A121571 -id:A121571 - cp's OEIS frontend

A001661 Largest number not the sum of distinct positive n-th powers.

128, 12758, 5134240, 67898771, 11146309947, 766834015734, 4968618780985762

Offset: 2

a(8) > 74^8. - Donovan Johnson, Nov 23 2010
Fuller and Nichols prove that a(6) = 11146309947 and that 2037573096 positive numbers cannot be written as the sum of distinct 6th powers. - Robert Nichols, Sep 09 2017
a(8) >= 83^8 ~ 2.25e15 since A030052(8) = 84. Similarly, a(9..15) >= (46^9, 62^10, 67^11, 80^12, 101^13, 94^14, 103^15) ~ (9.2e14, 8.4e17, 1.2e20, 6.9e22, 1.1e26, 4.2e27, 1.6e30), cf. formula. Most often a(n) will be closer to and even larger than A030052(n)^n. - In the literature, a(n)+1 is known as the anti-Waring number N(n,1). - M. F. Hasler, May 15 2020
a(9..16) > (1.55e17, 1.31e19, 1.64e21, 5.55e23, 1.32e26, 1.37e28, 2.09e30, 9.99e35). - Michael J. Wiener, Jun 10 2023

S. Lin, Computer experiments on sequences which form integral bases, pp. 365-370 of J. Leech, editor, Computational Problems in Abstract Algebra. Pergamon, Oxford, 1970.
Harry L. Nelson, The Partition Problem, J. Rec. Math., 20 (1988), 315-316.
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).

R. E. Dressler and T. Parker, 12,758, Math. Comp. 28 (1974), 313-314.
Shalosh B. Ekhad and Doron Zeilberger, Automating John P. D'Angelo's method to study Complete Polynomial Sequences, arXiv:2111.02832 [math.NT], 2021.
Mauro Fiorentini, Rappresentazione di interi come somma di potenze (in Italian).
C. Fuller and R. H. Nichols Jr., Generalized Anti-Waring Numbers, J. Int. Seq. 18 (2015), #15.10.5.
R. L. Graham, Complete sequences of polynomial values, Duke Math. J. 31 (1964), pp. 275-285.
D. Kim, On the largest integer that is not a sum of distinct nth powers of positive integers, arXiv:1610.02439 [math.NT], 2016-2017.
D. Kim, On the largest integer that is not a sum of distinct nth powers of positive integers, J. Int. Seq. 20 (2017), #17.7.5.
P. LeVan and D. Prier, Improved Bounds on the Anti-Waring Number, J. Int. Seq. 20 (2017), #17.8.7.
D. C. Mayer, Sharp bounds for the partition function of integer sequences, BIT 27 (1987), 98-110.
D. C. Mayer, Partition functions via bit list operations, 2009.
N. J. A. Sloane and R. E. Dressler, Correspondence, June 1974
R. Sprague, Über Zerlegungen in n-te Potenzen mit lauter verschiedenen Grundzahlen, Math. Z. 51 (1948) 466-468.
Eric Weisstein's World of Mathematics, Waring's Problem
M. J. Wiener, The Largest Integer Not the Sum of Distinct 8th Powers, J. Integer Sequences, 26 (2023), Article 23.5.4.
J. W. Wrench, Jr., Letter to N. J. A. Sloane, 10 Apr, 1974

Cf. A030052, A173563, A279529.

Cf. A121571 (primes instead of integers).

a(n) < d*2^(n-1)*(c*2^n + (2/3)*d*(4^n - 1) + 2*d - 2)^n + c*d, where c = n!*2^(n^2) and d = 2^(n^2 + 2*n)*c^(n-1) - 1, according to Kim [2016-2017]. - Danny Rorabaugh, Oct 11 2016

a(n) >= (A030052(n)-1)^n. - M. F. Hasler, May 15 2020

a(7) from Donovan Johnson, Nov 23 2010

a(8) from Michael J. Wiener, Jun 10 2023

A231407 Positive integers that are not the sum of distinct primes.

