A025528 -id:A025528 - cp's OEIS frontend

A375734 Indices of consecutive prime-powers (exclusive) differing by 1. Positions of 1's in A057820.

1, 2, 3, 5, 6, 10, 17, 43, 70, 1077, 6635, 12369, 43578, 105102700

Offset: 1

Gus Wiseman, Sep 04 2024

The corresponding prime-powers A246655(a(n)) are given by A006549.

From A006549, it is not known whether this sequence is infinite.

			The fifth prime-power is 7 and the sixth is 8, so 5 is in the sequence.

For nonprime numbers (A002808) we have A375926, differences A373403.
Positions of 1's in A057820.
First differences are A373671.
For nonsquarefree numbers we have A375709, differences A373409.
For non-prime-powers we have A375713.
For non-perfect-powers we have A375740.
For squarefree numbers we have A375927, differences A373127.
Prime-powers:
- terms: A000961, complement A024619.
- differences: A057820.
- runs: A373675, A373673, A373674, A174965
- anti-runs: A373576, A120430, A006549, A373671
Non-prime-powers:
- terms: A361102
- differences: A375708
- runs: A373678, A373676, A373677, A110969 (A373669, sorted A373670).
- anti-runs: A373679, A373575, A255346, A373672
A000040 lists all of the primes, differences A001223.
A025528 counts prime-powers up to n.
Cf. A007916, A014963, A027833, A046933, A053289, A093555, A375714.

Join@@Position[Differences[Select[Range[100],PrimePowerQ]],1]

Numbers k such that A246655(k+1) - A246655(k) = 1.

The inclusive version is a(n) + 1 shifted.

a(14) from Amiram Eldar, Sep 24 2024

A182908 Rank of 2^n when all prime powers (A246655) p^n, for n>=1, are jointly ranked.

Original entry on oeis.org

1, 3, 6, 10, 18, 27, 44, 70, 117, 198, 340, 604, 1078, 1961, 3590, 6635, 12370, 23150, 43579, 82267, 155921, 296347, 564688, 1078555, 2064589, 3958999, 7605134, 14632960, 28195586, 54403835, 105102701, 203287169, 393625231, 762951922, 1480223716, 2874422303

Offset: 1

Clark Kimberling, Dec 13 2010

nonn

			a(3)=6 because 2^3 has rank 6 in the sequence (2,3,4,5,7,8,9,...).

Ray Chandler, Table of n, a(n) for n = 1..92 (using b-file file from A007053)

Cf. A000720, A024622, A025528, A246655.

Row 1 of A182869. Complement of A182909.

T[i_,j_]:=Sum[Floor[j*Log[Prime[i]]/Log[Prime[h]]],{h,1,PrimePi[Prime[i]^j]}]; Flatten[Table[T[i,j],{i,1,1},{j,1,22}]]
f[n_] := Sum[ PrimePi[ Floor[2^(n/k)]], {k, n + 1}]; Array[f, 34] (* Robert G. Wilson v, Jul 08 2011 *)

from sympy import primepi, integer_nthroot
def A182908(n):
    x = 1<Chai Wah Wu, Nov 05 2024

a(n) = A182908(n) = A024622(n) - 1 for n>=1.
a(n) = Sum_{i=1..n} pi(floor(2^(n/i))), where pi(n) = A000720(n). - Ridouane Oudra, Oct 26 2020
a(n) = A025528(2^n). - Pontus von Brömssen, Sep 27 2024

Minor edits by Ray Chandler, Aug 20 2021

A376341 Position of first appearance of n in A057820, the sequence of first differences of prime-powers, or 0 if n does not appear.

