A158924 -id:A158924 - cp's OEIS frontend

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

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

A158923 a(1) = 2, a(n) = a(n-1) + round(log(a(n-1))) for n >= 2.

Original entry on oeis.org

2, 3, 4, 5, 7, 9, 11, 13, 16, 19, 22, 25, 28, 31, 34, 38, 42, 46, 50, 54, 58, 62, 66, 70, 74, 78, 82, 86, 90, 94, 99, 104, 109, 114, 119, 124, 129, 134, 139, 144, 149, 154, 159, 164, 169, 174, 179, 184, 189, 194, 199, 204, 209, 214, 219, 224, 229, 234, 239, 244, 249

Offset: 1

Daniel Forgues, Mar 30 2009

nonn

Each interval (a(n-1), a(n)] asymptotically contains one prime power on the average.

Daniel Forgues, Table of n, a(n) for n = 1..100000

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."
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."

NestList[# + Round@ Log[#] &, 2, 60] (* Michael De Vlieger, Nov 05 2020 *)

from math import log
print(2)
a_last = n = 2
while n >= 2:
    a = a_last + int(log(a_last) + 0.5)
    print(a)
    a_last = a
    n += 1 # Ya-Ping Lu, Oct 24 2020

cp's OEIS Frontend

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

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