A046017 -id:A046017 - cp's OEIS frontend

A060799 Erroneous version of A046017.

1, 9, 8, 7, 28, 18, 18, 46, 54, 82, 98, 108, 20, 91, 107, 133, 80, 172, 80, 90, 90, 90, 234, 252, 140, 306, 305, 90, 305, 396, 170, 388, 170, 387, 378, 388, 414, 468, 449, 250, 432, 280, 461, 280, 360, 360, 350, 370, 270, 685, 360, 625, 648, 370, 677, 684, 370

Offset: 1

dead

A152147 Irregular triangle in which row n lists k > 0 such that the sum of digits of k^n equals k.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 9, 1, 8, 17, 18, 26, 27, 1, 7, 22, 25, 28, 36, 1, 28, 35, 36, 46, 1, 18, 45, 54, 64, 1, 18, 27, 31, 34, 43, 53, 58, 68, 1, 46, 54, 63, 1, 54, 71, 81, 1, 82, 85, 94, 97, 106, 117, 1, 98, 107, 108, 1, 108, 1, 20, 40, 86, 103, 104, 106, 107, 126, 134, 135

Offset: 1

T. D. Noe, Nov 26 2008

Each row begins with 1 and has length A046019(n).

			1, 2, 3, 4, 5, 6, 7, 8, 9;
1, 9;
1, 8, 17, 18, 26, 27;              (A046459, with 0)
1, 7, 22, 25, 28, 36;              (A055575    "   )
1, 28, 35, 36, 46;                 (A055576    "   )
1, 18, 45, 54, 64;                 (A055577    "   )
1, 18, 27, 31, 34, 43, 53, 58, 68; (A226971    "   )
1, 46, 54, 63;
1, 54, 71, 81,
1, 82, 85, 94, 97, 106, 117,
1, 98, 107, 108, etc.

T. D. Noe, Rows n = 1..1000 of triangle, flattened

Cf. A046000, A046017, A046471, A133509.

def ok(k, r): return sum(map(int, str(k**r))) == k
def agen(rows, startrow=1, withzero=0):
  for r in range(startrow, rows + startrow):
    d, lim = 1, 1
    while lim < r*9*d: d, lim = d+1, lim*10
    yield from [k for k in range(1-withzero, lim+1) if ok(k, r)]
print([an for an in agen(13)]) # Michael S. Branicky, May 23 2021

A061210 Numbers which are the fourth powers of their digit sum.

Original entry on oeis.org

0, 1, 2401, 234256, 390625, 614656, 1679616

Offset: 1

Amarnath Murthy, Apr 21 2001

It can be shown that 1679616 = 36^4 is the largest such number.

			614656 = ( 6+1+4+6+5+6)^4 =28^4.

Amarnath Murthy, The largest and the smallest m-th power whose digit sum is the m-th root. (To be published)
Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, page 36.

Cf. A061209 (with cubes), A061211.

Cf. A046000, A076090, A046017; A252648 and references there.

Select[Range[0,17*10^5],#==Total[IntegerDigits[#]]^4&] (* Harvey P. Dale, Sep 22 2019 *)

isok(n) = n == sumdigits(n)^4; \\ Michel Marcus, Jan 22 2015

Corrected by Ulrich Schimke, Feb 11 2002

Initial 0 added by M. F. Hasler, Apr 12 2015

A061211 Largest number m such that m is the n-th power of the sum of its digits.

Original entry on oeis.org

9, 81, 19683, 1679616, 205962976, 68719476736, 6722988818432, 248155780267521, 150094635296999121, 480682838924478847449, 23316389970546096340992, 2518170116818978404827136, 13695791164569918553628942336, 4219782742781494680756610809856

Offset: 1

Amarnath Murthy, Apr 21 2001

Clearly m = 1 always works, so a(n) exists for all n. - Farideh Firoozbakht, Nov 23 2007

105 is the smallest number n such that a(n)=1. This means that if n<105 there exists at least one number m greater than 1 such that m is the n-th power of the sum of its digits while 1 is the only number m such that m is the 105th power of the sum of its digits. A133509 gives n such that a(n) = 1. - Farideh Firoozbakht, Nov 23 2007

			a(3) = 19683 = 27^3 and no bigger number can have this property. (This has been established in the Murthy reference.)
a(4) = 1679616 = (1+6+7+9+6+1+6)^4 = 36^4.

Amarnath Murthy, The largest and the smallest m-th power whose digits sum /product is its m-th root. To appear in Smarandache Notions Journal.
Amarnath Murthy, e-book, "Ideas on Smarandache Notions", manuscript.

