, 1277, 1283, 1301, 1307, 1373, 1427, 1451, 1493, 1499, 1523, 1571, 1619, 1637, 1667, 1733, 1787, 1877, 1901, 1907, 1931, 1949, 1973, 1979, 1997, 2027, 2069, 2099, 2141, 2213, 2237, 2243, 2267, 2309, 2333, 2339, 2357, 2459, 2477, 2531, 2549, 2579, 2621, 2693, 2699, 2741, 2789, 2819, 2837, 2843, 2861, 2909, 2939, 2957, 2963, 3011, 3083, 3203, 3299, 3323, 3347, 3371, 3413, 3461, 3467, 3491, 3533, 3539, 3557, 3581, 3659, 3677, 3701, 3779, 3797, 3803, 3851, 3917, 3923, 3947, 3989, 4013, 4019, 4091, 4133, 4139, 4157, 4229, 4253, 4259, 4283, 4349, 4373, 4397, 4451, 4493, 4517, 4547, 4637, 4691, 4787, 4877, 4973, 5003, 5051, 5099, 5147, 5171, 5189, 5261, 5309, 5333, 5387, 5477, 5483, 5501, 5507, 5573, 5651, 5693, 5717, 5741, 5813, 5843, 5939, 5987, 6011, 6029, 6053, 6101, 6131, 6173, 6197, 6203, 6269, 6299, 6317, 6323, 6389, 6491, 6653, 6659, 6701, 6779, 6803, 6827, 6869, 6899, 6917, 6947, 6971, 7013, 7019, 7043, 7109, 7187, 7211, 7229, 7253, 7283, 7307, 7331, 7349, 7451, 7499, 7517, 7523, 7541, 7547, 7589, 7643, 7691, 7757, 7829, 7853, 7877, 7883, 7901, 7907, 7949, 8069, 8093, 8117, 8123, 8147, 8171, 8219, 8237, 8243, 8291, 8363, 8387, 8429, 8573, 8597, 8627, 8669, 8693, 8699, 8741, 8747, 8819, 8837, 8861, 8867, 8933, 8963, 9011, 9029, 9059, 9173, 9203, 9221, 9227, 9293, 9323, 9341, 9371, 9419, 9437, 9467, 9491, 9533, 9539, 9587, 9629, 9677, 9749, 9803, 9851, 9923, 9941,...

I verified up to 9941 and I found that except for 3 and 5, for these primes the minimum value k such that k*n is a Niven (or Harshad) number, that is, a number divisible by the sum of its digits, is greater than or equal to 9.

What do you think?

See you soon

Davide

[Davide Rotondo]

P.S. I think several categories of primes have this property. For example, I analyzed Sophie Germain's primes, and they also seem to have this property.

See you soon

Davide

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[Jack Brennen]

For the numbers 1667 and 166667, wouldn't k=6 work?

1667*6 is 10002, divisible by digit sum 3.

166667*6 is 1000002, also divisible by digit sum 3.

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[Davide Rotondo]

Sorry i Miss 1667 in my check. My apolgies

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