Inder J. Taneja RGMIA Research Report Collection, 20(2017), pp.1-113, http://rgmia.org/v20.php Fixed and Flexible Powers Narcissistic Numbers with Division Inder J. Taneja1 Abstract This paper brings extension of narcissistic numbers with division. The extension is done in different sit- uations, such as, with positive and negative coefficients, fixed and flexible powers. Comparison with previous known numbers are also given. This is revised and enlarged version of author’s previous work [13]. Contents 1 Introduction 1 2 Fixed Power Narcissistic Numbers with Division 3 2.1 Fixed Power: Positive Coefficients . .3 2.2 Fixed Power: Positive and Negative Coefficients . .3 3 Flexible Power Narcissistic Numbers with Division 11 3.1 Flexible Power: Positive Coefficients . 11 3.2 Flexible Power: Positive and Negative Coefficients . 51 1 Introduction An n digit number that is the sum of the nth powers of its digits is called an n narcissistic number. It is ¡ ¡ also sometimes known as an Armstrong number, perfect digital invariant (Madachy 1966 [6]), or plus perfect number. Hardy in 1940 [2] (pg. 25) wrote, there are just four numbers of three digits, which are the sums of the cubes of their digits: 153 13 53 33 Æ Å Å 370 33 73 03 Æ Å Å 371 33 73 13 Æ Å Å 407 43 03 73 Æ Å Å The above four numbers have the same digits on both sides except the power 3 and are with 3 digits. There are four more numbers with 4 digits: 1634 : 14 64 34 44 Æ Å Å Å 4151 : 45 15 55 15 Æ Å Å Å 8208 : 84 24 04 84 Æ Å Å Å 9472 : 94 44 74 24 Æ Å Å Å 1Formerly, Professor of Mathematics, Universidade Federal de Santa Catarina, 88.040-900 Florianópolis, SC, Brazil. E-mail: [email protected]; Web-site: http://inderjtaneja.com; Twitter: @IJTANEJA. 1 Inder J. Taneja RGMIA Research Report Collection, 20(2017), pp.1-113, http://rgmia.org/v20.php Above numbers are with fixed power. Relaxing this condition, we many have much numbers with flexible power, for examples: 132 : 11 31 27 Æ Å Å 2353 : 25 32 53 37 Æ Å Å Å Moreover, instead of having positive coefficients, we can have positive-negative coefficients, for examples, 128 : 11 27 80 Æ¡ Å Å 9942 : 91 94 47 27 Æ¡ ¡ Å Å Let’s consider the following expression: abc... am1 bm2 cm3 ..., a,b,c,... {0,1,2,3,4,5,6,7,8,9}, a 0. (1) Æ § § § 2 6Æ For simplicity, let us the write the (1) as abc ... (a,b,d,...)(m1,m2,m3,...) (2) Æ Detailed study of numbers arising due to equation (1) or (2) are given in [13, 19]. If the powers and bases are of same digits with different permutations, then the equation (2) can be re-written as abc ... (a,b,d,...)