A014576 -id:A014576 - cp's OEIS frontend

A001694 Powerful numbers, definition (1): if a prime p divides n then p^2 must also divide n (also called squareful, square full, square-full or 2-powerful numbers).

1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 72, 81, 100, 108, 121, 125, 128, 144, 169, 196, 200, 216, 225, 243, 256, 288, 289, 324, 343, 361, 392, 400, 432, 441, 484, 500, 512, 529, 576, 625, 648, 675, 676, 729, 784, 800, 841, 864, 900, 961, 968, 972, 1000

Offset: 1

N. J. A. Sloane

Numbers of the form a^2*b^3, a >= 1, b >= 1.
In other words, if the prime factorization of n is Product_k p_k^e_k then all e_k are greater than 1.
Numbers n such that Sum_{d|n} phi(d)*phi(n/d)*mu(d) > 0; places of nonzero A300717. - Benoit Cloitre, Nov 30 2002
This sequence is closed under multiplication. The primitive elements are A168363. - Franklin T. Adams-Watters, May 30 2011
Complement of A052485. - Reinhard Zumkeller, Sep 16 2011
The number of terms less than or equal to 10^k beginning with k = 0: 1, 4, 14, 54, 185, 619, 2027, 6553, 21044, ...: A118896. - Robert G. Wilson v, Aug 11 2014
a(10^n): 1, 49, 3136, 253472, 23002083, 2200079025, 215523459072, 21348015504200, 2125390162618116, ... . - Robert G. Wilson v, Aug 15 2014
a(m) mod prime(n) > 0 for m < A258599(n); a(A258599(n)) = A001248(n) = prime(n)^2. - Reinhard Zumkeller, Jun 06 2015
From Des MacHale, Mar 07 2021: (Start)
A number m is powerful if and only if |R/Z(R)| = m, for some finite non-commutative ring R.
A number m is powerful if and only if |G/Z(G)| = m, for some finite nilpotent class two group G (Reference Aine Nishe). (End)
Numbers n such that Sum_{k=1..n} phi(gcd(n,k))*mu(gcd(n,k)) > 0. - Richard L. Ollerton, May 09 2021

			1 is a term because for every prime p that divides 1, p^2 also divides 1.
2 is not a term since 2 divides 2 but 2^2 does not.
4 is a term because 2 is the only prime that divides 4 and 2^2 does divide 4. - _N. J. A. Sloane_, Jan 16 2022

G. E. Hardy and M. V. Subbarao, Highly powerful numbers, Congress. Numer. 37 (1983), 277-307.
Aleksandar Ivić, The Riemann Zeta-Function, Wiley, NY, 1985, see p. 407.
Richard A. Mollin, Quadratics, CRC Press, 1996, Section 1.6.
Aine NiShe, Commutativity and Generalisations in Finite Groups, Ph.D. Thesis, University College Cork, 2000.
Paulo Ribenboim, Meine Zahlen, meine Freunde, 2009, Springer, 9.1 Potente Zahlen, pp. 241-247.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Gérald Tenenbaum, Introduction to analytic and probabilistic number theory, Cambridge University Press, 1995, p. 54, exercise 10 (in the third edition 2015, p. 63, exercise 70).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe, terms 1001..5000 from G. C. Greubel)
Paul T. Bateman and Emil Grosswald, On a theorem of Erdős and Szekeres, Illinois J. Math. 2:1 (1958), pp. 88-98.
Chris Bispels, Matthew Cohen, Joshua Harrington, Joshua Lowrance, Kaelyn Pontes, Leif Schaumann, and Tony W. H. Wong, A further investigation on covering systems with odd moduli, arXiv:2507.16135 [math.NT], 2025. See p. 3.
Valentin Blomer, Binary quadratic forms with large discriminants and sums of two squareful numbers II, Journal of the London Mathematical Society 71:1 (2005), pp. 69-84.
Chris K. Caldwell, Powerful Numbers.
Tsz Ho Chan, Arithmetic Progressions Among Powerful Numbers, J. Int. Seq., Vol. 26 (2023), Article 23.1.1.
Tsz Ho Chan, A note on three consecutive powerful numbers, arXiv:2503.21485 [math.NT], 2025.
Jean-Marie De Koninck, Nicolas Doyon, and Florian Luca, Powerful Values of Quadratic Polynomials, J. Int. Seq. 14 (2011), Article 11.3.3.
Paul Erdős and George Szekeres, Über die Anzahl der Abelschen Gruppen gegebener Ordnung und über ein verwandtes zahlentheoretisches Problem, Acta Sci. Math. (Szeged), 7 (1935), 95-102. [Zahlen i-ter Art, p. 101]
Solomon W. Golomb, Powerful numbers, Amer. Math. Monthly, Vol. 77 (1970), 848-852.
K. Schneider, PlanetMath.org, Squarefull Number.
Vladimir Shevelev, S-exponential numbers, Acta Arithmetica, Vol. 175(2016), 385-395.
D. Suryanarayana and R. Sita Rama Chandra Rao, The distribution of square-full integers, Ark. Mat., Volume 11, Number 1-2 (1973), 195-201.
Eric Weisstein's World of Mathematics, Powerful Number.
Eric Weisstein's World of Mathematics, Squareful.
Wikipedia, Powerful number.
Index entries for sequences related to powerful numbers.

