In fact, using Mathematica 8.0, we were able to obtain all positive integers for all in Table 11. Note that the results of in Table 11 are also presented in [[21], p.100, pp.103-104].
By Lemma 4.3, we deduce the corollary.
We list the following interesting remarks and examples.
-
(a)
= { 6, 12, 14, 20, 33, 44, 62, 92, 116, 138, 164, 212, 254, 280, 308, 320, 332, 356, 452, 524, 572, 692, 716, 764, 932, 956, 1,004, 1,124, 1,172, 1,436, 1,496, 1,562, 1,676, 1,724, 1,772, 1,964, 2,002, 2,036, 2,132, 2,372, 2,564, 2,598, 2,612, 2,636, 2,732, 2,876, 2,913, 2,972, 3,044, 3,228, 3,236, 3,344, 3,408, 3,644, 3,812, 4,052, 4,076, 4,124, 4,187, 4,196, 4,292, 4,412, 4,728, 4,892, 4,916, 5,156, 5,170, 5,636, 5,756, 5,804, 5,924, 5,996, 6,044, 6,236, 6,332, 6,404, 6,764, 6,932, 7,169, 7,244, 7,424, 7,556, 7,604, 7,724, 7,892, 8,012, 8,050, 8,156, 8,234, 8,252, 8,276, 8,516, 8,564, 8,930, 9,092, 9,356, 9,359, 9,404, 9,572, 9,596, 9,836, 10,172, 10,196, 10,772, 10,796, 10,964, 11,012, 11,276, 11,612, 11,756, 11,852, 11,876, 12,092, 12,212, 12,565, 13,196, 13,316, 13,436, 13,556, 13,652, 13,796, 13,964, 14,156, 14,372, 14,492, 15,044, 15,085, 15,116, 15,212, 15,284, 15,404, 15,452, 15,644, 16,076, 16,120, 16,292, 16,376, 16,382, 16,844, 17,084, 17,396, 17,492, 17,564, 17,636, 17,924, 18,170, 18,932, 19,172, 19,484, 19,676, 19,772, 20,012, 20,156, 20,204, 20,324, 20,684, 20,924, 21,116, 21,212, 21,332, 21,461, 21,518, 21,596, 21,764, 22,004, 22,556, 22,844, 22,964, 23,396, 23,612, 24,212, 24,404, 24,452, 24,524, 24,692, 24,881, 25,019, 25,052, 25,076, 25,292, 25,316, 25,338, 25,796, 25,964, 26,084, 26,204, 26,252, 26,324, 26,609, 27,044, 27,596, 27,932, 28,172, 28,316, 28,412, 28,484, 28,604, 28,772, 28,844, 29,396, 29,696, 29,732, 30,164, 30,572, 30,596, 30,764, 31,292, 31,364, 31,532, 31,604, 32,276, 32,372, 32,444, 32,804, 32,972, 33,092, 33,590, 34,052, 34,652, 34,772, 34,964, 35,358, 35,804, 35,876, 36,116, 36,236, 36,884, 37,172, 37,484, 37,676, 37,892, 37,916, 38,156, 38,516, 38,756, 39,164, 40,244, 40,364, 40,458, 40,652, 41,012, 41,084, 41,252, 41,324, 42,116, 42,356, 42,452, 42,764, 42,836, 42,932, 43,124, 43,196, 43,532, 44,684, 45,078, 45,284, 45,476, 45,572, 45,884, 46,076, 46,196, 46,316, 46,796, 46,863, 47,132, 47,204, 47,252, 47,324, 47,636, 47,756, 47,792, 47,978, 48,044, 48,164, 48,404, 48,476, 48,812, 49,052, 49,225, 49,316, 50,612, 50,684, 51,164, 51,284, 51,596, 51,692, 51,835, 51,836, 52,004, 52,196, 52,916, 53,058, 53,252, 53,804, 53,852, 53,963, 54,212, 54,476, 54,596, 55,052, 55,532, 55,604, 55,652, 55,748, 56,036, 