(a,b,c,...) (3) Æ Numbers arising due to (3) are defined as flexible power selfie numbers, and studied by author [15, 16, 17]. See below some examples, 23 : 22 33 Æ¡ Å 1364 : 16 31 64 43 Æ Å Å Å 3435 : 33 44 33 55 Æ Å Å Å 3045 : 34 03 40 55 Æ¡ Å Å Å 78205 : 72 80 25 08 57 Æ ¡ Å Å Å From above, we observe that the powers and bases are of same digits with different permutations. More study on numbers in different situations see author’s work [18]. Let’s extend the expression (1) with division: am1 bm2 cm3 ... abc... § § § , a,b,c,... {0,1,2,3,4,5,6,7,8,9}, a 0. (4) Æ an1 bn2 cn3 ... 2 6Æ § § § The above construction (4) , we call flexible power narcissistic numbers with division. The powers are al- ways natural numbers, and it may be flexible. Also, instead having positive coefficients, we may have positive and negative coefficients. In some situations, this general way include the previous known results. Relaxing some conditions, the above narcissistic numbers can also be re-written in terms of division. See below: 13 53 33 33 73 13 153 : Å Å 371 : Å Å Æ 10 50 30 Æ 30 70 10 Å ¡ Å ¡ 33 73 03 43 03 73 370 : Å Å 407 : Å Å Æ 30 70 00 Æ 40 00 70 Å ¡ Å ¡ 2 Inder J. Taneja RGMIA Research Report Collection, 20(2017), pp.1-113, http://rgmia.org/v20.php 14 64 34 44 84 24 04 84 1634 : Å Å Å 8208 : Å Å Å Æ 10 60 30 41 Æ 80 20 01 80 ¡ ¡ ¡ Å ¡ Å Å Å 45 15 55 15 94 44 74 24 4151 : Å Å Å 9472 : Å Å Å Æ 41 11 51 11 Æ 90 40 70 22 ¡ Å Å ¡ Å Å Å In this work, aim is to write narcissistic numbers with division according to (4) with fixed and flexible powers. In each case there are two situations: one with positive coefficients and another with positive and negative coefficients. 2 Fixed Power Narcissistic Numbers with Division Below are fixed power narcissistic number with division. This is again divided in two subsections. One with positive coefficients, and second with positive and negative coefficients. In this situation we have very few numbers. 2.1 Fixed Power: Positive Coefficients There are very few narcissistic numbers division having fixed power and positive coefficients, such as, 33 73 55 25 45 75 37 : Å 5247 : Å Å Å Æ 3 7 Æ 50 20 40 70 Å Å Å Å 43 83 85 25 05 05 48 : Å 8200 : Å Å Å Æ 4 8 Æ 80 20 00 00 Å Å Å Å 28 48 18 19 59 59 09 19 241 : Å Å 15501 : Å Å Å Å Æ 24 44 11 Æ 13 53 53 03 13 Å Å Å Å Å Å 45 15 55 17 47 27 77 47 07 415 : Å Å 142740 : Å Å Å Å Å Æ 4 1 5 Æ 10 40 20 70 40 00 Å Å Å Å Å Å Å 25 45 65 45 27 37 17 57 97 17 2464 : Å Å Å 231591 : Å Å Å Å Å . Æ 20 40 60 40 Æ 2 3 1 5 9 1 Å Å Å Å Å Å Å Å 45 75 15 45 4714 : Å Å Å Æ 40 70 10 40 Å Å Å 2.2 Fixed Power: Positive and Negative Coefficients Below are narcissistic numbers with division, having fixed power in numerator and denominator and coeffi- cients with positive and negative signs. 