Disjoint union of A062503 and A320966.
Cf. A007532 (Powerful numbers, definition (2)), A005934, A005188, A003321, A014576, A023052 (Powerful numbers, definition (3)), A046074, A013929, A076871, A258599, A001248, A112526, A168363, A224866, A261883, A300717.
Cf. A052485 (complement), A076446 (first differences), A376361, A376362.

a001694 n = a001694_list !! (n-1)
a001694_list = filter ((== 1) . a112526) [1..]
-- Reinhard Zumkeller, Nov 30 2012

isA001694 := proc(n) for p in ifactors(n)[2] do if op(2,p) = 1 then return false; end if; end do; return true; end proc:
A001694 := proc(n) option remember; if n = 1 then 1; else for a from procname(n-1)+1 do if isA001694(a) then return a; end if; end do; end if; end proc:
seq(A001694(n),n=1..20) ; # R. J. Mathar, Jun 07 2011

Join[{1}, Select[ Range@ 1250, Min@ FactorInteger[#][[All, 2]] > 1 &]]
(* Harvey P. Dale, Sep 18 2011; modified by Robert G. Wilson v, Aug 11 2014 *)
max = 10^3; Union@ Flatten@ Table[a^2*b^3, {b, max^(1/3)}, {a, Sqrt[max/b^3]}] (* Robert G. Wilson v, Aug 11 2014 *)
nextPowerfulNumber[n_] := Block[{r = Range[ Floor[1 + n^(1/3)]]^3}, Min@ Select[ Sort[ r*Floor[1 + Sqrt[n/r]]^2], # > n &]]; NestList[ nextPowerfulNumber, 1, 55] (* Robert G. Wilson v, Aug 16 2014 *)

isA001694(n)=n=factor(n)[,2];for(i=1,#n,if(n[i]==1,return(0)));1 \\ Charles R Greathouse IV, Feb 11 2011

list(lim,mn=2)=my(v=List(),t); for(m=1,sqrtnint(lim\1,3), t=m^3; for(n=1,sqrtint(lim\t), listput(v,t*n^2))); Set(v) \\ Charles R Greathouse IV, Jul 31 2011; edited Sep 22 2015

is=ispowerful \\ Charles R Greathouse IV, Nov 13 2012

from sympy import factorint
A001694 = [1]+[n for n in range(2,10**6) if min(factorint(n).values()) > 1]
# Chai Wah Wu, Aug 14 2014

from math import isqrt
from sympy import mobius, integer_nthroot
def A001694(n):
    def squarefreepi(n):
        return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        c, l = n+x, 0
        j = isqrt(x)
        while j>1:
            k2 = integer_nthroot(x//j**2,3)[0]+1
            w = squarefreepi(k2-1)
            c -= j*(w-l)
            l, j = w, isqrt(x//k2**3)
        c -= squarefreepi(integer_nthroot(x,3)[0])-l
        return c
    return bisection(f,n,n) # Chai Wah Wu, Sep 09 2024

sloane.A001694.list(54) # Peter Luschny, Feb 08 2015

A112526(a(n)) = 1. - Reinhard Zumkeller, Sep 16 2011
Bateman & Grosswald prove that there are zeta(3/2)/zeta(3) x^{1/2} + zeta(2/3)/zeta(2) x^{1/3} + O(x^{1/6}) terms up to x; see section 5 for a more precise error term. - Charles R Greathouse IV, Nov 19 2012
a(n) = A224866(n) - 1. - Reinhard Zumkeller, Jul 23 2013
Sum_{n>=1} 1/a(n) = zeta(2)*zeta(3)/zeta(6). - Ivan Neretin, Aug 30 2015
Sum_{n>=1} 1/a(n)^s = zeta(2*s)*zeta(3*s)/zeta(6*s), s > 1/2 (Golomb, 1970). - Amiram Eldar, Oct 02 2022

