A001661 -id:A001661 - cp's OEIS frontend

A364637 a(n) is the least k > 1 that can be represented as a sum of one or more distinct positive m-th powers for 1 <= m <= n.

2, 4, 9, 881, 7809, 134067, 12939267, 2029992385, 122120396036

Offset: 1

David A. Corneth and Peter Munn, Jul 30 2023

Sprague showed that for any m, all sufficiently large integers are the sum of distinct m-th powers. A001661(m) gives the largest number not of this form, so we can use A001661 to write an upper bound for the terms here.

			a(5) = 7809 as it can be written as a sum of one or more distinct positive m-th powers for 1 <= m <= 5 as follows. 1^5 + 2^5 + 6^5 = 2^4 + 6^4 + 7^4 + 8^4 = 3^3 + 5^3 + 14^3 + 17^3 = 1^2 + 8^2 + 88^2 = 7809^1 and no number less than 7809 can be written as such.

R. Sprague, Über Zerlegungen in n-te Potenzen mit lauter verschiedenen Grundzahlen, Math. Z. 51 (1948) 466-468.

Sequences giving solutions for related problems: A001661, A030052.

For n >= 2, a(n) <= 1 + Max_{m=2..n} A001661(m).

A173563 Number of positive integers not the sum of distinct positive n-th powers.

Original entry on oeis.org

0, 31, 2788, 889576, 13912682, 2037573096, 198526316569

Offset: 1

R. H. Hardin, Feb 21 2010

Fuller and Nichols prove that a(6) = 2037573096. - Robert Nichols, Sep 10 2017

Here, the "sum of n-th powers" includes the case where this sum consists in just one term. (For example, 1 is the sum of just 1^n, for any n; and 4 = 2^2 is considered to be a sum of distinct squares.) - M. F. Hasler, May 25 2020

			The list of the a(2) = 31 integers which are not the sum of distinct squares is given in A001422. - _M. F. Hasler_, May 25 2020

C. Fuller and R. H. Nichols Jr., Generalized Anti-Waring Numbers, J. Int. Seq. 18 (2015), #15.10.5.
Michael J. Wiener, The Largest Integer Not the Sum of Distinct 8th Powers, J. Integer Sequences, 26 (2023), Article 23.5.4.

Cf. A001661, A001422.

a(2..6) confirmed and a(7) added by Michael J. Wiener, Jun 18 2023

A279529 a(n) is a largest m such that coefficient [x^m] in Product_{k>=1} (1-x^(k^n)) is equal to zero.

Original entry on oeis.org

7169, 353684, 64674419

Offset: 2

Vaclav Kotesovec, Dec 14 2016

			a(2) = 7169 because A276516(7169) = 0 and A276516(m) <> 0 for m > 7169.
a(3) = 353684 because A279484(353684) = 0 and A279484(m) <> 0 for m > 353684.
a(4) = 64674419 because A279485(64674419) = 0 and A279485(m) <> 0 for m > 64674419.
a(2) = A276517(173) = 7169.
a(3) = A279486(5216) = 353684.
a(4) = A279487(1040799) = 64674419.

Cf. A001661.
Cf. A276517, A279486, A279487.
Cf. A276516, A279484, A279485.
Cf. A001422, A001476, A046039.

A332096 Irregular table where T(n,m) = min_{A subset {1..m-1}} |m^n - Sum_{x in A} x^n|, for 1 <= m <= A332098(n) = largest m for which this is nonzero.

Original entry on oeis.org

1, 1, 1, 3, 4, 2, 0, 1, 0, 1, 1, 7, 18, 28, 25, 0, 1, 8, 0, 7, 1, 1, 15, 64, 158, 271, 317, 126, 45, 17, 59, 14, 2, 15, 3, 0, 2, 1, 2, 1, 2, 2, 2, 1, 2, 0, 2, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 31, 210, 748, 1825, 3351, 4606, 3760, 398, 131, 299, 0, 318, 0, 8

Offset: 1

M. F. Hasler, Jul 20 2020

nonn

It is known (Sprague 1948, cf. A001661) that for any n, only a finite number of positive integers are not the sum of distinct positive n-th powers. Therefore each row is finite, their lengths are given by A332098.
The number of nonzero terms in row n is A332066(n).
The column of the first zero (exact solution m^n = Sum_{x in A} x^n) in each row is given by A030052, unless A030052(n) = A332066(n) + 1 = A332098(n) + 1.

