cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-9 of 9 results.

A046019 a(n) gives the number of different powers m^n for which the sum of the digits is equal to m.

Original entry on oeis.org

1, 9, 2, 6, 6, 5, 5, 9, 4, 4, 7, 4, 2, 12, 6, 8, 7, 5, 3, 10, 4, 4, 8, 4, 4, 14, 5, 3, 7, 6, 2, 11, 2, 8, 4, 6, 3, 9, 3, 3, 7, 2, 5, 10, 6, 4, 9, 9, 5, 12, 2, 4, 5, 5, 6, 3, 2, 7, 4, 5, 5, 6, 3, 4, 5, 5, 4, 9, 2, 6, 4, 3, 3, 6, 5, 6, 4, 4, 5, 9, 5, 3, 5, 5, 2, 6, 3, 7, 7, 4, 3, 8, 4, 4, 9, 6, 2, 8, 2, 5, 6, 3
Offset: 0

Views

Author

David W. Wilson

Keywords

Comments

Number of m >= 1 with m = sum of digits of m^n.

Examples

			a(7)=9 because:
1^7=1
18^7= 612220032 and 6+1+2+2+2+3+2=18
27^7= 10460353203 and 1+4+6+3+5+3+2+3=27
31^7= 27512614111 and 2+7+5+1+2+6+1+4+1+1+1=31
34^7= 52523350144 and 5+2+5+2+3+3+5+1+4+4=34
43^7= 271818611107 and 2+7+1+8+1+8+6+1+1+1+7=43
53^7= 1174711139837 and 1+1+7+4+7+1+1+1+3+9+8+3+7=53
58^7= 2207984167552 and 2+2+7+9+8+4+1+6+7+5+5+2=58
68^7= 6722988818432 and 6+7+2+2+9+8+8+8+1+8+4+3+2=68
a(9)=4 because:
1^9=1
54^9=3904305912313344 and 3+9+4+3+5+9+1+2+3+1+3+3+4+4=54
71^9=45848500718449031 and 4+5+8+4+8+5+7+1+8+4+4+9+3+1=71
81^9=150094635296999121 and 1+5+9+4+6+3+5+2+9+6+9+9+9+1+2+1=81

Links

Crossrefs

Cf. A124359, A152147 (table of m such that the sum of digits of m^n equals m)

Formula

a(n) = 1 + A046471(n). - T. D. Noe, Nov 26 2008

Extensions

Examples provided by Paolo P. Lava, Oct 30 2006
Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, May 27 2007

A124360 Numbers n for which the sum of the digits of n^k and the sum of the digits of n^(k+1), for some k, is equal to n.

Original entry on oeis.org

1, 9, 28, 36, 54, 90, 108, 135, 154, 181, 207, 225, 234, 307, 360, 388, 469, 523, 540, 685, 720
Offset: 0

Views

Author

Paolo P. Lava and Giorgio Balzarotti, Oct 27 2006

Keywords

Examples

			18^6=34012224 (3+4+0+1+2+2+2+4=18) and 18^7=612220032 (6+1+2+2+2+0+0+3+2=18)
Again 36^4=1679616 (1+6+7+9+6+1+6=36) and 36^5=60466176 (6+0+4+6+6+1+7+6=36)

Crossrefs

Cf. A124359.

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)).

Original entry on oeis.org

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

Views

Author

Paolo P. Lava and Giorgio Balzarotti, Jan 22 2007

Keywords

Comments

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

Examples

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

Crossrefs

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

Programs

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

A247889 Least number k > 0 such that digsum(n^k) = n, or 0 if no such k exists.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 3, 3, 0, 13, 0, 4, 0, 0, 4, 3, 3, 4, 0, 0, 7, 0, 0, 7, 5, 4, 0, 0, 0, 13, 0, 0, 7, 0, 6, 5, 0, 0, 0, 0, 0, 0, 7, 6, 0, 0, 0, 7, 0, 0, 0, 0, 8, 6, 0, 0, 0, 7, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0, 17, 9, 10, 0, 0, 10, 13, 0, 0, 0, 19, 14, 0, 0, 10, 0, 0, 10, 11, 0, 0
Offset: 0