More terms from Henry Bottomley, Mar 16 2000

Definition expanded by Jonathan Sondow, Jan 03 2016

A005188 Armstrong (or pluperfect, or Plus Perfect, or narcissistic) numbers: m-digit positive numbers equal to sum of the m-th powers of their digits.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 153, 370, 371, 407, 1634, 8208, 9474, 54748, 92727, 93084, 548834, 1741725, 4210818, 9800817, 9926315, 24678050, 24678051, 88593477, 146511208, 472335975, 534494836, 912985153, 4679307774, 32164049650, 32164049651

Offset: 1

N. J. A. Sloane, Robert G. Wilson v

A finite sequence, the 88th and last term being 115132219018763992565095597973971522401.
Let k = d_1 d_2 ... d_n in base 10; then k is in the sequence iff k = Sum_{i=1..n} d_i^n.
These are the fixed points in the "Recurring Digital Invariant Variant" described in A151543.
a(15) = A229381(3) = 8208 is the "Simpsons' narcissistic number".
If a(n) is a multiple of 10, then a(n+1) = a(n) + 1, and if a(n) == 1 (mod 10) then a(n-1) = a(n) - 1 except for n = 1, cf. Examples. - M. F. Hasler, Oct 18 2018
Named after Michael Frederick Armstrong (1941-2020), who used these numbers in his computing class at the University of Rochester in the mid 1960's. - Amiram Eldar, Mar 09 2024

			153 = 1^3 + 5^3 + 3^3,
8208 = 8^4 + 2^4 + 0^4 + 8^4,
4210818 = 4^7 + 2^7 + 1^7 + 0^7 + 8^7 + 1^7 + 8^7.
The eight terms 370, 24678050, 32164049650, 4338281769391370, 3706907995955475988644380, 19008174136254279995012734740, 186709961001538790100634132976990 and 115132219018763992565095597973971522400 end in a digit zero, therefore their successor a(n) + 1 is the next term a(n+1). This also yields the last term of the sequence. The initial a(1) = 1 is the only term ending in a digit 1 not preceded by a(n) - 1. - _M. F. Hasler_, Oct 18 2018

Jean-Marie De Koninck, Ces nombres qui nous fascinent, Entry 88, pp. 30-31, Ellipses, Paris 2008.
Lionel E. Deimel, Jr. and Michael T. Jones, Finding Pluperfect Digital Invariants: Techniques, Results and Observations, J. Rec. Math., 14 (1981), 87-108.
Jean-Pierre Lamoitier, Fifty Basic Exercises. SYBEX Inc., 1981.
Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 68.
Alfred S. Posamentier, Numbers: Their Tales, Types, and Treasures, Prometheus Books, 2015, pp. 242-244.
Joe Roberts, The Lure of the Integers, The Mathematical Association of America, 1992, page 36.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..88 (the full list of terms, from Winter)
Anonymous, Narcissistic number.
Michael F. Armstrong, A Brief Introduction to Armstrong Numbers.
Pat Ballew, The Cubic Attractiveness of 153, Pat's Blog, May 30, 2023.
Hans J. de Jong, Letter to N. J. A. Sloane, Mar 8 1988.
Lionel E. Deimel, Armstrong Numbers.
Lionel E. Deimel, Mystery Solved!, Lionel Deimel's Web Log, May 5, 2010.
Lionel E. Deimel, Narcissistic Numbers.
Martin Gardner & N. J. A. Sloane, Correspondence, 1973-74.
Shyam Sunder Gupta, Digital Invariants and Narcissistic Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 21, 513-526.
Harvey Heinz, Narcissistic Numbers (backup from March 2018 on web/archive.org: page no longer available), Sep. 1998, last updated in Sep. 2010.
History of Science and Mathematics StackExchange, Armstrong numbers - who is or was Armstrong?, 2021.
L. H. & W. Lopez, PlanetMath.Org, Armstrong number (latest backup on web.archive.org of ArmstrongNumber.html from 2012), published by L.H. not later than July 2007.
Gordon L. Miller and Mary T. Whalen, Armstrong Numbers: 153 = 1^3 + 5^3 + 3^3, Fibonacci Quarterly, 30-3 (1992), 221-224.
Tomas Antonio Mendes Oliveira e Silva (tos(AT)ci.ua.pt), Loneliness of the Factorions, gave the full sequence in a posting (Article 42889) to sci.math on May 09 1994.
Walter Schneider, Perfect Digital Invariants: Pluperfect Digital Invariants(PPDIs)
B. Shader, Armstrong number.
Eric Weisstein's World of Mathematics, Narcissistic Number.
Wikipedia, Narcissistic number.
Robert G. Wilson v, Letter to N. J. A. Sloane, Jan 23 1989.
D. T. Winter, Table of Armstrong Numbers (latest backup on web.archive.org from Jan. 2010; page no longer available), published not later than Aug. 2003.