			The table reads:
  n\ m=1   2    3    4     5     6     7     8    9   10   11  12   13
----+--------------------------------------------------------------------------
  1 |  1   1                                                  (A332098(1) = 2.)
  2 |  1   3    4    2     0     1     0     1                (A332098(2) = 8.)
  3 |  1   7   18   28    25     0     1     8    0    7    1
  4 |  1  15   64  158   271   317   126    45   17   59   14   2   15  3  0 ...
  5 |  1  31  210  748  1825  3351  4606  3760  398  131  299   0  318  0  8 ...
The first column is all ones (A000012), since {1..m-1} = {} for m = 1.
The second column is 2^n - 1 = A000225 \ {0}, since {1..m-1} = {1} for m = 2.
The third column is 3^n - 2^n - 1 = |A083321(n)| for n > 1.

R. Sprague, Über Zerlegungen in n-te Potenzen mit lauter verschiedenen Grundzahlen, Math. Z. 51 (1948) 466-468.

Cf. A001661, A332098, A332066, A030052, A332097, A078607.

Cf. A000012, A000225, A083321.

A332096(n,m,r=0)={if(r, (m<2||r<2^(n-1)) && return(r-1); my(E, t=1); while(m^n>=r, E=m--); E=abs(r-(m+!!E)^n); for(a=2,m, if(r=m && return(min(E,r-t)); while(m>=t && E, E=min(self()(n,m-1,r-m^n),E); E && E=min(self()(n,m-=1,r),E)); E, m < n/log(2)+1.5, m^n-sum(x=1,m-1,x^n), self()(n,m-1,m^n))}

For all n and m, T(n,m) <= A332097(n) = T(n,m*) with m* = A078607(n).

For m <= m* + 1, T(n,m) = m^n - Sum_{0 < x < m} x^n.

A332098 Largest m for which m^n = Sum_{x in S} x^n has no solution S subset of {1, ..., m-1}.

Original entry on oeis.org

2, 8, 11, 44, 27, 33, 42, 83, 51, 62, 72, 83

Offset: 1

M. F. Hasler, Apr 19 2020

Row n of table A332065 lists all s for which there is some S subset of {1,...,m-1} with s^n = Sum_{x in S} x^n. This is the case for all sufficiently large s (cf. reference there). Here we give the largest integer not in this list.

Sequence A030052 lists the smallest m for which there is a solution, so a(n) >= A030052(n) - 1. We have a(9) = 51 = A030052(9) + 4, a(10) = 62 = A030052(10) - 1, a(11) = 72 = A030052(11) + 4. - M. F. Hasler, May 14 2020, edited Jul 19 2020

			For n=1, we have m^n = (m-1)^n + 1^n, so S = {1, m-1} is a solution for all m > 2, but 2^n > 1^n and therefore no solution with m = 2 = a(1).
For n=2, we have a solution to m^n = Sum_{x in S} x^n for S subset of {1,...,m-1} for all m > 8 (cf. FORMULA in A332065), but no solution with m = 8 = a(2).

Cf. A332065, A030052, A332066.

a(n) = A030052(n) - 1 or a(n) > A030052(n).

a(n) < A001661(n)^(1/n).

a(8) - a(12) from M. F. Hasler, Jul 23 2020

A334360 Anti-Waring numbers: least number k such that k and all larger numbers can be expressed as the sum of one or more distinct n-th powers.