Original entry on oeis.org

1, 4, 6

Offset: 1

Jonathan Sondow, Nov 24 2013

Using elementary methods, Richert proved that 6 is the largest integer which is not the sum of distinct primes.

H.-E. Richert, Über Zerlegungen in paarweise verschiedene Zahlen, Norsk. Mat. Tidsskr., 31 (1949), 120-122.

R. E. Dressler, Addendum to "A stronger Bertrand’s postulate with an application to partitions", Proc. Am. Math. Soc., 38 (1973), 667.

Cf. A121571, A231408.

a(3) = A121571(1).

A213519 Numbers that are the sum of cubes of distinct primes.

Original entry on oeis.org

0, 8, 27, 35, 125, 133, 152, 160, 343, 351, 370, 378, 468, 476, 495, 503, 1331, 1339, 1358, 1366, 1456, 1464, 1483, 1491, 1674, 1682, 1701, 1709, 1799, 1807, 1826, 1834, 2197, 2205, 2224, 2232, 2322, 2330, 2349, 2357, 2540, 2548, 2567, 2575, 2665, 2673, 2692

Offset: 1

T. D. Noe, Jul 10 2012

The complement of this sequence is conjectured to have 483370 terms, the last one being 1866000 = A121571(3).

This conjecture was proved by Fuller and Nichols (see the link). - Robert Nichols, Sep 17 2017

T. D. Noe, Table of n, a(n) for n = 1..10000
C. Fuller and R. H. Nichols Jr., Generalized Anti-Waring Numbers, J. Int. Seq. 18, 2015, #15.10.5.

Cf. A121571.

lim = PrimePi[17]; s = {0}; Do[p = Prime[n]; s = Union[s, s + p^3], {n, lim}]; Select[s, # <= Prime[lim]^3 &]

A231408 Positive integers that are not the sum of distinct odd primes.

Original entry on oeis.org

1, 2, 4, 6, 9

Offset: 1

Jonathan Sondow, Nov 24 2013

Using elementary methods, Dressler proved that 9 is the largest integer which is not the sum of distinct odd primes.

R. E. Dressler, A stronger Bertrand's postulate with an application to partitions, Proc. Amer. Math. Soc., 33 (1972), 226-228.
R. E. Dressler, Addendum to "A stronger Bertrand’s postulate with an application to partitions", Proc. Am. Math. Soc., 38 (1973), 667.

Cf. A121571, A231407.

A351326 a(n) is the least number k such that k and all larger numbers can be expressed as the sum of n-th powers of distinct primes.

Original entry on oeis.org

7, 17164, 1866001

Offset: 1

Ilya Gutkovskiy, Mar 24 2022

C. Fuller and R. H. Nichols Jr., Generalized Anti-Waring Numbers, J. Int. Seq. 18 (2015), #15.10.5.

Cf. A048261, A121571, A213519, A334360.

a(n) = A121571(n) + 1.

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.

A352600 Number of positive integers that are not the sum of n-th powers of distinct primes.

Original entry on oeis.org

3, 2438, 483370

Offset: 1

Ilya Gutkovskiy, Mar 22 2022

C. Fuller and R. H. Nichols Jr., Generalized Anti-Waring Numbers, J. Int. Seq. 18 (2015), #15.10.5.

Cf. A121518, A121571, A173563, A231407.

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A001661 Largest number not the sum of distinct positive n-th powers.

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A231407 Positive integers that are not the sum of distinct primes.

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A213519 Numbers that are the sum of cubes of distinct primes.

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A231408 Positive integers that are not the sum of distinct odd primes.

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A351326 a(n) is the least number k such that k and all larger numbers can be expressed as the sum of n-th powers of distinct primes.

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

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A352600 Number of positive integers that are not the sum of n-th powers of distinct primes.

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