Cf. A001694, A007532, A005934, A003321, A014576, A046074.
Similar to but different from A023052.
Cf. A151543.
Cf. A010343 to A010354 (bases 4 to 9). - R. J. Mathar, Jun 28 2009

filter:= proc(k) local d;
d:= 1 + ilog10(k);
add(s^d, s=convert(k,base,10)) = k
end proc:
select(filter, [$1..10^6]); # Robert Israel, Jan 02 2015

f[n_] := Plus @@ (IntegerDigits[n]^Floor[ Log[10, n] + 1]); Select[ Range[10^7], f[ # ] == # &] (* Robert G. Wilson v, May 04 2005 *)
Select[Range[10^7],#==Total[IntegerDigits[#]^IntegerLength[#]]&] (* Harvey P. Dale, Sep 30 2011 *)

is(n)=my(v=digits(n));sum(i=1,#v,v[i]^#v)==n \\ Charles R Greathouse IV, Nov 20 2012

select( is_A005188(n)={n==vecsum([d^#n|d<-n=digits(n)])}, [0..9999]) \\ M. F. Hasler, Nov 18 2019

from itertools import combinations_with_replacement
A005188_list = []
for k in range(1,10):
    a = [i**k for i in range(10)]
    for b in combinations_with_replacement(range(10),k):
        x = sum(map(lambda y:a[y],b))
        if x > 0 and tuple(int(d) for d in sorted(str(x))) == b:
            A005188_list.append(x)
A005188_list = sorted(A005188_list) # Chai Wah Wu, Aug 25 2015

32164049651 from Amit Munje (amit.munje(AT)gmail.com), Oct 07 2006
In order to agree with the Definition, first comment modified by Jonathan Sondow, Jan 02 2015
Comment in name moved to comment section and links edited by M. F. Hasler, Oct 18 2018
"Positive" added to definition by N. J. A. Sloane, Nov 18 2019

A023052 Perfect Digital Invariants: numbers that are the sum of some fixed power of their digits.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 153, 370, 371, 407, 1634, 4150, 4151, 8208, 9474, 54748, 92727, 93084, 194979, 548834, 1741725, 4210818, 9800817, 9926315, 14459929, 24678050, 24678051, 88593477, 146511208, 472335975, 534494836, 912985153

Offset: 1

David W. Wilson

The old name was "Powerful numbers, definition (3)". Cf. A001694, A007532. - N. J. A. Sloane, Jan 16 2022.
Randle has suggested that these numbers be called "powerful", but this usually refers to a distinct property related to prime factorization, cf. A001694, A036966, A005934.
Numbers m such that m = Sum_{i=1..k} d(i)^s for some s, where d(1..k) are the decimal digits of m.
Superset of A005188 (Plusperfect, narcissistic or Armstrong numbers: s=k), A046197 (s=3), A052455 (s=4), A052464 (s=5), A124068 (s=6, 7), A124069 (s=8). - R. J. Mathar, Jun 15 2009, Jun 22 2009

			153 = 1^3 + 5^3 + 3^3, 4210818 = 4^7 + 2^7 + 1^7 + 0^7 + 8^7 + 1^7 + 8^7.