T. D. Noe, Table of n, a(n) for n = 1..105

Cf. A061209, A061210, A046000, A076090, A046017.

meanDigit = 9/2; translate = 900; upperm[1] = translate;
upperm[n_] := Exp[-ProductLog[-1, -Log[10]/(meanDigit*n)]] + translate;
a[n_] := (For[max = m = 1, m <= upperm[n], m++, If[m == Total[ IntegerDigits[ m^n ] ], max = m]]; max^n);
Array[a, 14] (* Jean-François Alcover, Jan 09 2018 *)

More terms from Ulrich Schimke, Feb 11 2002

Edited by N. J. A. Sloane at the suggestion of Farideh Firoozbakht, Dec 04 2007

A046000 a(n) is the largest number m equal to the sum of digits of m^n.

Original entry on oeis.org

1, 9, 9, 27, 36, 46, 64, 68, 63, 81, 117, 108, 108, 146, 154, 199, 187, 216, 181, 207, 207, 225, 256, 271, 288, 337, 324, 307, 328, 341, 396, 443, 388, 423, 463, 477, 424, 495, 469, 523, 502, 432, 531, 572, 603, 523, 592, 666, 667, 695, 685, 685, 739, 746, 739, 683, 684, 802, 754, 845, 793, 833, 865

Offset: 0

David W. Wilson and Patrick De Geest

Cases a(n) = 1 begin: 0, 105, 164, 186, 194, 206, 216, 231, 254, 282, 285, ... Cf. A133509. - Jean-François Alcover, Jan 09 2018

			a(3) = 27 because 27 is the largest number with 27^3 = 19683 and 1+9+6+8+3 = 27.
a(5) = 46 because 46 is the largest number with 46^5 = 205962976 and 2+0+5+9+6+2+9+7+6 = 46.

Amarnath Murthy, The largest and the smallest m-th power whose digits sum /product is its m-th root. To appear in Smarandache Notions Journal, 2003.
Amarnath Murthy, e-book, "Ideas on Smarandache Notions" MS.LIT
Joe Roberts, "Lure of the Integers", The Mathematical Association of America, 1992, p. 172.

T. D. Noe, Table of n, a(n) for n = 0..1000 (a(105), a(164), a(186), ... corrected by Mohammed Yaseen)

Cf. A046459, A055575, A055576, A055577.
Cf. A046017, A046019, A046471, A133509, A152147.
Cf. A061211, A076090.

meanDigit = 9/2; translate = 900; upperm[1] = translate;
upperm[n_] := Exp[-ProductLog[-1, -Log[10]/(meanDigit*n)]] + translate;
(* assuming that upper bound of m fits the implicit curve m = Log[10, m^n]*9/2 *)
a[0] = 1; a[n_] := (For[max = m = 0, m <= upperm[n], m++, If[m == Total[IntegerDigits[m^n]], max = m]]; max);
Table[a[n], {n, 0, 1000}] (* Jean-François Alcover, Jan 09 2018, updated Jul 07 2022 *)

def ok(k, n): return sum(map(int, str(k**n))) == k
def a(n):
    d, lim = 1, 1
    while lim < n*9*d: d, lim = d+1, lim*10
    return next(k for k in range(lim, 0, -1) if ok(k, n))
print([a(n) for n in range(63)]) # Michael S. Branicky, Jul 06 2022

a(n) = A061211(n)^(1/n), for n > 0.

More terms from Asher Auel, Jun 01 2000
More terms from Franklin T. Adams-Watters, Sep 01 2006
Edited by N. J. A. Sloane at the suggestion of David Wasserman, Dec 12 2007

A133509 Numbers k such that m=1 is the only number for which the sum of digits of m^k equals m.

Original entry on oeis.org

0, 105, 164, 186, 194, 206, 216, 231, 254, 282, 285, 302, 314, 324, 374, 386, 402, 416, 456, 468, 491, 504, 521, 552, 588, 606, 610, 615, 629, 651, 656, 657, 696, 759, 794, 830, 842, 854, 870, 903, 906, 954, 956, 981, 998, 1029, 1064, 1079, 1082, 1109, 1112, 1131

Offset: 1

Farideh Firoozbakht, Dec 04 2007

Carole Dubois, Table of n, a(n) for n = 1..226 (terms 1..100 from Michael S. Branicky)
Carole Dubois, Pin plot of A133509

Cf. A046000, A046017, A046019, A046471, A061211, A152147.

def ok(n):
    d, lim = 1, 1
    while lim < n*9*d: d, lim = d+1, lim*10
    return not any(sum(map(int, str(k**n))) == k for k in range(2, lim+1))
for k in range(195):
    if ok(k): print(k, end=", ") # Michael S. Branicky, Jul 06 2022

If t is a term, A046000(t)=1, A046017(t)=0, A046019(t)=1, A046471(t)=0 and A061211(t)=1. - Mohammed Yaseen, Jun 29 2022

Description improved by T. D. Noe, Nov 26 2008
Extension by T. D. Noe, Nov 26 2008
Edited by Charles R Greathouse IV, Aug 02 2010
a(1) = 0 and a(46) and beyond from Michael S. Branicky, Jul 06 2022

A072408 Least number > 1 which equals n-th power of the sum of its digits in decimal base.

