A158923 -id:A158923 - cp's OEIS frontend

A158924 Number of prime powers - 1 in interval (A158923(n-1), A158923(n)] expressing the excess or deficit relative to the asymptotic average of 1.

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, -1, 1, 0, 1, 0, 0, 0, 0, 1, 0, -1, 1, 0, 0, 1, -1, 1, 0, 0, -1, 0, 1, 1, 0, -1, 0, 2, 0, 1, -1, 0, 0, 0, 0, 1, 0, 0, 0, -1, 1, 1, -1, -1, 0, -1, 0, 1, 0, 0, 1, -1, 0, 1, 0, 1, 0, 1, 0, 0, -1, 0, 1, 0, -1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, -1, 1, 0, 0, 0, 0, 1, 0

Offset: 1

Daniel Forgues, Mar 31 2009

sign

The first interval is assumed to be (1, A158923(1)].

Daniel Forgues, Table of n, a(n) for n=1..9696

Cf. A158923: a(1) = 2, a(n) = a(n-1) + round(log(a(n-1))), n >= 2, for which each (a(n-1), a(n)] interval asymptotically contains one prime power on average.
Cf. A158925: Accumulated excess or deficit of prime powers in (1, A158924(n)] (Partial sums of A158924).
Cf. A000961 Prime powers p^k (p prime, k >= 0).
Cf. A025528 Number of prime powers <= n with exponents >0.

Corrected and edited by Daniel Forgues, Apr 21 2009

A338396 The prime numbers in A158923.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 19, 31, 109, 139, 149, 179, 199, 229, 239, 683, 739, 809, 823, 907, 977, 991, 1019, 1033, 1061, 1103, 1117, 1187, 1201, 1229, 1327, 1439, 1453, 1481, 1523, 1579, 1607, 1621, 1663, 1733, 1747, 1789, 4931, 4967, 5003, 5021, 5039, 5147, 5237

Offset: 1

Ya-Ping Lu, Oct 23 2020

nonn

This sequence is consists of the prime numbers in A158923, in which A158923(m) is the sum of the previous term A158923(m-1) and the average prime gap log(A158923(m-1)) rounded to the nearest integer with A158923(1) = 2.

Cf. A000040, A158923.

from sympy import isprime
from math import log
print(2)
a_last = p_last = m = 2
n = 1
while m >= 2:
    a = a_last + int(log(a_last) + 0.5)
    if isprime(a) == 1:
        print(a)
        n += 1
    a_last = a
    m += 1

A158925 Accumulated excess or deficit of prime powers in (1, A158924(n)].

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 1, 2, 2, 3, 3, 3, 3, 3, 4, 4, 3, 4, 4, 4, 5, 4, 5, 5, 5, 4, 4, 5, 6, 6, 5, 5, 7, 7, 8, 7, 7, 7, 7, 7, 8, 8, 8, 8, 7, 8, 9, 8, 7, 7, 6, 6, 7, 7, 7, 8, 7, 7, 8, 8, 9, 9, 10, 10, 10, 9, 9, 10, 10, 9, 9, 9, 9, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 10, 11, 10, 11, 11, 11

Offset: 1

Daniel Forgues, Mar 31 2009

sign

Partial sums of A158924.

First term < 0: a(9503) = -1. - Sidney Cadot, Sep 23 2015

Daniel Forgues, Table of n, a(n) for n=1..9696

Cf. A158924, "Number of prime powers - 1 in interval (A158923(n-1), A158923(n)] expressing the excess or deficit relative to the asymptotic average of 1."

Corrected and edited by Daniel Forgues, Apr 22 2009

A338404 Primes p in A158932 such that p = prime(k) = A158932(k) for some k.

Original entry on oeis.org

2, 3, 397217, 412193, 418927, 421163, 421501, 423763, 426077, 431797, 454859, 456367, 456523, 458993, 475529, 480989, 482393, 484733, 501451, 1003133, 1003469, 1003763, 1003819, 1003931, 1007599, 1007711, 1392851, 1393103, 1393159, 1393187, 1393229, 1393313

Offset: 1

Ya-Ping Lu, Oct 24 2020

nonn

Sequence A158923, in which every term is the sum of the previous term and the average prime gap, is a "simulation" of the prime number sequence A000040. This sequence lists the terms in A158923 that match those in A000040 both in value and in position, or A158923(m) = A000040(m).

There are 68 matches found for m up to 1073741824 (prime(1073741824)=24563311309), with a(68) = 12496469849.

Cf. A000040, A158923, A338396.

lista(nn) = {my(a = 2); for (n=1, nn, if (a == prime(n), print1(a, ", ")); a += round(log(a)););} \\ Michel Marcus, Oct 31 2020

from sympy import nextprime
from math import log
print(2)
a_last = b_last = m = 2
n = 1
while m >= 2:
    a = a_last + int(log(a_last) + 0.5)
    b = nextprime(b_last)
    if a == b:
        n += 1
        print (m)
    a_last = a
    b_last = b
    m += 1

a(n) = A158923(m), where m is the n-th index in A158923 such that A158923(m) = A000040(m).

cp's OEIS Frontend

A158924 Number of prime powers - 1 in interval (A158923(n-1), A158923(n)] expressing the excess or deficit relative to the asymptotic average of 1.

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A338396 The prime numbers in A158923.

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A158925 Accumulated excess or deficit of prime powers in (1, A158924(n)].

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A338404 Primes p in A158932 such that p = prime(k) = A158932(k) for some k.

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