56,324, 56,612, 56,636, 56,996, 57,212, 57,284, 57,956, 58,056, 58,244, 58,484, 58,676, 58,796, 58,964, 59,132, 59,324, 59,516, 59,756, 59,987, 60,404, 60,644, 60,692, 60,932, 61,076, 61,604, 61,832, 62,276, 62,516, 63,092, 63,164, 63,212, 63,692, 64,004, 64,364, 65,012, 65,204, 65,684, 65,972, 66,212, 66,692, 67,244, 67,292, 67,532, 67,724, 68,636, 68,732, 69,164, 69,332, 69,404, 70,316, 70,676, 70,724, 71,756, 71,924, 72,164, 72,524, 72,596, 72,764, 72,836, 72,932, 73,364, 73,772, 73,844, 74,924, 75,092, 75,108, 75,212, 75,596, 76,652, 77,057, 77,204, 77,276, 77,492, 77,564, 77,732, 78,212, 78,236, 78,644, 78,836, 79,004, 79,292, 79,556, 79,652, 79,676, 79,964, 80,252, 80,996, 81,055, 81,476, 81,572, 81,644, 81,764, 82,772, 83,012, 83,036, 83,084, 83,156, 83,516, 83,684, 83,756, 83,852, 84,044, 84,356, 84,596, 84,716, 84,884, 85,364, 85,532, 85,676, 86,170, 86,444, 86,804, 86,852, 87,212, 87,572, 88,052, 88,316, 88,532, 89,036, 89,084, 89,372, 89,396, 89,636, 89,732, 89,876, 89,924, 90,164, 91,004, 91,412, 91,772, 92,396, 93,116, 93,284, 93,356, 93,836, 94,244, 94,412, 94,676, 95,012, 95,276, 95,636, 95,924, 96,812, 96,956, 97,124, 97,892, 98,036, 98,204, 98,444, 98,732, 98,996, 99,638, 99,884 }.
-
(b)
= { 12, 24, 28, 40, 51, 66, 88, 115, 124, 184, 232, 276, 319, 328, 424, 508, 560, 616, 640, 664, 712, 904, 1,003, 1,048, 1,144, 1,384, 1,432, 1,528, 1,864, 1,912, 2,008, 2,248, 2,344, 2,585, 2,872, 2,992, 3,124, 3,352, 3,448, 3,544, 3,928, 4,004, 4,072, 4,183, 4,195, 4,264, 4,744, 5,128, 5,196, 5,224, 5,272, 5,464, 5,752, 5,826, 5,944, 5,959, 6,088, 6,456, 6,472, 6,688, 6,816, 7,288, 7,624, 8,104, 8,152, 8,248, 8,374, 8,392, 8,584, 8,824, 9,456, 9,784, 9,832, 10,312, 10,340, 11,272, 11,512, 11,608, 11,659, 11,848, 11,992, 12,088, 12,367, 12,472, 12,561, 12,664, 12,808, 13,528, 13,581, 13,864, 14,338, 14,488, 14,848, 15,112, 15,208, 15,365, 15,448, 15,784, 16,024, 16,100, 16,312, 16,468, 16,504, 16,552, 17,032, 17,128, 17,860, 18,184, 18,712, 18,718, 18,808, 19,144, 19,192, 19,672, 20,344, 20,392, 20,541, 21,544, 21,592, 21,928, 22,024, 22,552, 23,224, 23,512, 23,704, 23,752, 24,184, 24,424, 25,130, 26,392, 26,632, 26,872, 27,112, 27,304, 27,592, 27,928, 28,312, 28,744, 28,984, 29,393, 30,088, 30,170, 30,232, 30,424, 30,568, 30,808, 30,904, 31,288, 32,152, 32,240, 32,584, 32,665, 32,752, 32,764, 33,688, 34,168, 34,792, 34,984, 35,128, 35,272, 35,848, 36,340, 37,864, 38,344, 38,968, 39,352, 39,544, 39,913, 40,024, 40,312, 40,408, 40,648, 41,368, 41,848, 42,232, 42,423, 42,424, 42,664, 42,922, 43,036, 43,192, 43,528, 44,008, 45,112, 