A252648 -id:A252648 - cp's OEIS frontend

A262091 Amicable digital pairs: The smaller number of a pair (x,y) with x <> y such that, in decimal notation and with an appropriate number of leading zeros prepended, x=(x_m...x_1x_0){10}, y=(y_m...y_1y_0){10}, x = y_m^m + ... + y_1^m + y_0^m, and y = x_m^m + ... + x_1^m + x_0^m.

136, 919, 2178, 58618, 89883, 63804, 2755907, 8139850, 144839908, 277668893, 304162700, 4370652168, 21914086555935085, 187864919457180831, 13397885590701080090, 19095442247273220984552, 108493282045082839040458, 1553298727699254868304830

Offset: 1

Don Knuth, Sep 10 2015

If we allow x to be equal to y we get numbers such as 1, 153, 370, 371, 407, ... See A252648. - Chai Wah Wu, Jan 04 2016

			a(1) is amicably paired to 244, because 1^3 + 3^3 + 6^3 = 244 and 2^3 + 4^3 + 4^3 = 136.

D. Knuth, Table of a(n) and its mate for n=1..36
K. Oséki, A problem of number theory, Proceedings of the Japan Academy 36 (1960), 578-587.

A262092 has the larger element of each pair. Cf. A252648.

# print pairs with leading zeros
from _future_ import print_function
from itertools import combinations_with_replacement
for m in range(2,11):
    fs = '0'+str(m+1)+'d'
    for c in combinations_with_replacement(range(10),m+1):
        n = sum(d**m for d in c)
        r = sum(int(q)**m for q in str(n))
        rlist = sorted(int(d) for d in str(r))
        rlist = [0]*(m+1-len(rlist))+rlist
        if n < r and rlist == list(c):
            print(format(n,fs),format(r,fs)) # Chai Wah Wu, Jan 04 2016

Definition clarified by Chai Wah Wu, Jan 04 2016

A123253 Sum of 7th powers of digits of n.

Original entry on oeis.org

0, 1, 128, 2187, 16384, 78125, 279936, 823543, 2097152, 4782969, 1, 2, 129, 2188, 16385, 78126, 279937, 823544, 2097153, 4782970, 128, 129, 256, 2315, 16512, 78253, 280064, 823671, 2097280, 4783097, 2187, 2188, 2315, 4374, 18571, 80312, 282123

Offset: 0

Zerinvary Lajos, Nov 06 2006

Fixed points are listed in A124068 = row n=7 of A252648. - M. F. Hasler, Apr 12 2015

G. C. Greubel, Table of n, a(n) for n = 0..5000

Cf. A003132, A055012, A055013, A055014, A055015, A210840.

[0] cat [&+[d^7: d in Intseq(n)]: n in [1..40]]; // Bruno Berselli, Feb 01 2013

A123253 := proc(n)
        add(d^7,d=convert(n,base,10)) ;
end proc: # R. J. Mathar, Jan 16 2013

Table[Sum[DigitCount[n][[i]] i^7, {i, 9}], {n, 0, 40}] (* Bruno Berselli, Feb 01 2013 *)

A123253(n)=sum(i=1,#n=digits(n),n[i]^7) \\ M. F. Hasler, Apr 12 2015

A255668 Number of perfect digital invariants of order n, i.e., numbers equal to the sum of n-th powers of their digits.

Original entry on oeis.org

1, 10, 2, 6, 5, 8, 3, 7, 5, 6, 3, 10, 2, 3, 3, 2, 4, 6, 2, 6, 3, 4, 2, 7, 5, 10, 2, 9, 2, 9, 2, 6, 3, 5, 3, 6, 3, 5, 5, 7, 2, 2, 4, 9, 6, 9, 5, 7, 2, 3, 2, 4, 2, 3, 6, 4, 5, 4, 2, 4, 4, 4, 3, 7, 3, 6, 3, 4, 3, 3, 4, 3, 4, 5, 3, 4, 5, 5, 3, 3, 2, 3, 2, 4, 3, 8, 3, 5, 2, 7, 3

Offset: 0

M. F. Hasler, Apr 14 2015

Row lengths of the table A252648.

For a number with d digits, the sum of n-th powers cannot exceed d*9^n, but the number is not less than 10^(d-1). Therefore there is only a finite number of possible perfect digital invariants for any n, the largest of which has at most d* digits, where d* = 1+(n*log(9)+log d*)/log(10).

			a(0)=1 because 1 is the only number equal to the sum of 0th powers of its digits.
a(1)=10 because { 0, 1, ... 9 } are the only numbers equal to the sum of their digits (taken to the power 1).
a(2)=2 because 0 and 1 are the only numbers equal to the sum of the squares of their digits.
a(3)=6 because { 0, 1, 153, 370, 371, 407 } is the set of all numbers equal to the sum of the 3rd powers of their digits, cf. A046197.
For more examples, see the table A252648.

Don Knuth, Table of n, a(n) for n = 0..172
Table of a(n) for n=0..172 [From _Don Knuth_, Sep 09 2015]

Cf. A252648, A003321, A046197, A052455, A052464, A124068, A124069, A226970.

Reap@ For[n = 0, n < 6, n++, Sow@ Length@ Select[Range[0, 10^(n + 1)], Plus @@ (IntegerDigits[#]^n) == # &]] // Flatten // Rest (* Michael De Vlieger, Apr 14 2015 *)

a(n) >= 2 for all n > 0, since 0 and 1 are digital invariants for any power n > 0.

a(10)-a(90) from Don Knuth, Sep 09 2015

cp's OEIS Frontend

A262091 Amicable digital pairs: The smaller number of a pair (x,y) with x <> y such that, in decimal notation and with an appropriate number of leading zeros prepended, x=(x_m...x_1x_0){10}, y=(y_m...y_1y_0){10}, x = y_m^m + ... + y_1^m + y_0^m, and y = x_m^m + ... + x_1^m + x_0^m.

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A123253 Sum of 7th powers of digits of n.

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A255668 Number of perfect digital invariants of order n, i.e., numbers equal to the sum of n-th powers of their digits.

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Mathematica

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