A364637 -id:A364637 - cp's OEIS frontend

A030052 Smallest number whose n-th power is a sum of distinct smaller positive n-th powers.

3, 5, 6, 15, 12, 25, 40, 84, 47, 63, 68, 81, 102, 95, 104, 162, 123

Offset: 1

Richard C. Schroeppel

Sprague has shown that for any n, all sufficiently large integers are the sum of distinct n-th powers. Sequence A001661 lists the largest number not of this form, so we know that a(n) is less than or equal to the next larger n-th power. - M. F. Hasler, May 25 2020

a(18) <= 200, a(19) <= 234, a(20) <= 242; for more upper bounds see the Al Zimmermann's Programming Contests link: The "Final Report" gives exact solutions for n = 16 through 30; those for n = 16 and 17 have been confirmed to be minimal by Jeremy Sawicki. - M. F. Hasler, Jul 20 2020

			3^1 = 2^1 + 1^1, and there is no smaller solution given that the r.h.s. is the smallest possible sum of distinct positive powers.
For n = 2, one sees immediately that 3 is not a solution (3^2 > 2^2 + 1^2) and one can check that 4^2 isn't equal to Sum_{x in A} x^2 for any subset A of {1, 2, 3}. Therefore, the well known hypotenuse number 5 (cf. A009003) with 5^2 = 4^2 + 3^2 provides the smallest possible solution.
a(17) = 123 since 123^17 = Sum {3, 5, 7, 8, 9, 11, 13, 16, 17, 19, 30, 33, 34, 35, 38, 40, 41, 43, 51, 52, 54, 55, 58, 59, 60, 63, 66, 69, 70, 71, 72, 73, 75, 76, 81, 86, 87, 88, 89, 90, 92, 95, 98, 106, 107, 108, 120}^17, with obvious notation. [Solution found by Jeremy Sawicki on July 3, 2020, see Al Zimmermann's Programming Contests link.] - _M. F. Hasler_, Jul 18 2020
For more examples, see the link.

R. Sprague, Über Zerlegungen in n-te Potenzen mit lauter verschiedenen Grundzahlen, Math. Z. 51 (1948) 466-468.
Various authors, How each power is decomposed
Eric Weisstein's World of Mathematics, Diophantine Equation.
Al Zimmermann, Sum Of Powers: Final Report, Al Zimmermann's Programming Contests, April-July 2020

Cf. A078607, A332065, A332097, A332101.

Other sequences defined by sums of distinct n-th powers: A001661, A364637.

A030052(n, m=n\/log(2)+1, s=0)={if(!s, until(A030052(n, m, (m+=1)^n),), s < 2^n || s > (m+n+1)*m^n\(n+1), m=s<2, m=min(sqrtnint(s, n), m); s==m^n || until( A030052(n, m-1, s-m^n) || (s>=(m+n)*(m-=1)^n\(n+1) && !m=0), )); m} \\ Does exhaustive search to find the least solution m. Use optional 2nd arg to specify a starting value for m. Calls itself with nonzero 3rd (optional) argument: in this case, returns nonzero iff s is the sum of powers <= m^n. - For illustration only: takes very long already for n = 8 and n >= 10. - M. F. Hasler, May 25 2020

a(n) <= A001661(n)^(1/n) + 1. - M. F. Hasler, May 25 2020

a(n) >= A332101(n) = A078607(n)+2 (conjectured). - M. F. Hasler, May 25 2020

a(8)-a(10) found by David W. Wilson
a(11) from Al Zimmermann, Apr 07 2004
a(12) from Al Zimmermann, Apr 13 2004
a(13) from Manol Iliev, Jan 04 2010
a(14) and a(15) from Manol Iliev, Apr 28 2011
a(16) and a(17) due to Jeremy Sawicki, added by M. F. Hasler, Jul 20 2020

A003999 Sums of distinct nonzero 4th powers.

Original entry on oeis.org

1, 16, 17, 81, 82, 97, 98, 256, 257, 272, 273, 337, 338, 353, 354, 625, 626, 641, 642, 706, 707, 722, 723, 881, 882, 897, 898, 962, 963, 978, 979, 1296, 1297, 1312, 1313, 1377, 1378, 1393, 1394, 1552, 1553, 1568, 1569, 1633, 1634, 1649, 1650, 1921, 1922

Offset: 1

N. J. A. Sloane

5134240 is the largest positive integer not in this sequence. - Jud McCranie

If we tightened the sequence requirement so that the sum was of more than one 4th power, we would remove exactly 32 4th powers from the terms: row 4 of A332065 indicates which 4th powers would remain. After a(1) = 1, the next number in this sequence that is in the analogous sequences for cubes and squares is a(24) = 881 = A364637(4). - Peter Munn, Aug 01 2023

The Penguin Dictionary of Curious and Interesting Numbers, David Wells, entry 5134240.