13 33 23 63 43 83 93 13 13 : ¡ Å 26 : ¡ Å 48 : Å 91 : ¡ Æ 10 30 Æ 21 61 Æ 41 81 Æ 91 11 Å Å Å ¡ 26 16 33 73 63 33 21 : ¡ 37 : Å 63 : ¡ Æ 21 11 Æ 31 71 Æ 61 31 Å Å ¡ 16 36 26 14 54 24 24 24 44 132 : ¡ Å Å 152 : ¡ Å ¡ 224 : ¡ ¡ Å Æ 11 31 21 Æ 11 51 21 Æ 20 20 40 Å Å Å ¡ Å ¡ 16 36 46 13 53 33 24 44 04 134 : ¡ Å Å 153 : Å Å 240 : ¡ Å ¡ Æ 13 33 43 Æ 10 50 30 Æ 20 40 00 ¡ ¡ Å Å ¡ Å ¡ 3 Inder J. Taneja RGMIA Research Report Collection, 20(2017), pp.1-113, http://rgmia.org/v20.php 24 44 14 33 73 03 44 44 84 241 : ¡ Å Å 370 : Å ¡ 448 : ¡ ¡ Å Æ 20 40 10 Æ 30 70 00 Æ 41 41 81 Å ¡ Å ¡ ¡ Å 24 64 44 33 73 13 54 44 74 264 : Å ¡ 371 : Å Å 547 : Å Å Æ 21 61 41 Æ 30 70 10 Æ 51 41 71 Å ¡ Å ¡ ¡ Å Å 23 83 43 43 03 73 76 66 36 284 : ¡ Å Å 407 : ¡ Å 763 : ¡ Å Æ 21 81 41 Æ 40 00 70 Æ 72 62 32 ¡ Å ¡ Å ¡ Å Å 33 13 73 45 15 55 84 04 34 317 : ¡ Å Å 415 : Å Å 803 : ¡ ¡ Æ 30 10 70 Æ 41 11 51 Æ 81 01 31 Å ¡ Å Å ¡ ¡ 16 26 56 36 34 54 94 34 67 17 87 17 1253 : ¡ ¡ Å Å 3593 : Å Å ¡ 6181 : ¡ ¡ Å ¡ Æ 12 22 52 32 Æ 30 50 90 30 Æ 63 13 83 13 ¡ Å ¡ Å Å ¡ ¡ ¡ Å ¡ 14 24 94 34 34 64 14 74 1293 : ¡ Å ¡ 3617 : ¡ Å Å Å 66 66 16 36 Æ 11 21 91 31 Æ 31 61 11 71 6613 : Å ¡ ¡ ¡ Å ¡ Å ¡ ¡ Æ 61 61 11 31 14 34 64 84 35 65 85 55 Å ¡ Å 1368 : ¡ Å Å Å 3685 : ¡ Å ¡ Æ 10 30 60 80 Æ 31 61 81 51 66 86 66 46 ¡ Å ¡ ¡ Å Å Å ¡ Å Å ¡ 6864 : 1 1 1 1 15 45 55 55 34 94 64 94 Æ 6 8 6 4 1455 : Å Å Å 3969 : Å ¡ Å Å Å Å Æ 11 41 51 51 Æ 31 91 61 91 Å Å ¡ ¡ Å Å ¡ 68 98 38 98 6 6 6 6 5 5 5 5 Å Å ¡ 1 5 4 3 4 1 5 1 6939 : 3 3 3 3 1543 : Å ¡ ¡ 4151 : Å Å Å Æ 6 9 3 9 Æ 11 51 41 31 Æ 41 11 51 11 Å Å ¡ Å Å ¡ ¡ Å Å ¡ 8 8 8 8 6 6 6 6 67 97 57 97 2 0 4 4 4 1 5 6 Å ¡ ¡ 2044 : ¡ ¡ Å Å 4156 : ¡ ¡ Å Å 6959 : 1 1 1 1 Æ 26 06 46 46 Æ 41 11 51 61 Æ 6 9 5 9 ¡ Å ¡ ¡ Å Å Å Å Å 26 06 66 36 45 75 15 45 5 5 5 5 ¡ Å Å Å Å Å 6 9 7 2 2063 : 2 2 2 2 4714 : 0 0 0 0 6972 : Å Å Å Æ 2 0 6 3 Æ 4 7 1 4 Æ 61 91 71 21 ¡ ¡ Å ¡ Å Å Å ¡ Å Å Å 27 17 47 57 55 25 45 75 ¡ ¡ Å Å Å Å Å 7 7 7 7 2145 : 2 2 2 2 5247 : 0 0 0 0 7 3 6 9 Æ 2 1 4 5 Æ 5 2 4 7 7369 : ¡ ¡ Å Å ¡ Å Å Å Å Å Æ 73 33 63 93 25 35 75 15 54 24 94 54 ¡ ¡ Å Å ¡ Å Å ¡ ¡ Å ¡ 2371 : 1 1 1 1 5295 : 1 1 1 1 Æ 2 3 7 1 Æ 5 2 9 5 85 15 25 35 ¡ Å Å ¡ ¡ Å Å ¡ 8123 : ¡ ¡ ¡ 25 45 65 45 56 46 36 26 Æ 80 10 20 30 2464 : Å Å Å 5432 : ¡ ¡ Å Å Å Å Æ 20 40 60 40 Æ 50 40 30 20 Å Å Å Å Å ¡ 85 25 05 05 25 45 65 85 55 45 95 05 8200 : Å ¡ ¡ 2468 : ¡ ¡ Å Å 5490 : ¡ ¡ Å ¡ Æ 80 20 00 00 Æ 21 41 61 81 Æ 51 41 91 01 Å Å Å ¡ Å Å Å ¡ Å ¡ 24 94 54 24 54 94 24 14 85 45 45 85 2952 : ¡ Å ¡ ¡ 5921 : ¡ Å ¡ Å 8448 : Å Å Å Æ 20 90 50 20 Æ 51 91 21 11 Æ 81 41 41 81 Å Å ¡ ¡ Å ¡ ¡ Å Å ¡ 16 36 76 86 76 19 59 59 09 19 26 16 46 36 86 13787 : ¡ Å ¡ Å ¡ 15501 : Å Å ¡ Å 21438 : ¡ Å ¡ ¡ Å Æ 11 31 71 81 71 Æ 13 53 53 03 13 Æ 21 11 41 31 81 ¡ ¡ Å ¡ Å Å Å ¡ Å Å Å ¡ Å 16 46 56 66 26 16 66 56 26 76 26 36 36 66 06 14562 : ¡ ¡ Å Å Å 16527 : ¡ Å ¡ Å Å 23360 : Å ¡ Å ¡ Æ 11 41 51 61 21 Æ 11 61 51 21 71 Æ 21 31 31 61 01 Å ¡ Å ¡ ¡ Å ¡ Å Å Å Å ¡ ¡ 15 45 75 05 55 28 18 38 58 18 28 48 28 08 38 14705 : ¡ Å Å ¡ ¡ 21351 : ¡ ¡ Å ¡ 24203 : Å Å ¡ Å Æ 10 40 70 00 50 Æ 22 12 32 52 12 Æ 20 40 20 00 30 Å Å ¡ ¡ ¡ ¡ Å ¡ Å Å Å ¡ 4 Inder J.