Original entry on oeis.org

129, 12759, 5134241, 67898772, 11146309948

Offset: 2

Charles R Greathouse IV, Apr 24 2020

Sprague finds a(2) in 1948 and proves that a(n) exists for all n >= 2 in the same year. Graham finds a(3) in 1964 with a paper "to appear" with details; Dressler & Parker give an independent proof in 1974. Lin finds a(4) in 1970. Patterson finds a(5) in 1992. Fuller & Nichols, Jr. find a(6) in 2020.

			129 = 2^2 + 5^2 + 10^2, but no subset of {1^2, 2^2, ..., 11^2} sums to 128, so a(2) >= 129.
a(3) = 5^3 + 6^3 + 7^3 + 11^3 + 14^3 + 20^3, but a(3) - 1 = 12758 cannot be so represented.
a(4) = 2^4 + 6^4 + 7^4 + 14^4 + 28^4 + 46^4
a(5) = 2^5 + 3^5 + 6^5 + 8^5 + 9^5 + 10^5 + 13^5 + 14^5 + 19^5 + 22^5 + 27^5 + 29^5 + 30^5

S. Lin, Computer experiments on sequences which form integral bases, in J. Leech, ed., Computational Problems in Abstract Algebra, Pergamon Press, 1970, pp. 365-370.

R. Dressler and T. Parker, 12,758, Mathematics of Computation 28:125 (1974), pp. 313-314.
Chris Fuller and Robert H. Nichols, Jr., Generalized anti-Waring numbers, Journal of Integer Sequences 18 (2015), Article 15.10.5.
R. L. Graham, Complete sequences of polynomial values, Duke Math. J. 31 (1964), pp. 275-285.
C. Patterson, The Derivation of a High Speed Sieve Device, Ph.D. thesis, University of Calgary, 1992. [See 2.2.3.2, Complete Sequences, pp. 18-23.]
R. Sprague, Über Zerlegung in ungleiche Quadratzahlen, Mathematische Zeitschrift 51 (1948), pp. 289-290.
R. Sprague, Über Zerlegungen in n-te Potenzen mit lauter verschiedenen Grundzahlen, Mathematische Zeitschrift, 51 (1948), pp. 466-468.

Cf. A001661.

sumOf(n,k,e,xmax=n)=my(t); if(k==1, my(t); if(ispower(n,e,&t) && t<=xmax, return([t]), return(0))); xmax=min(sqrtnint(n,e),xmax); forstep(x=xmax,k,-1, t=sumOf(n-x^e,k-1,e,x-1); if(t, return(concat(t,x)))); 0
bestPowerRep(n,e)=my(k,t); while((t=sumOf(n,k++,e))==0,); t \\ Finds a representation for n as a sum of distinct e-th powers; Charles R Greathouse IV, May 04 2020

a(n) = A001661(n) + 1. - Ilya Gutkovskiy, Mar 24 2022

A352586 a(n) is the largest prime that is not the sum of n-th powers of distinct primes.

Original entry on oeis.org

12601, 1656517

Offset: 2

Ilya Gutkovskiy, Mar 21 2022

The corresponding indices of primes are 1505, 125100.

Cf. A001661, A121571, A244637.

cp's OEIS Frontend

A364637 a(n) is the least k > 1 that can be represented as a sum of one or more distinct positive m-th powers for 1 <= m <= n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A173563 Number of positive integers not the sum of distinct positive n-th powers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A279529 a(n) is a largest m such that coefficient [x^m] in Product_{k>=1} (1-x^(k^n)) is equal to zero.

Views

Author

Keywords

Examples

Crossrefs

A332096 Irregular table where T(n,m) = min_{A subset {1..m-1}} |m^n - Sum_{x in A} x^n|, for 1 <= m <= A332098(n) = largest m for which this is nonzero.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A332098 Largest m for which m^n = Sum_{x in S} x^n has no solution S subset of {1, ..., m-1}.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

Extensions

A334360 Anti-Waring numbers: least number k such that k and all larger numbers can be expressed as the sum of one or more distinct n-th powers.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

PARI

Formula

A352586 a(n) is the largest prime that is not the sum of n-th powers of distinct primes.

Views

Author

Keywords

Comments

Crossrefs