Jerome Raulin, Table of n, a(n) for n = 1..345 (terms 1..255 from Joseph Myers)
Encyclopaedia Britannica, Perfect digital invariant, article "Number patterns and curiosities" online since July 26, 1999, revised Aug 25, 2000.
Hans Havermann, Extended table of values for A023052 and A046074
Donald E. Knuth, The Art of Computer Programming, Volume 4, Pre-Fascicle 9B A Potpourri of Puzzles
J. Randle, Powerful numbers, Note 3208, Math. Gaz. 52 (1968), 383.
J. Randle, Powerful numbers, Note 3208, Math. Gaz. 52 (1968), 383. [Annotated scanned copy]
Eric Weisstein's World of Mathematics, Narcissistic Number
Index entries for sequences related to powerful numbers

Cf. A001694 (powerful numbers: p|n => p^2|n), A005934 (highly powerful numbers).
Cf. A003321, A007532, A014576, A046074, A046761, A053540, A161752.
Cf. A005188 (here the power must be equal to the number of digits).
In other bases: A162216 (base 3), A162219 (base 4), A162222 (base 5), A162225 (base 6), A162228 (base 7), A162231 (base 8), A162234 (base 9).

Select[Range[0, 10^5], Function[m, AnyTrue[Function[k, Total@ Map[Power[#, k] &, IntegerDigits@ m]] /@ Range@ 10, # == m &]]] (* Michael De Vlieger, Feb 08 2016, Version 10 *)

is(n)=if(n<10, return(1)); my(d=digits(n),m=vecmax(d)); if(m<2, return(0)); for(k=3,logint(n,m), if(sum(i=1,#d,d[i]^k)==n, return(1))); 0 \\ Charles R Greathouse IV, Feb 06 2017

select( is_A023052(n,b=10)={nn|| return(t==n))}, [0..10^5]) \\  M. F. Hasler, Nov 21 2019

Computed to 10^50 by G. N. Gusev (GGN(AT)rm.yaroslavl.ru)
Computed to 10^74 by Xiaoqing Tang
A-number typo corrected by R. J. Mathar, Jun 22 2009
Computed to 10^105 by Joseph Myers
Cross-references edited by Joseph Myers, Jun 28 2009
Edited by M. F. Hasler, Nov 21 2019

A003321 Smallest n-th order perfect digital invariant or PDI: smallest number > 1 equal to sum of n-th powers of its digits, or 0 if no such number exists.

Original entry on oeis.org

2, 0, 153, 1634, 4150, 548834, 1741725, 24678050, 146511208, 4679307774, 32164049650, 0, 564240140138, 28116440335967, 0, 4338281769391370, 233411150132317, 0, 1517841543307505039, 63105425988599693916

Offset: 1

N. J. A. Sloane

Except for the initial term, this is the third column of A252648. - M. F. Hasler, Feb 16 2015

a(n) = 0 if n>1 and in A262094. - Dmitry Kamenetsky, Jun 05 2020

			1^3 + 5^3 + 3^3 = 153.
1*0^17 + 5*1^17 + 2*2^17 + 4*3^17 + 1*4^17 + 1*5^17 + 1*7^17 = 233411150132317.

M. Gardner, The Magic Numbers of Dr Matrix. Prometheus, Buffalo, NY, 1985, p. 249.
J. S. Madachy, Mathematics on Vacation, Thomas Nelson and Sons Ltd. 1966, p. 164.
J. S. Madachy, Madachy's Mathematical Recreations, Dover, p. 164.
C. A. Pickover, Keys to Infinity. New York: W. H. Freeman, pp. 169-170, 1995.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Joseph Myers, Table of n, a(n) for n=1..109
L. E. Deimel, Narcissistic Numbers
H. Heinz, Narcissistic Numbers
Eric Weisstein's World of Mathematics, Narcissistic Number.

Cf. A001694, A007532, A005934, A005188, A014576, A023052, A046074, A046761.

In other bases: A033835 (base 3), A033836 (base 4), A033837 (base 5), A033838 (base 6), A033839 (base 7), A033840 (base 8), A033841 (base 9).

a(n)=m=1;while(m*9^n>=10^m,m++);for(k=2,10^m,d=digits(k);s=sum(i=1,#d,d[i]^n);if(s==k,return(k)));0
n=1;while(n<10,print1(a(n),", ");n++) \\ Derek Orr, Dec 19 2014

Additional comments from Lekraj Beedassy, May 23 2001

Extended and cross-references edited by Joseph Myers, Jun 28 2009

A005934 Highly powerful numbers: numbers with record value of the product of the exponents in prime factorization (A005361).

