A231407 -id:A231407

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A121571

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

+10
7

6, 17163, 1866000

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

REFERENCES

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

LINKS

Table of n, a(n) for n=1..3.

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.

FORMULA

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

EXAMPLE

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

CROSSREFS

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).

KEYWORD

nonn,hard,more,bref

AUTHOR

T. D. Noe, Aug 08 2006

STATUS

approved

A231408

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

+10
3

1, 2, 4, 6, 9

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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

LINKS

Table of n, a(n) for n=1..5.

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.

CROSSREFS

Cf. A121571, A231407.

KEYWORD

fini,full,nonn

AUTHOR

Jonathan Sondow, Nov 24 2013

STATUS

approved

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.

+10
1

9, 121, 55, 205, 161

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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.

REFERENCES

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

LINKS

Table of n, a(n) for n=1..5.

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.

EXAMPLE

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.

CROSSREFS

Cf. A231407, A231408.

KEYWORD

nonn,more

AUTHOR

Jonathan Sondow, Dec 28 2013

STATUS

approved

A234321

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

+10
0

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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

LINKS

Table of n, a(n) for n=1..48.

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

CROSSREFS

Cf. A104272, A231407, A231408.

KEYWORD

fini,full,nonn

AUTHOR

Jonathan Sondow, Dec 28 2013

STATUS

approved

A352600

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

+10
0

3, 2438, 483370

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Table of n, a(n) for n=1..3.

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

CROSSREFS

Cf. A121518, A121571, A173563, A231407.

KEYWORD

nonn,bref,hard,more

AUTHOR

Ilya Gutkovskiy, Mar 22 2022

STATUS

approved

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