45,688, 45,928, 46,792, 47,224, 47,841, 48,424, 48,808, 48,904, 49,048, 49,384, 49,762, 50,038, 50,104, 50,152, 50,435, 50,584, 50,632, 50,676, 51,592, 51,928, 52,168, 52,408, 52,504, 52,648, 53,218, 54,088, 55,192, 55,864, 56,344, 56,632, 56,824, 56,968, 57,208, 57,544, 57,688, 58,792, 59,392, 59,464, 60,328, 61,144, 61,192, 61,528, 62,584, 62,728, 63,064, 63,208, 64,552, 64,744, 64,888, 65,608, 65,944, 66,184, 67,180, 68,104, 69,304, 69,544, 69,928, 70,716, 71,608, 71,752, 72,232, 72,472, 73,768, 74,344, 74,968, 75,352, 75,784, 75,832, 76,312, 77,032, 77,512, 78,328, 80,488, 80,728, 80,916, 81,304, 82,024, 82,168, 82,504, 82,648, 84,232, 84,712, 84,904, 85,528, 85,672, 85,864, 86,248, 86,392, 87,064, 88,303, 89,368, 90,156, 90,568, 90,952, 91,144, 91,768, 92,152, 92,392, 92,599, 92,632, 93,592, 93,726, 94,264, 94,408, 94,504, 94,648, 95,272, 95,512, 95,584, 95,956, 96,088, 96,328, 96,808, 96,952, 97,624, 98,104, 98,450, 98,632}.
-
(c)
= { 15, 24, 48, 56, 69, 80, 87, 102, 132, 175, 176, 230, 248, 368, 464, 552, 638, 656, 689, 848, 1,016, 1,120, 1,127, 1,232, 1,280, 1,328, 1,349, 1,424, 1,808, 2,006, 2,096, 2,288, 2,768, 2,864, 3,056, 3,728, 3,824, 4,016, 4,496, 4,688, 5,170, 5,744, 5,984, 6,248, 6,704, 6,896, 7,088, 7,856, 8,008, 8,144, 8,366, 8,390, 8,528, 9,488, 10,256, 10,392, 10,448, 10,544, 10,928, 11,504, 11,652, 11,888, 11,918, 12,176, 12,912, 12,944, 13,376, 13,632, 14,576, 15,248, 16,208, 16,304, 16,496, 16,748, 16,784, 17,168, 17,648, 18,912, 19,511, 19,568, 19,664, 19,829, 20,624, 20,680, 22,544, 23,024, 23,216, 23,318, 23,696, 23,984, 24,176, 24,597, 24,734, 24,944, 25,122, 25,328, 25,616, 27,056, 27,162, 27,728, 28,676, 28,976, 29,696, 30,224, 30,416, 30,730, 30,896, 31,568, 32,048, 32,200, 32,624, 32,936, 33,008, 33,104, 34,064, 34,256, 35,720, 36,368, 37,391, 37,424, 37,436, 37,616, 37,901, 38,288, 38,384, 39,344, 40,688, 40,784, 41,082, 43,088, 43,184, 43,856, 44,048, 45,104, 45,925, 46,448, 47,024, 47,408, 47,487, 47,504, 48,368, 48,848, 50,260, 52,784, 53,264, 53,744, 54,224, 54,608, 55,184, 55,856, 56,624, 57,488, 57,968, 58,786, 59,125, 60,176, 60,340, 60,464, 60,848, 61,053, 61,136, 61,616, 61,808, 62,576, 64,304, 64,480, 65,168, 65,330, 65,504, 65,528, 67,376, 68,336, 69,584, 69,968, 70,256, 70,544, 71,696, 72,680, 75,728, 76,688, 77,936, 78,704, 79,088, 79,826, 80,048, 80,624, 80,816, 81,296, 81,989, 82,736, 83,696, 84,464, 84,846, 84,848, 85,328, 85,844, 86,072, 86,384, 87,056, 88,016, 90,224, 91,376, 91,856, 93,584, 94,448, 95,682, 96,848, 97,557, 97,616, 97,808, 98,096, 98,768, 99,524, 99,827}.
