27662 is a numbers whose arithmetic and multiplicative digital roots are not digits of the number itself but whose arithmetic root is equal to the number of digits in the number. Here the arithmetic digital root is 5 and the multiplicative digital root is 0. The number of digits in the number (5) corresponds to the arithmetic digital root.

The sequence can be generated as follows (permalink):

INPUT

L=[]

for n in [1..40000]:

  b=10

  if sum(n.digits(b))%(b-1)!=0:

    dr=sum(n.digits())%(b-1)

  else:

    dr=9

  number=n

  while len(number.digits())>1:

    number=prod(number.digits())

  if dr not in n.digits() and (number not in n.digits() and dr==len(n.digits())):

    L.append(n)

print(L)

print(len(L))

OUTPUT (from 27374 to 40000 shown)

[ ... 27662, 27671, 27689, 27698, 27716, 27734, 27743, 27761, 27779, 27788, 27797, 27833, 27869, 27878, 27887, 27896, 27914, 27923, 27932, 27941, 27968, 27977, 27986, 28112, 28121, 28148, 28166, 28184, 28211, 28229, 28238, 28283, 28292, 28328, 28337, 28346, 28364, 28373, 28382, 28418, 28436, 28463, 28481, 28499, 28616, 28634, 28643, 28661, 28679, 28688, 28697, 28733, 28769, 28778, 28787, 28796, 28814, 28823, 28832, 28841, 28868, 28877, 28886, 28922, 28949, 28967, 28976, 28994, 29111, 29129, 29147, 29174, 29192, 29219, 29228, 29237, 29246, 29264, 29273, 29282, 29291, 29327, 29372, 29399, 29417, 29426, 29444, 29462, 29471, 29489, 29498, 29624, 29642, 29669, 29678, 29687, 29696, 29714, 29723, 29732, 29741, 29768, 29777, 29786, 29822, 29849, 29867, 29876, 29894, 29912, 29921, 29939, 29948, 29966, 29984, 29993, 31127, 31136, 31163, 31172, 31199, 31217, 31226, 31244, 31262, 31271, 31316, 31334, 31343, 31361, 31379, 31388, 31397, 31424, 31433, 31442, 31469, 31478, 31487, 31496, 31613, 31622, 31631, 31649, 31667, 31676, 31694, 31712, 31721, 31739, 31748, 31766, 31784, 31793, 31838, 31847, 31874, 31883, 31919, 31937, 31946, 31964, 31973, 31991, 32117, 32126, 32144, 32162, 32171, 32216, 32234, 32243, 32261, 32279, 32288, 32297, 32324, 32342, 32378, 32387, 32414, 32423, 32432, 32441, 32468, 32477, 32486, 32612, 32621, 32648, 32666, 32684, 32711, 32729, 32738, 32747, 32774, 32783, 32792, 32828, 32837, 32846, 32864, 32873, 32882, 32927, 32972, 32999, 33116, 33134, 33143, 33161, 33179, 33188, 33197, 33224, 33242, 33278, 33287, 33314, 33341, 33368, 33377, 33386, 33413, 33422, 33431, 33449, 33467, 33476, 33494, 33611, 33638, 33647, 33674, 33683, 33719, 33728, 33737, 33746, 33764, 33773, 33782, 33791, 33818, 33827, 33836, 33863, 33872, 33881, 33899, 33917, 33944, 33971, 33989, 33998, 34124, 34133, 34142, 34169, 34178, 34187, 34196, 34214, 34223, 34232, 34241, 34268, 34277, 34286, 34313, 34322, 34331, 34349, 34367, 34376, 34394, 34412, 34421, 34439, 34448, 34466, 34484, 34493, 34619, 34628, 34637, 34646, 34664, 34673, 34682, 34691, 34718, 34727, 34736, 34763, 34772, 34781, 34799, 34817, 34826, 34844, 34862, 34871, 34889, 34898, 34916, 34934, 34943, 34961, 34979, 34988, 34997, 36113, 36122, 36131, 36149, 36167, 36176, 36194, 36212, 36221, 36248, 36266, 36284, 36311, 36338, 36347, 36374, 36383, 36419, 36428, 36437, 36446, 36464, 36473, 36482, 36491, 36617, 36626, 36644, 36662, 36671, 36689, 36698, 36716, 36734, 36743, 36761, 36779, 36788, 36797, 36824, 36833, 36842, 36869, 36878, 36887, 36896, 36914, 36941, 36968, 36977, 36986, 37112, 37121, 37139, 37148, 37166, 37184, 37193, 37211, 37229, 37238, 37247, 37274, 37283, 37292, 37319, 37328, 37337, 37346, 37364, 37373, 37382, 37391, 37418, 37427, 37436, 37463, 37472, 37481, 37499, 37616, 37634, 37643, 37661, 37679, 37688, 37697, 37724, 37733, 37742, 37769, 37778, 37787, 37796, 37814, 37823, 37832, 37841, 37868, 37877, 37886, 37913, 37922, 37931, 37949, 37967, 37976, 37994, 38138, 38147, 38174, 38183, 38228, 38237, 38246, 38264, 38273, 38282, 38318, 38327, 38336, 38363, 38372, 38381, 38399, 38417, 38426, 38444, 38462, 38471, 38489, 38498, 38624, 38633, 38642, 38669, 38678, 38687, 38696, 38714, 38723, 38732, 38741, 38768, 38777, 38786, 38813, 38822, 38831, 38849, 38867, 38876, 38894, 38939, 38948, 38966, 38984, 38993, 39119, 39137, 39146, 39164, 39173, 39191, 39227, 39272, 39299, 39317, 39344, 39371, 39389, 39398, 39416, 39434, 39443, 39461, 39479, 39488, 39497, 39614, 39641, 39668, 39677, 39686, 39713, 39722, 39731, 39749, 39767, 39776, 39794, 39839, 39848, 39866, 39884, 39893, 39911, 39929, 39938, 39947, 39974, 39983, 39992]

27662 is a happy and untouchable number.

27662 is unprimeable meaning it is composite and changing any single digit doesn't alter that.

27662 has a sum of DISTINCT prime factors that is a prime number, namely 13833.

27662 has not reached a home prime after 25 steps.

27662 can be rendered as a digit equation as follows:

2 * 7 - 6 = 6 + 2