Original entry on oeis.org

1, 5, 10, 13, 19, 25, 199, 35, 118, 48, 28195587, 61, 3745011205066703, 80, 6635, 312, 1079, 207, 3249254387600868788, 179, 43580, 216, 21151968922, 615, 762951923, 403, 1962, 466, 12371, 245, 1480223716, 783, 494890212533313, 1110, 2064590, 1235, 375744164943287809536

Offset: 1

Gus Wiseman, Sep 22 2024

nonn

For odd n either a(n) or a(n)+1 is in A024622 (unless a(n) = 0), corresponding to cases where the smaller or the larger term in the pair of consecutive prime powers, respectively, is a power of 2. - Pontus von Brömssen, Sep 27 2024

			a(4) = 13, because the first occurrence of 4 in A057820 is at index 13. The corresponding first pair of consecutive prime powers with difference 4 is (19, 23), and a(4) = A025528(23) = 13.
a(61) = A024622(96), because the first pair of consecutive prime powers with difference 61 is (2^96, 2^96+61), and A025528(2^96+61) = A024622(96).

Pontus von Brömssen, Table of n, a(n) for n = 1..60

For compression instead of first appearances we have A376308.
For run-lengths instead of first appearances we have A376309.
For run-sums instead of first appearances we have A376310.
For squarefree numbers instead of prime-powers we have A376311.
The sorted version is A376340.
A000040 lists the prime numbers, differences A001223.
A000961 and A246655 list prime-powers, first differences A057820.
A024619 and A361102 list non-prime-powers, first differences A375708.
A003242 counts compressed compositions, ranks A333489.
A005117 lists squarefree numbers, differences A076259.
A116861 counts partitions by compressed sum, by compressed length A116608.
Cf. A001597, A006549, A007916, A024622, A025475, A025528, A037201, A038664, A053289, A110969, A120430, A174965, A373400, A375706.

mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
q=Differences[Select[Range[100],#==1||PrimePowerQ[#]&]];
Table[Position[q,k][[1,1]],{k,mnrm[q]}]

A057820(a(n)) = n whenever a(n) > 0. - Pontus von Brömssen, Sep 24 2024

Definition modified by Pontus von Brömssen, Sep 26 2024

More terms from Pontus von Brömssen, Sep 27 2024

A302778 Number of "Fermi-Dirac primes" (A050376) <= n.

Original entry on oeis.org

0, 1, 2, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 8, 9, 10, 10, 11, 11, 11, 11, 12, 12, 13, 13, 13, 13, 14, 14, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 17, 17, 18, 18, 18, 18, 19, 19, 20, 20, 20, 20, 21, 21, 21, 21, 21, 21, 22, 22, 23, 23, 23, 23, 23, 23, 24, 24, 24, 24, 25, 25, 26, 26, 26, 26, 26, 26, 27, 27, 28, 28, 29, 29, 29, 29, 29, 29, 30

Offset: 1

Antti Karttunen, Apr 16 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Partial sums of A302777. A left inverse of A050376.
Cf. A302785, A302786.
Differs from A203967 for the first time at n=64, where a(64) = 23, while A203967(64) = 24.
Cf. also A000720, A025528.

s[n_] := Boole[n > 1 && Length[(f = FactorInteger[n])] == 1 && (e = f[[;; , 2]]) == 2^IntegerExponent[e, 2]]; Accumulate @ Array[s, 100] (* Amiram Eldar, Nov 27 2020 *)

A209229(n) = (n && !bitand(n,n-1));
A302777(n) = A209229(isprimepower(n));
s=0; for(n=1,105,s+=A302777(n); print1(s,", "));

from sympy import primepi, integer_nthroot
def A302778(n): return sum(primepi(integer_nthroot(n,1<Chai Wah Wu, Feb 18-19 2025

a(1) = 0; for n > 1, a(n) = A302777(n) + a(n-1).

For all n >= 1, a(A050376(n)) = n.

A024620 Positions of primes among the powers of primes (A000961).