Original entry on oeis.org

2, 81, 512, 2401, 17210368, 34012224, 612220032, 20047612231936, 3904305912313344, 13744803133596058624, 8007313507497959524352, 2518170116818978404827136, 81920000000000000, 2670419511272061205254504361, 2759031540715333904109053133443

Offset: 1

Labos Elemer, Jun 17 2002

			a(1) = 2 > 1;
a(9) = 3904305912313344 = 54^9 = (3+9+0+4+3+0+5+9+1+2+3+1+3+3+4+4)^9.

Michael De Vlieger, Table of n, a(n) for n = 1..274
Leo Moser, Digits of Powers, Recreational Mathematics, Curiosa 145. Scripta Mathematica Vol. XIII, March-June 1947, Nos. 1-2, page 117.

Cf. A007953, A046017, A072081-A072083.

Array[Block[{k = 2}, While[k^# != Total[IntegerDigits[k^#]]^#, k++]; k^#] &, 13] (* Michael De Vlieger, Nov 05 2020 *)

a(n) = Min{ x; x=SumDigit(x)^n} = Min{x; x=A007953(x)^n}

a(n) = A046017(n)^n. - Michael De Vlieger, Nov 05 2020

A076090 a(1) = 1; for n > 1, smallest number > 1 which is the n-th power of the sum of its digits, or 0 if no such number exists.

Original entry on oeis.org

1, 81, 512, 2401, 17210368, 34012224, 612220032, 20047612231936, 3904305912313344, 13744803133596058624, 8007313507497959524352, 2518170116818978404827136, 81920000000000000

Offset: 1

Amarnath Murthy, Oct 07 2002

			a(3) = 512 = 8^3 and no smaller number (>1) can have this property. (This has been established in the reference.)

Amarnath Murthy, The largest and the smallest m-th power whose digits sum /product is its m-th root. To appear in Smarandache Notions Journal.
Amarnath Murthy, e-book, " Ideas on Smarandache Notions " MS.LIT

Cf. A046017.

For n > 1, a(n) = A046017(n)^n. - David Wasserman, Mar 25 2005

a(n)=A072408(n), n>1. [From R. J. Mathar, Sep 23 2008]

More terms from David Wasserman, Mar 25 2005

A212667 Numbers n such that the sum of digits of n equals the concatenation of the distinct prime divisors of n.

Original entry on oeis.org

2, 3, 5, 7, 2401, 4913, 655360, 3906250, 6553600, 39062500, 41943040, 65536000, 390625000, 419430400, 655360000, 3906250000, 4194304000, 6553600000, 27512614111, 39062500000, 41943040000, 65536000000, 271818611107, 390625000000, 419430400000

Offset: 1

Michel Lagneau, May 23 2012

The sequence is infinite because 3906250 = 2*5^9 is in the sequence => 2^(1+p) * 5^(9+p) = 39062500….0 is also in the sequence.

The prime numbers of A046017 are included in this sequence. For example A046017(4) = 7 => 7^4 = 2401 is in this sequence.

			655360 is in the sequence because 655360 = 2^17 * 5 => the concatenation of the prime divisors is the number 25 and 6+5+5+3+6+0 = 25.

Cf. A046017.

with(numtheory):for n from 1 to 10^8 do: V:=convert(n, base, 10): n1:=nops(V): s1:=sum(‘V[m]’, ‘m’=1..n1):x:=factorset(n):n1:=nops(x): s:=0:s0:=0:for i from n1 by -1 to 1 do: a:=x[i]:b:=length(a):s:=s+a*10^s0:s0:=s0+b:od: if s=s1 then print(n):else fi:od:

cp's OEIS Frontend

A060799 Erroneous version of A046017.

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A152147 Irregular triangle in which row n lists k > 0 such that the sum of digits of k^n equals k.

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A061210 Numbers which are the fourth powers of their digit sum.

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A061211 Largest number m such that m is the n-th power of the sum of its digits.

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A046000 a(n) is the largest number m equal to the sum of digits of m^n.

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A133509 Numbers k such that m=1 is the only number for which the sum of digits of m^k equals m.

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A072408 Least number > 1 which equals n-th power of the sum of its digits in decimal base.

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A076090 a(1) = 1; for n > 1, smallest number > 1 which is the n-th power of the sum of its digits, or 0 if no such number exists.

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