-
(d)
= { 30, 48, 55, 96, 112, 138, 160, 174, 204, 205, 264, 350, 352, 355, 460, 496, 736, 928, 1,104, 1,276, 1,293, 1,312, 1,378, 1,696, 2,032, 2,240, 2,254, 2,464, 2,560, 2,656, 2,698, 2,848, 3,277, 3,616, 3,669, 4,012, 4,192, 4,576, 5,536, 5,728, 6,112, 7,456, 7,648, 8,032, 8,992, 9,376, 9,853, 10,340, 11,488, 11,968, 12,496, 12,549, 13,408, 13,792, 13,899, 14,176, 14,857, 15,712, 16,016, 16,288, 16,732, 16,780, 17,056, 18,976, 20,512, 20,784, 20,896, 21,088, 21,856, 23,008, 23,287, 23,304, 23,776, 23,836, 24,352, 25,824, 25,888, 26,077, 26,752, 27,264, 29,152, 30,496, 32,416, 32,608, 32,992, 33,496, 33,568, 34,336, 35,296, 36,669, 37,824, 39,022, 39,136, 39,187, 39,328, 39,658, 41,248, 41,360, 41,525, 45,088, 46,048, 46,432, 46,636, 47,392, 47,968, 48,352, 49,194, 49,468, 49,888, 50,244, 50,656, 51,232, 54,112, 54,324, 55,456, 56,743, 57,352, 57,952, 59,392, 60,448, 60,832, 61,460, 61,792, 63,136, 64,096, 64,400, 64,963, 65,248, 65,872, 66,016, 66,208, 68,128, 68,512, 71,440, 72,736, 73,321, 74,782, 74,848, 74,872, 75,232, 75,802, 76,576, 76,768, 78,688, 81,376, 81,568, 82,164, 85,839, 86,176, 86,368, 87,712, 88,096, 90,208, 90,637, 91,850, 92,896, 94,048, 94,816, 94,974, 95,008, 95,113, 96,193, 96,736, 97,696}.
-
(e)
= { 60, 96, 110, 177, 192, 224, 276, 303, 320, 348, 408, 410, 528, 605, 700, 704, 710, 749, 920, 992, 1,045, 1,157, 1,472, 1,856, 2,208, 2,552, 2,567, 2,586, 2,624, 2,756, 3,392, 4,064, 4,480, 4,508, 4,533, 4,928, 5,120, 5,312, 5,396, 5,696, 6,554, 7,232, 7,338, 7,697, 8,024, 8,384, 9,152, 10,547, 11,072, 11,456, 12,224, 13,199, 14,912, 15,296, 16,064, 16,345, 17,984, 18,752, 19,706, 20,381, 20,680, 21,197, 21,797, 22,976, 23,936, 24,992, 25,075, 25,098, 26,816, 27,584, 27,798, 28,352, 29,321, 29,365, 29,714, 31,424, 32,032, 32,576, 32,849, 33,464, 33,560, 34,112, 37,929, 37,952, 41,024, 41,568, 41,792, 42,176, 43,712, 46,016, 46,574, 46,608, 47,552, 47,672, 48,704, 51,648, 51,776, 52,154, 53,504, 54,528, 58,304, 60,992, 61,705, 64,832, 65,216, 65,984, 66,992, 67,136, 68,672, 70,049, 70,592, 73,338, 75,648, 78,044, 78,272, 78,374, 78,656, 79,316, 82,496, 82,720, 83,050, 85,769, 89,033, 90,176, 92,096, 92,864, 93,272, 94,784, 95,936, 96,704, 98,388, 98,936, 99,776}.
We conclude this paper by the following remark.