A024620 -id:A024620 - cp's OEIS frontend

A366833 Number of times n appears in A362965 (number of primes <= the n-th prime power).

1, 2, 1, 3, 1, 2, 1, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Paolo Xausa, Oct 25 2023

nonn

Conjecture: a(n) can be only 1, 2, or 3 (with the first occurrences of 3 appearing at n = 4, 9, 30, 327 and 3512).

One less than the number of prime powers between prime(n) and prime(n+1), inclusive. - Gus Wiseman, Jan 09 2025

Paolo Xausa, Table of n, a(n) for n = 1..10000
Paolo Xausa, 1200 X 1200 raster image of a(n), n = 1..1440000, read left to right, top to bottom, showing a(n) = 1 in blue, a(n) = 2 in white and a(n) = 3 in red.

Run lengths of A362965.
Subtracting one gives A080101.
For non prime powers we have A368748.
Positions of terms > 1 are A377057.
Positions of 1 are A377286.
Positions of 2 are A377287.
For perfect powers we have A377432.
For squarefree we have A373198.
A000015 gives the least prime power >= n, difference A377282.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820.
A024619 and A361102 list the non prime powers, differences A375708 and A375735.
A031218 gives the greatest prime power <= n, difference A276781.
A046933(n) counts the interval from A008864(n) to A006093(n+1).
A246655 lists the prime powers not including 1.
A366835 counts primes between prime powers.
Cf. A053607, A053706, A065514, A068435, A080102, A304521, A345531, A377289.
Cf. A024620, A027883, A067871, A075526, A080769, A151800, A377436, A379157.

With[{upto=1000},Map[Length,Most[Split[PrimePi[Select[Range[upto],PrimePowerQ]]]]]] (* Considers prime powers up to 1000 *)

a(n) = A080101(n) + 1. - Gus Wiseman, Jan 09 2025

A027883 Positions of primes in sequence (A246655) of primes and prime powers {p^i, i >= 1}.

Original entry on oeis.org

1, 2, 4, 5, 8, 9, 11, 12, 13, 16, 17, 19, 20, 21, 22, 24, 25, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 43, 45, 46, 47, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 69, 71, 72, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83, 84, 85, 87, 88, 89

Offset: 1

N. J. A. Sloane

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A024620, A246655.

from sympy import prime, primepi, integer_nthroot
def A027883(n):
    p = prime(n)
    return n+sum(primepi(integer_nthroot(p,k)[0]) for k in range(2,p.bit_length())) # Chai Wah Wu, Dec 08 2024

a(n) = A024620(n) - 1.

More terms from Erich Friedman.

Name clarified by Ilya Gutkovskiy, Mar 12 2020

A024621 Positions of nonprimes among the powers of primes (A000961).

Original entry on oeis.org

1, 4, 7, 8, 11, 15, 16, 19, 24, 28, 33, 42, 43, 45, 54, 69, 71, 79, 87, 92, 118, 121, 137, 153, 171, 188, 199, 245, 248, 293, 314, 341, 360, 361, 364, 393, 446, 483, 526, 559, 605, 651, 699, 719, 750, 857, 894, 930, 935, 1050, 1079, 1215, 1305, 1348, 1436, 1479, 1514

Offset: 1

Clark Kimberling

nonn

Complement of A024620.

lista(nn) = {vec = vector(nn, i, i); powpr = select((i->((omega(i)==1) || (i==1))), vec); for (i = 1, #powpr, if (! isprime(powpr[i]), print1(i, ", ")););} \\ Michel Marcus, Oct 02 2013

from sympy import primepi, integer_nthroot
def A024621(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return int(n+x-1-sum(primepi(integer_nthroot(x,k)[0]) for k in range(2,x.bit_length())))
    return n+primepi(bisection(f,n,n)) # Chai Wah Wu, Nov 05 2024

A182909 Ranks of composite numbers when all prime powers p^n for n>=1 are jointly ranked.

Original entry on oeis.org

3, 6, 7, 10, 14, 15, 18, 23, 27, 32, 41, 42, 44, 53, 68, 70, 78, 86, 91, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142

Offset: 1

Clark Kimberling, Dec 13 2010

nonn

The complement of A027883.

			In the sequence A000961 (2,3,4,5,7,8,9,11,13,16,17,19,23,25,27,...) of prime powers p^n for n>=1, the composites 4,8,9,16,25,27,... occur with ranks 3,6,7,10,14,15...

Cf. A000961, A024620, A027883.

T[i_,j_]:=Sum[Floor[j*Log[Prime[i]]/Log[Prime[h]]],{h,1,PrimePi[Prime[i]^j]}]; Complement[Range[200],Flatten[Table[T[i,j],{i,1,80},{j,1,1}]]]

A344948 Primes whose position among the powers of primes (A000961) is also prime.

Original entry on oeis.org

2, 3, 5, 19, 29, 47, 67, 73, 101, 113, 137, 167, 193, 199, 239, 263, 313, 349, 389, 419, 431, 449, 461, 487, 571, 599, 641, 701, 719, 751, 797, 823, 857, 887, 911, 977, 991, 1019, 1097, 1193, 1223, 1231, 1277, 1301, 1307, 1399, 1439, 1481, 1511, 1531, 1571, 1601

Offset: 1

Michel Marcus, Jun 03 2021

nonn

			The position of 2 in A000961 is 2; so 2 is a term.
The position of 19 in A000961 is 13; so 19 is a term.

Michel Marcus, Table of n, a(n) for n = 1..10000

Cf. A000040, A000961, A024620.

pow = Select[Range[1600], # == 1 || PrimePowerQ[#] &]; Select[pow[[Select[Range @ Length[pow], PrimeQ]]], PrimeQ] (* Amiram Eldar, Jun 03 2021 *)

allmps(nn) = {my(map = Map()); mapput(map, 1, 1); my(nb=1); for (n=2, nn, if (isprimepower(n), nb++; mapput(map, n, nb));); map;}
lista(nn) = {my(nb = prime(nn), map = allmps(nb)); forprime (p=1, nn, if( isprime(mapget(map, p)), print1(p, ", ")););}

cp's OEIS Frontend

A366833 Number of times n appears in A362965 (number of primes <= the n-th prime power).

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A027883 Positions of primes in sequence (A246655) of primes and prime powers {p^i, i >= 1}.

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A024621 Positions of nonprimes among the powers of primes (A000961).

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A182909 Ranks of composite numbers when all prime powers p^n for n>=1 are jointly ranked.

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Mathematica

A344948 Primes whose position among the powers of primes (A000961) is also prime.

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Mathematica

PARI