Donovan Johnson, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A046039 (complement).
Cf. A003995, A003997, A194768, A194769 (analogs for squares, cubes, 5th and 6th powers).
Cf. A001661, A332065, A364637.
A217844 is a subsequence.

(1+x)*(1+x^16)*(1+x^81)*(1+x^256)*(1+x^625)*(1+x^1296)*(1+x^2401)*(1+x^4096)*(1+x^6561)*(1+x^10000)

max = 2000; f[x_] := Product[1 + x^(k^4), {k, 1, 10}]; Exponent[#, x]& /@ List @@ Normal[Series[f[x], {x, 0, max}]] // Rest (* Jean-François Alcover, Nov 09 2012, after Charles R Greathouse IV *)

upto(lim)={
    lim\=1;
    my(v=List(),P=prod(n=1,lim^(1/4),1+x^(n^4),1+O(x^(lim+1))));
    for(n=1,lim,if(polcoeff(P,n),listput(v,n)));
    Vec(v)
}; \\ Charles R Greathouse IV, Sep 02 2011

For n > 4244664, a(n) = n + 889576. - Charles R Greathouse IV, Sep 02 2011

A194768 Sum of distinct positive fifth powers.

Original entry on oeis.org

1, 32, 33, 243, 244, 275, 276, 1024, 1025, 1056, 1057, 1267, 1268, 1299, 1300, 3125, 3126, 3157, 3158, 3368, 3369, 3400, 3401, 4149, 4150, 4181, 4182, 4392, 4393, 4424, 4425, 7776, 7777, 7808, 7809, 8019, 8020, 8051, 8052, 8800, 8801, 8832, 8833, 9043, 9044, 9075, 9076

Offset: 1

Charles R Greathouse IV, Sep 02 2011

nonn

From Peter Munn, Aug 02 2023: (Start)
67898771 = A001661(5) is the largest number not in the sequence.
After a(1) = 1, the next term that is in all the analogous sequences for smaller powers is a(35) = 7809 = A364637(5).
If we tightened the sequence requirement so that the sum was of more than one 5th power, we would remove exactly 24 5th powers from the terms: row 5 of A332065 indicates which 5th powers would remain.
(End)

Robert Israel, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A000584 (5th powers), A001661, A332065, A364637.
Cf. A003997, A003999, A194769 (analogs for 3rd, 4th and 6th powers).
A217845 is a subsequence.

N:= 2*10^4: # to get all terms <= N
S:= {0}:
for i from 1 while i^5 <= N do
  S:= select(`<=`, map(`+`,S,i^5),N) union S
od:
sort(convert(S minus {0},list)); # Robert Israel, Jun 26 2019

upto(lim)={
    lim\=1;
    my(v=List(),P=prod(n=1,lim^(1/5),1+x^(n^5),1+O(x^(lim+1))));
    for(n=1,lim,if(polcoeff(P,n),listput(v,n)));
    Vec(v)
}; \\ Charles R Greathouse IV, Sep 02 2011

For n > 53986089, a(n) = n + 13912682. [Charles R Greathouse IV, Sep 02 2011]

Name qualified by Peter Munn, Aug 02 2023

A194769 Sum of distinct nonzero sixth powers.

Original entry on oeis.org

1, 64, 65, 729, 730, 793, 794, 4096, 4097, 4160, 4161, 4825, 4826, 4889, 4890, 15625, 15626, 15689, 15690, 16354, 16355, 16418, 16419, 19721, 19722, 19785, 19786, 20450, 20451, 20514, 20515, 46656, 46657, 46720, 46721, 47385, 47386, 47449, 47450, 50752, 50753, 50816

Offset: 1

Charles R Greathouse IV, Sep 02 2011

See A001661 for a proof of the formula. - M. F. Hasler, May 15 2020
From Peter Munn, Aug 02 2023: (Start)
11146309947 = A001661(6) is the largest number not in the sequence.
After a(1) = 1, the next term that is in all the analogous sequences for smaller powers is a(86) = 134067 = A364637(6).
If we tightened the sequence requirement so that the sum was of more than one 6th power, we would remove exactly 30 6th powers from the terms: row 6 of A332065 indicates which 6th powers would remain.
(End)

David A. Corneth, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A001014, A001661, A332065, A364637.
A217846 is a subsequence.
Cf. A003997, A003999, A194768 (analogs for 3rd, 4th and 5th powers).

upto(lim)={
    lim\=1;
    my(v=List(),P=prod(n=1,lim^(1/6),1+x^(n^6),1+O(x^(lim+1))));
    for(n=1,lim,if(polcoeff(P,n),listput(v,n)));
    Vec(v)
}

For n > 9108736851, a(n) = n + 2037573096.

More terms from David A. Corneth, Apr 21 2020

Name qualified by Peter Munn, Aug 02 2023

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A030052 Smallest number whose n-th power is a sum of distinct smaller positive n-th powers.

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A003999 Sums of distinct nonzero 4th powers.

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A194768 Sum of distinct positive fifth powers.

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A194769 Sum of distinct nonzero sixth powers.

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