Original entry on oeis.org

1, 4, 8, 16, 32, 64, 128, 144, 216, 288, 432, 864, 1296, 1728, 2592, 3456, 5184, 7776, 10368, 15552, 20736, 31104, 41472, 62208, 86400, 108000, 129600, 194400, 216000, 259200, 324000, 432000, 518400, 648000, 972000, 1296000, 1944000, 2592000

Offset: 1

N. J. A. Sloane

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Amiram Eldar, Table of n, a(n) for n = 1..609 (terms 1..300 from T. D. Noe)
R. K. Guy, Letter to N. J. A. Sloane with attachment, Jun. 1991.
G. E. Hardy and M. V. Subbarao, Highly powerful numbers, Congress. Numer., Vol. 37 (1983), pp. 277-307. (Annotated scanned copy)
C. B. Lacampagne and J. L. Selfridge, Large highly powerful numbers are cubeful, Proc. Amer. Math. Soc., Vol. 91, No. 2 (1984), pp. 173-181.
Wikipedia, Highly powerful number.
Index entries for sequences related to powerful numbers

Cf. A036965, A001694, A007532, A005361, A005188, A003321, A014576, A023052, A046074.

Cf. A085629.

a = {1}; b = {1}; f[n_] := Times @@ Last /@ FactorInteger[n]; Do[If[f@ n > Max[b], And[AppendTo[b, f@ n], AppendTo[a, n]]], {n, 1000000}]; a (* Michael De Vlieger, Aug 28 2015 *)
With[{s = Array[Times @@ FactorInteger[#][[All, -1]] &, 3*10^6]}, Map[FirstPosition[s, #][[1]] &, Union@ FoldList[Max, s]]] (* Michael De Vlieger, Oct 15 2017 *)
DeleteDuplicates[Table[{n,Times@@FactorInteger[n][[All,2]]},{n,26*10^5}],GreaterEqual[#1[[2]],#2[[2]]]&][[All,1]] (* Harvey P. Dale, May 13 2022 *)

{prdex(n)=local(s,fac); s=1; fac=factor(n); for(k=1,matsize(fac)[1],s=s*fac[k,2]); return(s)} {hp(m)=local(rec); rec=0; for(n=1,m,if(prdex(n)>rec,rec=prdex(n); print1(n",")))}

For n = Product p_i^e_i, let b(n) = Product e_i; then n is highly powerful if b(n) sets a new record.

Hardy and Subbarao give an extensive table.

Corrected and extended by Jason Earls, Jul 10 2003

A007532 Handsome numbers: sum of positive powers of its digits; a(n) = Sum_{i=1..k} d[i]^e[i] where d[1..k] are the decimal digits of a(n), e[i] > 0.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 24, 43, 63, 89, 132, 135, 153, 175, 209, 224, 226, 262, 264, 267, 283, 332, 333, 334, 357, 370, 371, 372, 373, 374, 375, 376, 377, 378, 379, 407, 445, 463, 518, 598, 629, 739, 794, 849, 935, 994, 1034

Offset: 1

N. J. A. Sloane, Robert G. Wilson v

The previous name was "Powerful numbers, Definition (2). Cf. A001694, A023052. - N. J. A. Sloane, Jan 16 2022

J. Randle has suggested the name "powerful numbers" for the perfect digital invariants A023052, equal to the sum of a fixed power of the digits. However, "powerful" usually refers to a prime factorization related property, cf. A001694 (and references there as well as on the MathWorld page). C. Rivera has suggested the name "handsome" for these numbers (in view of narcissistic numbers A005188) in his prime puzzle #15: see also contributed comments concerning terminology on that page. - M. F. Hasler, Nov 21 2019

			43 = 4^2 + 3^3 is OK; 254 = 2^7 + 5^3 + 4^0 is not OK since one of the powers is 0.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

David W. Wilson, Table of n, a(n) for n = 1..10000
Giovanni Resta, d-powerful numbers, the 30067 terms and sums up to 10^6.
Carlos Rivera, Puzzle 15.- Narcissistic and Handsome Primes, The Prime Puzzles and Problems Connection.
Eric Weisstein's World of Mathematics, Powerful Number.
Index entries for sequences related to powerful numbers

Cf. A001694, A005934, A005188, A003321, A014576, A023052, A046074, A050240 (>= 2 reps.), A050241.

