A231407 -id:A231407 - cp's OEIS frontend

A121571 Largest number that is not the sum of n-th powers of distinct primes.

6, 17163, 1866000

Offset: 1

T. D. Noe, Aug 08 2006

As stated by Sierpinski, H. E. Richert proved a(1) = 6. Dressler et al. prove a(2) = 17163.
Fuller & Nichols prove T. D. Noe's conjecture that a(3) = 1866000. They also prove that 483370 positive numbers cannot be written as the sum of cubes of distinct primes. - Robert Nichols, Sep 08 2017
Noe conjectures that a(4) = 340250525752 and that 31332338304 positive numbers cannot be written as the sum of fourth powers of distinct primes. - Charles R Greathouse IV, Nov 04 2017

			a(1) = 6 because only the numbers 1, 4 and 6 are not the sum of distinct primes.

W. Sierpinski, Elementary Theory of Numbers, Warsaw, 1964, p. 143-144.

R. E. Dressler, Addendum to "A stronger Bertrand’s postulate with an application to partitions", Proc. Am. Math. Soc., 38 (1973), 667.
Robert E. Dressler, Louis Pigno and Robert Young, Sums of squares of primes, Nordisk Mat. Tidskr. 24 (1976), 39-40.
C. Fuller and R. H. Nichols Jr., Generalized Anti-Waring Numbers, J. Int. Seq. 18, (2015), #15.10.5.
H. E. Richert, Über Zerfällungen in ungleiche Primzahlen, Math. Z. 52 no. 1 (1948), 342-343.

Cf. A231407 (numbers that are not the sum of distinct primes).
Cf. A121518 (numbers that are not the sum of squares of distinct primes).
Cf. A213519 (numbers that are the sum of cubes of distinct primes).
Cf. A001661 (integers instead of primes).

a(1) = A231407(3), a(2) = A121518(2438). - Jonathan Sondow, Nov 26 2013

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.

A234320 Largest number that is not the sum of distinct primes of the form 2k+1, 4k+1, 4k+3, 6k+1, 6k+5, ...; or 0 if none exists.

Original entry on oeis.org

9, 121, 55, 205, 161

Offset: 1

Jonathan Sondow, Dec 28 2013

Largest number that is not the sum of distinct primes of the form 2nk+r for fixed n > 0 and 0 < r < 2n with gcd(2n,r) = 1.
n = 1: Dressler proved that 9 is the largest integer which is not the sum of distinct odd primes.
n = 2 and 3: Makowski proved that the largest integer that is not the sum of distinct primes of the form 4k+1, 4k+3, 6k+1, 6k+5 is 121, 55, 205, 161, respectively.
n = 6: Dressler, Makowski, and Parker proved that the largest integer that is not the sum of distinct primes of the form 12k+1, 12k+5, 12k+7, 12k+11 is 1969, 1349, 1387, 1475.
For n = 4, 5, 7, 8, 9, ..., the largest number that is not the sum of distinct primes of the form 2nk+r seems to be unknown.

			The positive integers that are not the sum of distinct odd primes are A231408 = 1, 2, 4, 6, 9, so a(1) = A231408(5) = 9.

A. Makowski, Partitions into unequal primes, Bull. Acad. Polon. Sci. Sér. Math. Astronom. Phys., 8 (1960), 125-126.

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.
R. E. Dressler, A. Makowski, and T. Parker, Sums of Distinct Primes from Congruence Classes Modulo 12, Math. Comp., 28 (1974), 651-652.
T. Kløve, Sums of Distinct Elements from a Fixed Set, Math. Comp., 29 (1975), 1144-1149.

Cf. A231407, A231408.

A234321 Positive integers that are not the sum of distinct Ramanujan primes.

Original entry on oeis.org

1, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 23, 24, 25, 26, 27, 32, 33, 34, 35, 36, 37, 38, 39, 44, 45, 50, 51, 53, 55, 56, 62, 63, 65, 68, 74, 79, 85, 91, 92, 94, 122

Offset: 1

Jonathan Sondow, Dec 28 2013

Paksoy showed that every integer > 122 is the sum of distinct Ramanujan primes (A104272).

M. B. Paksoy, Derived Ramanujan primes: R'_n, arXiv:1210.6991 [math.NT], 2012.

Cf. A104272, A231407, A231408.

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.

cp's OEIS Frontend

A121571 Largest number that is not the sum of n-th powers of distinct primes.

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

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A234320 Largest number that is not the sum of distinct primes of the form 2k+1, 4k+1, 4k+3, 6k+1, 6k+5, ...; or 0 if none exists.

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A234321 Positive integers that are not the sum of distinct Ramanujan 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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