A231408 -id:A231408 - cp's OEIS frontend

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).

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.

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A231407 Positive integers that are not the sum of distinct 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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