Different from A061862.

a007532 n = a007532_list !! (n-1)
a007532_list = filter f [1..] where
   f x = g x 0 where
     g 0 v = v == x
     g u v = if d <= 1 then g u' (v + d) else v <= x && h d
             where h p = p <= x && (g u' (v + p) || h (p * d))
                   (u', d) = divMod u 10
-- Reinhard Zumkeller, Jun 02 2013

N:= 10000; # to get all entries <= N
Sums:= proc(L,N)
  option remember;
  local x1,L1;
  x1:= L[1];
  if x1 = 1 then L1:= {1}
  else L1:= {seq(x1^j,j=1..floor(log[x1](N)))};
  fi;
  if nops(L) = 1 then L1
  else select(`<=`,{seq(seq(a+b,a=L1),b=Sums(L[2..-1],N))},N)
  fi
end proc;
filter:= proc(x,N)
   local L;
   L:= sort(subs(0=NULL,convert(x,base,10))) ;
   member(x, Sums(L,N));
end proc;
A007532:= select(filter,[$1..N],N); # Robert Israel, Apr 13 2014

Select[Range@1000,(s=#;MemberQ[Total/@(a^#&/@Tuples[Range@If[#==1||#==0,1,Floor[Log[#,s]]]&/@(a=IntegerDigits[s])]),s])&] (* Giorgos Kalogeropoulos, Aug 18 2021 *)

from itertools import count, takewhile
def cands(n, d):
    return takewhile(lambda x: x<=n, (d**i for i in count(1)))
def handsome(s, t):
    if s == "":
        return t == 0
    if s[0] in "01":
        return handsome(s[1:], t - int(s[0]))
    return any(handsome(s[1:], t - p) for p in cands(t, int(s[0])))
def ok(n):
    return n and handsome(str(n), n)
print(list(filter(ok, range(1035)))) # Michael S. Branicky, Aug 18 2021

If n = d_1 d_2 ... d_k in decimal, then there are integers m_1, m_2, ..., m_k > 0 such that n = d_1^m_1 + ... + d_k^m_k.

A134703 Powerful numbers (2b): a sum of nonnegative powers of its digits.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 24, 43, 63, 89, 132, 135, 153, 175, 209, 224, 226, 254, 258, 262, 263, 264, 267, 283, 308, 332, 333, 334, 347, 357, 370, 371, 372, 373, 374, 375, 376, 377, 378, 379, 407, 445, 463, 472, 518, 538, 598, 629, 635, 653, 675, 730, 731, 732, 733, 734

Offset: 1

David W. Wilson, Sep 05 2009

Here 0 digits may be used, with the convention that 0^0 = 1. Of course 0^1 = 0, so one is free to use the 0 digit to get an extra 1, or not.

			43 = 4^2 + 3^3; 254 = 2^7 + 5^3 + 4^0 = 128 + 125 + 1.
209 = 2^7 + 0^1 + 9^2.
732 = 7^0 + 3^6 + 2^1.

David Wilson, Table of n, a(n) for n = 1..10000
Index entries for sequences related to powerful numbers

Cf. A001694, A005934, A005188, A003321, A014576, A023052, A046074.

Different from A007532 and A061862, which are variations.

If n = d_1 d_2 ... d_k in decimal then there are integers m_1 m_2 ... m_k >= 0 such that n = d_1^m_1 + ... + d_k^m_k.

A061862 Powerful numbers (2a): a sum of nonnegative powers of its digits.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 24, 43, 63, 89, 132, 135, 153, 175, 209, 224, 226, 254, 258, 262, 263, 264, 267, 283, 332, 333, 334, 347, 357, 370, 371, 372, 373, 374, 375, 376, 377, 378, 379, 407, 445, 463, 472, 518, 538, 598, 629, 635, 653, 675, 730, 731, 732

Offset: 1

Erich Friedman, Jun 23 2001

Zero digits cannot be used in the sum. - N. J. A. Sloane, Aug 31 2009

More precisely, digits 0 do not contribute to the sum, in contrast to A134703 where it is allowed to use 0^0 = 1. - M. F. Hasler, Nov 21 2019

			43 = 4^2 + 3^3; 254 = 2^7 + 5^3 + 4^0 = 128 + 125 + 1.
209 = 2^7 + 9^2.
732 = 7^0 + 3^6 + 2^1.

Cf. A001694, A005934, A005188, A003321, A014576, A023052, A046074.

