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Numbers k such that k concatenated with itself gives the product of two numbers which differ by 7.
+10
5
4, 94, 210, 294, 994, 5880, 9994, 52888, 99994, 127044, 414180, 999994, 8264470, 9999994, 12456750, 41868508, 99999994, 112670544, 441341880, 468144040, 669421494, 702338994, 715976338, 750005718, 960645294, 999999994
k times k+7 gives the concatenation of two numbers m and m-1.
+10
5
7, 30, 64, 42753, 57241, 75423, 425072, 574922, 979528, 4301393, 5698601, 7028666, 4925000747, 5074999247, 7748266574, 8511881484, 8814851184, 7059602159672, 7106167933828, 7439286611621, 7485852385777
PROG
(Python)
def ok(n):
s = str(n*(n+7)); h = (len(s)+1)//2; return int(s[:h])-1 == int(s[h:])
n times n+6 gives the concatenation of a number m with itself.
+10
5
5, 95, 462, 533, 715, 819, 995, 3425, 6570, 9995, 90904, 99995, 980199, 999995, 3636358, 6363637, 9999995, 41176465, 58823530, 99999995, 413533835, 426573427, 428571423, 432620006, 567379989, 571428572, 573426568
n times n+8 gives the concatenation of a number m with itself.
+10
5
3, 93, 377, 616, 707, 902, 993, 8760, 9993, 45455, 54538, 99993, 693062, 999993, 8181811, 9999993, 88235287, 99999993, 327935223, 330669331, 363636364, 418318516, 428571429, 461538454, 538461539, 571428564, 581681477
n times n+7 gives the concatenation of two numbers m and m+1.
+10
5
6, 699, 776, 790, 867, 42337, 57657, 96883, 44121666, 55878328, 85298137, 36680703009575609347721358493, 63319296990424390652278641501, 69346342454876071597336150481, 81501115172242572470460459683
EXAMPLE
85298137 * 85298144 = 72757727//72757728, where // denotes
concatenation.
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