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Cubes (n * n * n) in carryless arithmetic mod 10.
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0, 1, 8, 7, 4, 5, 6, 3, 2, 9, 1000, 1331, 1628, 1977, 1284, 1555, 1886, 1173, 1422, 1739, 8000, 8261, 8448, 8647, 8864, 8005, 8266, 8443, 8642, 8869, 7000, 7791, 7468, 7117, 7844, 7555, 7246, 7913, 7662, 7399, 4000, 4821, 4688, 4487, 4224, 4005, 4826, 4683
PROG
(PARI) a(n) = fromdigits(Vec(Pol(digits(n))^3)%10); \\ Seiichi Manyama, Mar 09 2023
Fourth powers (n * n * n * n) in carryless arithmetic mod 10.
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0, 1, 6, 1, 6, 5, 6, 1, 6, 1, 10000, 14641, 18426, 12481, 16666, 10005, 14646, 18421, 12486, 16661, 60000, 62481, 64646, 66661, 68426, 60005, 62486, 64641, 66666, 68421, 10000, 18421, 16666, 14641, 12486, 10005, 18426, 16661, 14646, 12481, 60000, 66661, 62486
PROG
(PARI) a(n) = fromdigits(Vec(Pol(digits(n))^4)%10); \\ Seiichi Manyama, Mar 09 2023
Numbers consisting of either all even digits or just 5's and 0's.
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0, 2, 4, 5, 6, 8, 20, 22, 24, 26, 28, 40, 42, 44, 46, 48, 50, 55, 60, 62, 64, 66, 68, 80, 82, 84, 86, 88, 200, 202, 204, 206, 208, 220, 222, 224, 226, 228, 240, 242, 244, 246, 248, 260, 262, 264, 266, 268, 280, 282, 284, 286, 288, 400, 402, 404, 406
COMMENTS
These are all the divisors of zero in carryless arithmetic mod 10. E.g. 5 * 44 = 0.
MATHEMATICA
With[{upto=410}, Select[Union[Join[Select[Range[upto], And@@EvenQ[ IntegerDigits[#]]&], FromDigits/@Tuples[{5, 0}, Ceiling[Log[ 10, upto]]]]], #<=upto&]] (* Harvey P. Dale, Aug 05 2011 *)
Primes in carryless arithmetic mod 10.
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21, 23, 25, 27, 29, 41, 43, 45, 47, 49, 51, 52, 53, 54, 56, 57, 58, 59, 61, 63, 65, 67, 69, 81, 83, 85, 87, 89, 201, 209, 227, 229, 241, 243, 261, 263, 287, 289, 403, 407, 421, 427, 443, 449, 463, 469, 481, 487, 551, 553, 557, 559, 603, 607, 623, 629, 641, 647, 661, 667, 683, 689, 801, 809, 821, 823, 847, 849, 867, 869, 881, 883
COMMENTS
Define the units in carryless arithmetic mod 10 to be the numbers 1, 3, 7 and 9 (these divide any number). A prime is a number N, not a unit, whose only factorizations are of the form N = u * M, where u is a unit.
EXAMPLE
Examples of nonprimes: 2 = 2*51, 4 = 2*2, 10 = 52*85, 11 = 57*83, 101 = 13*17, 102 = 58 * 254 = 502 * 801, 103 = 53 * 251 = 507 * 809, 107 = 53 * 259 = 507 * 801, 108 = 58 * 256 = 502 * 809, 111 = 227 * 553.
Primitive primes in carryless arithmetic mod 10.
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21, 23, 25, 27, 29, 51, 56, 201, 209, 227, 229, 241, 243, 261, 263, 287, 289, 551, 2023, 2027, 2043, 2047, 2061, 2069, 2081, 2089, 2207, 2209, 2221, 2223, 2263, 2267, 2281, 2287, 2401, 2407, 2421, 2423, 2441, 2449, 2483, 2489, 2603, 2609
COMMENTS
Define the units in carryless arithmetic mod 10 to be the numbers 1, 3, 7 and 9 (these divide any number). A prime is a number N, not a unit, whose only factorizations are of the form N = u * M, where u is a unit.
A prime is primitive if it not the carryless product of a smaller prime and a unit.
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