A321578 -id:A321578 - cp's OEIS frontend

A323701 a(n) is the number of primes p such that A007504(n) <= p < A007504(n+1).

2, 2, 2, 3, 3, 4, 5, 4, 6, 6, 7, 7, 8, 7, 9, 10, 10, 8, 12, 12, 11, 12, 12, 15, 14, 14, 14, 14, 17, 17, 16, 17, 19, 19, 22, 16, 24, 21, 20, 20, 20, 28, 22, 26, 21, 24, 28, 23, 31, 23, 30, 28, 28, 32, 28, 31, 30, 27, 35, 30, 32, 31, 38, 34, 38, 36, 36, 37, 35, 35

Offset: 1

David James Sycamore, Jan 24 2019

nonn

Corresponds to the number of terms in A321578 which are equal to n.

			a(1)=2 because 2 <= 2, 3 < 5;
a(5)=3 because 28 <= 29, 31, 37 < 41.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000040, A007504, A321578, A061568.

nn = 1375; MapIndexed[Set[s[First[#2]], #1] &, Accumulate[Prime@ Range[nn]]]; c = 0; j = k = 1; Rest@ Reap[Do[p = Prime[n]; While[s[k] <= p, k++]; If[j == k, c++, Sow[c + 1]; c = 0]; j = k, {n, nn}] ][[-1, 1]] (* Michael De Vlieger, Jun 29 2025 *)

use ntheory ':all'; sub a { my $p = nth_prime($[0]+1); my $s = sum_primes($p); prime_count($s-$p, $s-1) }; print join(", ", map { a($) } 1..100), "\n"; # Daniel Suteu, Jan 25 2019

More terms from Daniel Suteu, Jan 25 2019

A386515 a(n) is the largest number of distinct primes in a partition of prime(n) into primes.

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 4, 4, 4, 4, 5, 5, 6, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 15, 15, 15

Offset: 1

Alexander R. Povolotsky, Aug 21 2025

For each prime number prime(n) find all sums of smaller prime numbers which add up to this prime number. Among those sums find the largest number of distinct primes.

			Examples of such partitions for n = 3..11:
  prime(3) = 5 = 2 + 3 which gives a(3) = 2;
  prime(4) = 7 = 2 + 5 which gives a(4) = 2;
  prime(5) = 11 = 2 + 2 + 2 + 2 + 3 = 3 + 3 + 5 which gives a(5)=2;
  prime(6) = 13 = 2 + 3 + 5 + 3 which gives a(6)=3;
  prime(7) = 17 = 2 + 3 + 5 + 7 which gives a(7)=4;
  prime(8) = 19 = 2 + 3 + 5 + 7 + 2 which gives a(8)=4;
  prime(9) = 23 = 2 + 3 + 5 + 13 which gives a(9)=4;
  prime(10) = 29 = 2 + 3 + 5 + 19 which gives a(10)=4;
  prime(11) = 31 = 2 + 3 + 5 + 7 + 7 + 7 which gives a(11)=4.

Cf. A000040, A229219, A321578.

a(n) <= A321578(n). - David A. Corneth, Aug 22 2025

More terms from David A. Corneth, Aug 22 2025

cp's OEIS Frontend

A323701 a(n) is the number of primes p such that A007504(n) <= p < A007504(n+1).

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