A061568 -id:A061568 - cp's OEIS frontend

A321578 a(n) is the maximum value of k such that A007504(k) <= prime(n).

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

Offset: 1

David James Sycamore, Nov 12 2018

nonn

Let A be A007504. The number of distinct values of k such that a(k)=r is the number of primes p in the interval A(r) <= p < A(r+1); namely: 2,2,2,3,3,4,5,4,6,6,... (see A323701). Let b(n) be the smallest r such that a(r)=n, namely: 1,3,5,7,10,13,17,22,26,... For given n, if k is the index of the smallest prime >= A(n), then b(n)=k. (The equality applies when n is a term of A013916.)

			a(1)=1 since prime(1)=2 and 1 is max k such that A007504(k) <= 2.
a(5)=3 since prime(5)=11 and 3 is max k such that A007504(k) <= 11.
n=4 (in A013916). A(4)=17=prime(7), so b(4)=7.
n=7 (not in A013916). A(7)=58 < 59=prime(17), so b(7)=17.

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

Cf. A000040, A007504, A013916, A323701, A061568.

nn = 75; MapIndexed[Set[s[First[#2]], #1] &, Accumulate[Prime@ Range[nn]]]; k = 1; Table[p = Prime[n]; While[s[k] <= p, k++]; k - 1, {n, nn}] (* Michael De Vlieger, Jun 29 2025 *)

a(n) = my(k=0, p=0, s=0); while(s <= prime(n), k++; p=nextprime(p+1); s+=p); k-1; \\ Michel Marcus, Feb 19 2019

use ntheory ':all'; sub a { my $p = nth_prime($[0]); my($s, $q) = (0, 2); while ($s <= $p) { $s += $q; $q = next_prime($q) }; prime_count($q-1)-1 }; print join(", ", map { a($) } 1..100), "\n"; # Daniel Suteu, Jan 26 2019

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

Original entry on oeis.org

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

cp's OEIS Frontend

A321578 a(n) is the maximum value of k such that A007504(k) <= prime(n).

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