A337979 -id:A337979 - cp's OEIS frontend

A337978 a(n) = n + pi(n) - pi(n + pi(n)), where pi(n) is the prime count of n (n>=1).

1, 1, 2, 3, 4, 5, 6, 7, 7, 8, 10, 10, 11, 12, 13, 14, 15, 16, 18, 19, 19, 20, 21, 22, 23, 24, 25, 25, 27, 28, 29, 29, 30, 31, 32, 32, 34, 35, 36, 37, 38, 39, 41, 42, 42, 43, 44, 45, 46, 47, 48, 48, 50, 51, 51, 52, 52, 53, 55, 56, 57, 58, 59, 60, 60, 61, 63

Offset: 1

Ya-Ping Lu, Oct 06 2020

nonn

It seems that this is a nondecreasing sequence and a(n) < n for n >= 2.

Proofs of the above observations are provided in the Links below.

Robert Israel, Table of n, a(n) for n = 1..10000
Ya-Ping Lu, Proofs of the two observations in the Comments section

Cf. A000720, A062298, A095117, A337979.

f:= n -> n + numtheory:-pi(n) - numtheory:-pi(n + numtheory:-pi(n)):
map(f, [$1..100]); # Robert Israel, Feb 12 2024

pc[n_]:=With[{c=PrimePi[n]},n+c-PrimePi[n+c]]; Array[pc,70] (* Harvey P. Dale, Jan 18 2024 *)

a(n) = {my(x = n + primepi(n)); x - primepi(x); } \\ Michel Marcus, Oct 06 2020

from sympy import primepi
print(1)
n = 2
for n in range(2, 10001):
    n_f = n + primepi(n)
    a = n_f - primepi(n_f)
    print(a)

a(n) = n + pi(n) - pi(n + pi(n)).

A338260 a(n) is the number of nodes with depth of n in a binary tree defined as: root = 1 and a child (C) of a node (N) is such that A337978(C) = N. For nodes with two children, the smaller child is assigned as the left child and the bigger one as the right child. A child of a one-child node is assigned as the left child.

Original entry on oeis.org

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

Offset: 0

Ya-Ping Lu, Oct 19 2020

nonn

The binary tree, read from left to right in the order of increasing depth n, is the positive integer sequence A000027. The first 66 numbers are shown in the figure below.
                         1
                        2
                       3
                      4
                     5
                    6
                   7
                  8 \_(9)
                10
               11 \_12
              13   14
             15   16
           (17)  18
                19
               20 \_21
              22   23
             24   25
           (26)  27 \______28
                29        30
               31 \_32  (33)
              34   35 \______36
             37   38        39
           (40)  41        42
                43        44 \_45
               46        47   48
             (49)       50   51 \______52
                       53  (54)\_55   56 \______57
                      58        59   60        61
                    (62)       63   64 \_65  (66)
All right children are composite numbers and all prime numbers are left children.
a(n) in this sequence is the number of terms with value of n in A337979.

Cf. A000027, A062298, A095117, A337978, A337979.

from sympy import primepi
def depth(k):
    d = 0
    while k > 1:
        k += primepi(k)
        k -= primepi(k)
        d += 1
    return d
m = 1
for n in range (0, 101):
    a = 0
    while depth(m + a) == n:
        a += 1
    print(a)
    m += a

cp's OEIS Frontend

A337978 a(n) = n + pi(n) - pi(n + pi(n)), where pi(n) is the prime count of n (n>=1).

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