Views

Author

Derek Orr, Sep 25 2014

Keywords

Comments

a(10^n) = 0 for all n > 0.
a(n) = 0 if and only if n is in A124367, complement of A124359.
The PARI code uses that, if sumdigit(n^k) <> n until sumdigits(n^k) > 2n for the first time, then sumdigits(n^k) <> n for all larger k. This might not be true for all n, although statistically sumdigits(n^k) ~ 4.5*log_10(n')*k within a few % (n' = n with trailing 0's removed), when k is not too small. - M. F. Hasler, May 18 2017

Links

Crossrefs

Cf. A124359 (a(n) > 0), A124367 (a(n) = 0), A007953, A152147.

Programs

  • PARI
    A247889(n,L=2*n)= if(n<10||vecmin(digits(n-1))==9,return(n<10));k=1;while(L >= s=sumdigits(n^k),if(s,return(k));k++) \\ Renamed to A247889 for use in A124359, A124367 (and elsewhere?), and minor edits by M. F. Hasler, May 18 2017
apply(A247889,[0..100])

Extensions

Data corrected (initial 1 removed, terminal 0 added) by Hans Havermann, Jul 13 2018

A124365 Numbers that raised to only one specific exponent gives a result for which the sum of its digits is equal to number itself.

Original entry on oeis.org

2, 3, 4, 5, 6, 17, 20, 22, 25, 26, 31, 34, 35, 40, 43, 45, 53, 58, 63, 64, 68, 71, 81, 82, 85, 86, 91, 94, 97, 98, 103, 104, 117
Offset: 1

Views

Author

Paolo P. Lava and Giorgio Balzarotti, Oct 30 2006

Keywords

Comments

Subset of A124359

Examples

			31^7=27512614111 and 2+7+5+1+2+6+1+4+1+1+1=31. This is possible only with 7 as exponent.
97^10=73742412689492826049 and 7+3+7+4+2+4+1+2+6+8+9+4+9+2+8+2+6+4+9=97. This is possible only with 10 as exponent.

Crossrefs

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

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

Views

Author

Paolo P. Lava and Giorgio Balzarotti, Oct 30 2006

Keywords

Comments

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

Crossrefs

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

Programs

Extensions

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

Views

Author

Paolo P. Lava and Giorgio Balzarotti, Oct 30 2006

Keywords

Examples

			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)

Crossrefs

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

Views

Author

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

Keywords

Comments

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)

Examples

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

Crossrefs

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

Programs

  • Maple
    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

A273844 Least number k such that j*Sd(k) = Sd(k^j) for 1 <= j <= n, where Sd(k) is the sum of the digits of k.

Original entry on oeis.org

1, 2, 27, 36, 405, 11682, 33651, 669024, 25274655, 40809168, 59716233, 7932651138, 367732181646
Offset: 1

Views

Author

Paolo P. Lava, Jun 01 2016

Keywords

Comments

a(14) > 10^12. - Giovanni Resta, Jun 02 2016

Examples

			For n=2, 2*Sd(2) = 2*2 = 4 and Sd(2^2) = Sd(4) = 4.
For n=3, 2*Sd(27) = 2*9 = 18 and Sd(27^2) = Sd(729) = 18;
     and 3*Sd(27) = 3*9 = 27 and Sd(27^3) = Sd(19683) = 27.

Crossrefs

Cf. A007953, A124359.

Programs

  • Maple
    T:=proc(w) local x, y, z; x:=w; y:=0; for z from 1 to ilog10(x)+1 do
y:=y+(x mod 10); x:=trunc(x/10); od; y; end:
P:=proc(q) local a,j,k,n,ok,t; t:=1; for k from 1 to q do
for n from t to q do ok:=1; a:=T(n);
for j from 2 to k do if j*a<>T(n^j) then ok:=0; break; fi; od;
if ok=1 then t:=n; print(n); break; fi; od; od; end: P(10^20);

Extensions

a(12)-a(13) from Giovanni Resta, Jun 02 2016
Showing 1-9 of 9 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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