Original entry on oeis.org

2, 3, 5, 6, 9, 10, 12, 13, 14, 17, 18, 20, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 34, 35, 36, 37, 38, 39, 40, 41, 44, 46, 47, 48, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 70, 72, 73, 74, 75, 76, 77, 78, 80, 81, 82, 83, 84, 85, 86, 88, 89, 90, 91, 93, 94

Offset: 1

Clark Kimberling

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Complement of A024621.
Cf. A001222 (bigomega), A025474, A056604, A027883.
Cf. A025528, A065515, A000040.

a024620 n = a024620_list !! (n-1)
a024620_list = filter ((== 1) . a025474) [1..]
-- Reinhard Zumkeller, May 01 2015

a[n_] := PrimeOmega[LCM @@ Range@Prime@n] + 1; Array[a, 100] (* Amiram Eldar, Dec 02 2018 *)

lista(nn) = my(powpr = select((i->((omega(i)==1) || (i==1))), [1..nn])); for (i = 1, #powpr, if (isprime(powpr[i]), print1(i, ", ")); ); \\ Michel Marcus, Jun 03 2021

from sympy import prime, primepi, integer_nthroot
def A024620(n):
    x = prime(n)
    return n+1+sum(primepi(integer_nthroot(x,k)[0]) for k in range(2,x.bit_length())) # Chai Wah Wu, Nov 05 2024

A025474(a(n)) = 1. - Reinhard Zumkeller, May 01 2015
a(n) = A001222(A056604(n)) + 1. - Eric Desbiaux, Dec 02 2018
From Ridouane Oudra, Oct 18 2020: (Start)
a(n) = A027883(n) + 1;
a(n) = A025528(A000040(n)) + 1;
a(n) = A065515(A000040(n)). (End)

A024622 Position of 2^n among the powers of primes (A000961).

Original entry on oeis.org

1, 2, 4, 7, 11, 19, 28, 45, 71, 118, 199, 341, 605, 1079, 1962, 3591, 6636, 12371, 23151, 43580, 82268, 155922, 296348, 564689, 1078556, 2064590, 3959000, 7605135, 14632961, 28195587, 54403836, 105102702, 203287170, 393625232, 762951923, 1480223717, 2874422304

Offset: 0

Clark Kimberling

nonn

Number of prime powers <= 2^n. - Jon E. Schoenfield, Nov 06 2016

A000961(a(n)) = A000079(n); also position of record values in A192015: A001787(n) = A192015(a(n)). - Reinhard Zumkeller, Jun 26 2011

Ray Chandler, Table of n, a(n) for n = 0..92 (using b-file from A007053, first 61 terms from Hiroaki Yamanouchi)

Cf. A000079, A000720, A000961, A001787, A007053, A025528, A182908, A192015.

{1}~Join~Flatten[1 + Position[Select[Range[10^6], PrimePowerQ], k_ /; IntegerQ@ Log2@ k ]] (* Michael De Vlieger, Nov 14 2016 *)

lista(nn) = {v = vector(2^nn, i, i); vpp = select(x->ispp(x), v); print1(1, ", "); for (i=1, #vpp, if ((vpp[i] % 2) == 0, print1(i, ", ")););} \\ Michel Marcus, Nov 17 2014

a(n)=sum(k=1,n,primepi(sqrtnint(2^n,k)))+1 \\ Charles R Greathouse IV, Nov 21 2014

a(n)=my(s=0);for(i=1, 2^n, isprimepower(i) && s++);s+1 \\ Dana Jacobsen, Mar 23 2021

use ntheory ":all"; for my $n (0..20) { my $s=1; is_prime_power($) && $s++ for 1..2**$n; print "$n $s\n" } # _Dana Jacobsen, Mar 23 2021

use ntheory ":all"; for my $n (0..64) { my $s = ($n < 1) ? 1 : vecsum(map{prime_count(rootint(powint(2,$n)-1,$))}1..$n)+2; print "$n $s\n"; } # _Dana Jacobsen, Mar 23 2021