Different from A007532 and A134703, which are variations.

a061862 n = a061862_list !! (n-1)
a061862_list = filter f [0..] where
   f x = g x 0 where
     g 0 v = v == x
     g u v = if d <= 1 then g u' (v + d) else v <= x && h 1
             where h p = p <= x && (g u' (v + p) || h (p * d))
                   (u', d) = divMod u 10
-- Reinhard Zumkeller, Jun 02 2013

f[ n_ ] := Module[ {}, a=IntegerDigits[ n ]; e=g[ Length[ a ] ]; MemberQ[ Map[ Apply[ Plus, a^# ] &, e ], n ] ] g[ n_ ] := Map[ Take[ Table[ 0, {n} ]~Join~#, -n ] &, IntegerDigits[ Range[ 10^n ], 10 ] ] For[ n=0, n >= 0, n++, If[ f[ n ], Print[ n ] ] ]

If n = d_1 d_2 ... d_k in decimal then there are integers m_1 m_2 ... m_k >= 0 such that n = d_1^m_1 + ... + d_k^m_k.

A192636 Powerful sums of two powerful numbers.

Original entry on oeis.org

8, 9, 16, 25, 32, 36, 64, 72, 81, 100, 108, 121, 125, 128, 144, 169, 196, 200, 216, 225, 243, 256, 288, 289, 324, 343, 361, 392, 400, 432, 441, 484, 500, 512, 576, 625, 648, 675, 676, 729, 784, 800, 841, 864, 900, 961, 968, 972, 1000, 1024, 1089, 1125, 1152, 1156, 1225

Offset: 1

Charles R Greathouse IV, Jul 06 2011

nonn

Browning & Valckenborgh conjecture that a(n) ~ kn^2 with k approximately 0.139485255. See their Conjecture 1 and equation (14). Their Theorems 1 and 2 establish upper and lower asymptotic bounds.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..5000 from Charles R Greathouse IV)
Tim D. Browning and K. Van Valckenborgh, Sums of three squareful numbers, Experimental Mathematics, Vol. 21, No. 2 (2012), pp. 204-211; arXiv preprint, arXiv:1106.4472 [math.NT], 2011.

Subsequence of A001694 and of A076871.

Cf. A001694, A007532, A005934, A005188, A003321, A014576, A023052, A046074, A013929, A076871, A143813. - Jonathan Vos Post, Jul 10 2011

With[{m = 1225}, pow = Select[Range[m], # == 1 || Min[FactorInteger[#][[;; , 2]]] > 1 &]; Intersection[pow, Plus @@@ Tuples[pow, {2}]]] (* Amiram Eldar, Feb 12 2023 *)

isPowerful(n)=if(n>3,vecmin(factor(n)[,2])>1,n==1)
sumset(a,b)={
  my(c=vectorsmall(#a*#b));
  for(i=1,#a,
    for(j=1,#b,
      c[(i-1)*#b+j]=a[i]+b[j]
    )
  );
  vecsort(c,,8)
}; selfsum(a)={
  my(c=vectorsmall(binomial(#a+1,2)),k);
  for(i=1,#a,
    for(j=i,#a,
      c[k++]=a[i]+a[j]
    )
  );
  vecsort(c,,8)
};
list(lim)={
  my(v=select(isPowerful, vector(floor(lim),i,i)));
  select(n->n<=lim && isPowerful(n), Vec(selfsum(v)))
};

Numbers k such that there exists some a, b, c with A001694(a) + A001694(b) = k = A001694(c).

Corrected (on the advice of Donovan Johnson) by Charles R Greathouse IV, Sep 25 2012

cp's OEIS Frontend

A001694 Powerful numbers, definition (1): if a prime p divides n then p^2 must also divide n (also called squareful, square full, square-full or 2-powerful numbers).

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A005188 Armstrong (or pluperfect, or Plus Perfect, or narcissistic) numbers: m-digit positive numbers equal to sum of the m-th powers of their digits.

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A023052 Perfect Digital Invariants: numbers that are the sum of some fixed power of their digits.

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A003321 Smallest n-th order perfect digital invariant or PDI: smallest number > 1 equal to sum of n-th powers of its digits, or 0 if no such number exists.

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A005934 Highly powerful numbers: numbers with record value of the product of the exponents in prime factorization (A005361).

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A007532 Handsome numbers: sum of positive powers of its digits; a(n) = Sum_{i=1..k} d[i]^e[i] where d[1..k] are the decimal digits of a(n), e[i] > 0.

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