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Least nontrivial multiple of the n-th prime beginning with 1.
+10
9
10, 12, 10, 14, 110, 104, 102, 114, 115, 116, 124, 111, 123, 129, 141, 106, 118, 122, 134, 142, 146, 158, 166, 178, 194, 1010, 1030, 1070, 1090, 1017, 1016, 1048, 1096, 1112, 1043, 1057, 1099, 1141, 1002, 1038, 1074, 1086, 1146, 1158, 1182, 1194, 1055
FORMULA
a(n) = ceiling(10^d/prime(n))*prime(n) where prime(n) has d digits. - Robert Israel, Jul 21 2020
MAPLE
g:= proc(n) ceil(10^(ilog10(n)+1)/n)*n end proc:
MATHEMATICA
f[n_] := Block[{k = 2, m = n}, While[ IntegerDigits[k*m][[1]] != 1, k++ ]; k*m]; Table[ f[ Prime[n]], {n, 1, 55}]
Least nontrivial multiple of the n-th prime beginning with 3.
+10
9
30, 30, 30, 35, 33, 39, 34, 38, 322, 319, 310, 333, 328, 301, 329, 318, 354, 305, 335, 355, 365, 316, 332, 356, 388, 303, 309, 321, 327, 339, 381, 393, 3014, 3058, 3129, 302, 314, 326, 334, 346, 358, 362, 382, 386, 394, 398, 3165, 3122, 3178, 3206, 3029
MAPLE
g := proc (n) local m, k;
for m from 0 do
k := max(2, ceil(3*10^m/n));
if k*n < 4*10^m then return k*n end if
end do
end proc:
MATHEMATICA
f[n_] := Block[{k = 2, m = n}, While[ IntegerDigits[k*m][[1]] != 3, k++ ]; k*m]; Table[ f[ Prime[n]], {n, 1, 55}]
PROG
(PARI) a(n) = {my(k=2, p=prime(n)); while(digits(k*p)[1] != 3, k++); k*p; } \\ Michel Marcus, Feb 12 2018
Least nontrivial multiple of the n-th prime beginning with 2.
+10
8
20, 21, 20, 21, 22, 26, 204, 209, 207, 203, 217, 222, 205, 215, 235, 212, 236, 244, 201, 213, 219, 237, 249, 267, 291, 202, 206, 214, 218, 226, 254, 262, 274, 278, 298, 2114, 2041, 2119, 2004, 2076, 2148, 2172, 2101, 2123, 2167, 2189, 2110, 2007, 2043, 2061
MATHEMATICA
f[n_] := Block[{k = 2, m = n}, While[ IntegerDigits[k*m][[1]] != 2, k++ ]; k*m]; Table[ f[ Prime[n]], {n, 1, 55}]
Least nontrivial multiple of the n-th prime beginning with 4.
+10
8
4, 42, 40, 42, 44, 403, 408, 418, 46, 406, 403, 407, 410, 430, 423, 424, 413, 427, 402, 426, 438, 474, 415, 445, 485, 404, 412, 428, 436, 452, 4064, 4061, 411, 417, 447, 453, 471, 489, 4008, 4152, 4117, 4163, 4011, 4053, 4137, 4179, 422, 446, 454, 458, 466
MATHEMATICA
f[n_] := Block[{k = 2, m = n}, While[ IntegerDigits[k*m][[1]] != 4, k++ ]; k*m]; Table[ f[ Prime[n]], {n, 1, 55}]
Least nontrivial multiple of the n-th prime beginning with 6.
+10
8
6, 6, 60, 63, 66, 65, 68, 608, 69, 609, 62, 629, 615, 602, 611, 636, 649, 610, 603, 639, 657, 632, 664, 623, 679, 606, 618, 642, 654, 678, 635, 655, 685, 695, 6109, 604, 628, 652, 668, 692, 6086, 6154, 6112, 6176, 6107, 6169, 633, 669, 681, 687, 699, 6214
MATHEMATICA
f[n_] := Block[{k = 2, m = n}, While[ IntegerDigits[k*m][[1]] != 6, k++ ]; k*m]; Table[ f[ Prime[n]], {n, 1, 55}]
PROG
(Python)
from sympy import prime
def a(n):
pn = prime(n)
m = 2*pn
while str(m)[0] != '6': m += pn
return m
Least nontrivial multiple of the n-th prime beginning with 7.
+10
8
70, 72, 70, 70, 77, 78, 714, 76, 713, 725, 713, 74, 738, 731, 705, 742, 708, 732, 737, 710, 730, 711, 747, 712, 776, 707, 721, 749, 763, 791, 762, 786, 7124, 7089, 745, 755, 785, 7009, 7014, 7093, 716, 724, 764, 772, 788, 796, 7174, 7136, 7037, 7099, 7223
MATHEMATICA
f[n_] := Block[{k = 2, m = n}, While[ IntegerDigits[k*m][[1]] != 7, k++ ]; k*m]; Table[ f[ Prime[n]], {n, 1, 55}]
Least nontrivial multiple of the n-th prime beginning with 8.
+10
8
8, 81, 80, 84, 88, 806, 85, 817, 805, 87, 806, 814, 82, 86, 846, 848, 826, 854, 804, 852, 803, 869, 830, 801, 873, 808, 824, 856, 872, 8023, 889, 8122, 822, 834, 894, 8003, 8007, 815, 835, 865, 895, 8145, 8022, 8106, 8077, 8159, 844, 892, 8172, 8015, 8155
MATHEMATICA
f[n_] := Block[{k = 2, m = n}, While[ IntegerDigits[k*m][[1]] != 8, k++ ]; k*m]; Table[ f[ Prime[n]], {n, 1, 55}]
Least nontrivial multiple of the n-th prime beginning with 9.
+10
8
90, 9, 90, 91, 99, 91, 901, 95, 92, 928, 93, 925, 902, 903, 94, 901, 944, 915, 938, 923, 949, 948, 913, 979, 970, 909, 927, 963, 981, 904, 9017, 917, 959, 973, 9089, 906, 942, 978, 9018, 9169, 9129, 905, 955, 965, 985, 995, 9073, 9143, 908, 916, 932, 956, 964
MATHEMATICA
f[n_] := Block[{k = 2, m = n}, While[ IntegerDigits[k*m][[1]] != 9, k++ ]; k*m]; Table[ f[ Prime[n]], {n, 1, 55}]
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