A124365 -id:A124365 - cp's OEIS frontend

A125526 Numbers k for which the sum of the digits of k raised to the sum of the digits of k itself is equal to k. If "sumdigit" denotes the sum of the digits of a number then these are the numbers k such that k = sumdigit(k^sumdigit(k)).

1, 22, 34, 43, 54, 81, 82, 169, 187

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, Jan 22 2007

There are no other terms. Proof: Assume the next term has d digits. 10^d > k >= 10^(d-1); sumdigit(k) >= 9d; k^sumdigit(k) < (10^d)^(9d) < 10^(9d^2); 9*(9d^2+1) > sumdigit(k^sumdigit(k)); 9*(9d^2+1) > k 9*(9d^2+1) > 10^(d-1). So d < 5. - Fung Cheok Yin (cheokyin_restart(AT)yahoo.com.hk), Mar 11 2007

			a(2)=22 because 2 + 2 = 4, 22^4 = 234256, 2 + 3 + 4 + 2 + 5 + 6 = 22.

Cf. A124053, A124359, A124360, A046019, A124365, A124366, A124367.

P:=proc(n) local i,j,k,w; for i from 1 by 1 to n do w:=0; k:=i; while k>0 do w:=w+k-trunc(k/10)*10; k:=trunc(k/10); od; k:=i^w; w:=0; while k>0 do w:=w+k-trunc(k/10)*10; k:=trunc(k/10); od; if (i=w) then print(w); fi; od; end: P(200);
sod := proc(n,b) convert(convert(n,base,b),`+`) end; b:=10: L:=[]: for w to 1 do for n from 1 to 10^3 do x:=sod(n^sod(n,b),b); if x=n then print(n); L:=[op(L),n]; fi; od od; L; # Walter Kehowski, Feb 12 2007
sd:=proc(n) local nn: nn:=convert(n,base,10): sum(nn[j],j=1..nops(nn)) end: a:=proc(n) if sd(n^sd(n))=n then n else fi end: seq(a(n),n=1..500); # Emeric Deutsch, Feb 16 2007

Select[Range[200],Total[IntegerDigits[#^Total[IntegerDigits[#]]]]==#&] (* Harvey P. Dale, Jul 26 2019 *)

A124367 Numbers that raised to any exponent do not produce a number whose sum of digits is equal to the initial number.

Original entry on oeis.org

10, 11, 12, 13, 14, 15, 16, 19, 21, 23, 24, 29, 30, 32, 33, 37, 38, 39, 41, 42, 44, 47, 48, 49, 50, 51, 52, 55, 56, 57, 59, 60, 61, 62, 65, 66, 67, 69, 70, 72, 73, 74, 75, 76, 77, 78, 79, 83, 84, 87, 88, 89, 92, 93, 95, 96, 99, 100, 101, 102, 105

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, Oct 30 2006

Complement of A124359. Numbers for which A247889 is zero.

Most of the values are conjectural, so far not much is really proved about the function A247889. - M. F. Hasler and Robert Israel, May 18 2017

Cf. A124359, A124360, A046019, A124365, A124366, A247889.

for(n=0,199,if(!A247889(n),print1(n,", "))) \\ Derek Orr, Sep 25 2014, edited by M. F. Hasler, May 18 2017

106 removed by Robert Israel, May 18 2017

A124366 Consecutive numbers n and (n+1) that raised to the same exponent m produce two numbers for which the sum of their digits gives n and (n+1).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 17, 18, 35, 36, 103, 104, 106, 107, 108, 134, 135, 256, 257, 295, 296, 298, 299, 306, 307, 386, 387, 421, 422, 468, 469, 575, 576, 792, 793, 865, 866, 962, 963, 1008, 1009, 1061, 1062, 1476, 1477, 1495, 1496, 2032, 2033, 2376, 2377

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, Oct 30 2006

			17^3=4913 (4+9+1+3=17) and 18^3=5832 (5+8+3+2=18)
306^26= 42536043213832457558766474492498614961439017908885402928656941056
(sum of the digits equal to 306) and
307^26= 46301788027092145989912680349353041288862842956233592928809850249
(sum of the digits equal to 307)

Cf. A124359, A124360, A124361, A124362, A046019, A124364, A124365.

A124053 Numbers n that can be expressed as the sum of the digits of both m^k and k^m for distinct numbers m and k which are not both equal to powers of 10.

Original entry on oeis.org

7, 18, 45, 61, 72, 85, 90, 145, 270, 306, 315, 367, 376, 448, 477, 540, 547, 585, 667, 733, 756, 765, 943, 1152, 1377, 1899, 1971, 2106, 2133, 2155, 2215, 2224, 2349, 2628, 2822, 2871, 2968, 3123, 3139, 3181, 3204, 3355, 3546, 3553, 3775, 3780, 4131, 4455

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, Nov 03 2006, Nov 29 2006

If "sumdigit" denotes the sum of the digits of a number then these are the numbers n such that n=sumdigit(m^k)=sumdigit(k^m).

Two banal cases are not considered: 1) m=k because m^k=k^m and the sum of the digits is automatically equal for both the numbers; 2) powers of 10 because sumdigit(10^a)=1 for any integer a. The same number can be generated by different pairs: 477 cames from sumdigit(54^63)=sumdigit(63^54) and sumdigit(90^120)=sumdigit(120^90) 2349 cames from sumdigit(216^222)=sumdigit(222^216), sumdigit(216^225)=sumdigit(225^216) and sumdigit(219^222)=sumdigit(222^219)

			270 = sumdigit(36^39) = sumdigit(39^36);
1152 = sumdigit(114^126) = sumdigit(126^114);
2133 = sumdigit(204^213) = sumdigit(213^204).

Cf. A124359, A124360, A046019, A124365, A124366, A124367.

P:=proc(n)local i,j,k,w,x,y; for i from 1 by 1 to n do for j from 1 by 1 to n do w:=0; x:=0; k:=i^j; y:=j^i; while k>0 do w:=w+k-trunc(k/10)*10; k:=trunc(k/10); od; while y>0 do x:=x+y-trunc(y/10)*10; y:=trunc(y/10); od; if (w=x) and (w<>1) and (i

cp's OEIS Frontend

A125526 Numbers k for which the sum of the digits of k raised to the sum of the digits of k itself is equal to k. If "sumdigit" denotes the sum of the digits of a number then these are the numbers k such that k = sumdigit(k^sumdigit(k)).

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Mathematica

A124367 Numbers that raised to any exponent do not produce a number whose sum of digits is equal to the initial number.

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PARI

Extensions

A124366 Consecutive numbers n and (n+1) that raised to the same exponent m produce two numbers for which the sum of their digits gives n and (n+1).

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A124053 Numbers n that can be expressed as the sum of the digits of both m^k and k^m for distinct numbers m and k which are not both equal to powers of 10.

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Maple