# with b-file for pi(2^n)
perl -Mntheory=:all -nE 'my($n,$pc)=split; say "$n ", addint($pc,vecsum( map{prime_count(rootint(powint(2,$n),$))} 2..$n )+1);'  b007053.txt  # _Dana Jacobsen, Mar 23 2021

from sympy import primepi, integer_nthroot
def A024622(n):
    x = 1<Chai Wah Wu, Nov 05 2024

def a(n): return sum(prime_pi(ZZ(2^n).nth_root(k+1,truncate_mode=1)[0]) for k in range(n))+1 # Dana Jacobsen, Mar 23 2021

From Ridouane Oudra, Oct 26 2020: (Start)
a(n) = 1 + Sum_{i=1..n} pi(floor(2^(n/i))), where pi(n) = A000720(n);
a(n) = 1 + A182908(n). (End)
a(n) = A025528(2^n)+1. - Pontus von Brömssen, Sep 28 2024

a(28)-a(36) from Hiroaki Yamanouchi, Nov 21 2014

a(46)-a(53) corrected by Hiroaki Yamanouchi, Nov 15 2016

A069637 Number of prime powers <= n with exponents > 1.

Original entry on oeis.org

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

Offset: 1

Amarnath Murthy, Mar 27 2002

nonn

Counts A025475 without 1 = prime^0: a(n) = A085501(n) - 1. - Reinhard Zumkeller, Jul 03 2003

Counts the prime powers (A246655) without the primes. - Peter Luschny, Nov 18 2019

H. Sahu, K. Kar and B.S.K.R. Somayajulu, On the average order of pi*(n) - pi(n), Acta Cienc. Indica Math., Vol. 11 (1985), pp. 165-168.
József Sándor, Dragoslav S. Mitrinovic and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter VII, p. 237.

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

Cf. A025528, A000720, A025475, A085501, A246655.

Partial sums of A268340.

with(numtheory);
A069637 := proc(N) local ct,i; ct:=0;
for i from 1 to N do if not isprime(i) and nops(factorset(i))=1 then ct:=ct+1; fi; od; ct; end; # N. J. A. Sloane, Jun 05 2022

Table[Sum[PrimePi[n^(1/k)], {k, Log[2, n]}]-PrimePi[n],{n,94}] (* Stefano Spezia, Jun 05 2022 *)

from sympy import primepi, integer_nthroot
def A069637(n): return sum(primepi(integer_nthroot(n,k)[0]) for k in range(2,n.bit_length())) # Chai Wah Wu, Aug 15 2024

[A025528(n) - prime_pi(n)  for n in (1..100)] # Peter Luschny, Nov 18 2019

a(n) = A025528(n) - A000720(n) = A000720([n^(1/2)]) + A000720([n^(1/3)]) + ... . - Max Alekseyev, May 11 2009

Sum_{k=1..n} a(k) ~ (4/3) * n^(3/2)/log(n) + O(n^(3/2)/log(n)^2) (Sahu et al., 1985). - Amiram Eldar, Mar 07 2021

A082997 a(n) = card{ x <= n : omega(x) = 2 }.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 5, 5, 6, 6, 7, 8, 9, 9, 10, 10, 11, 11, 12, 12, 12, 12, 12, 13, 14, 15, 16, 16, 17, 18, 19, 19, 19, 19, 20, 21, 22, 22, 23, 23, 24, 25, 26, 26, 27, 28, 29, 30, 31, 31, 31, 31, 32, 33, 33, 34, 34, 34, 35, 36, 36, 36, 37, 37, 38, 39, 40, 41

Offset: 1

Benoit Cloitre, May 30 2003

nonn

G. Tenenbaum, Introduction à la théorie analytique et probabiliste des nombres, p. 203, Publications de l'Institut Cartan, 1990.

Daniel Suteu, Table of n, a(n) for n = 1..10000

Cf. A001221, A007774, A025528, A072000, A082998, A346621.

Partial sums of A215480.

a:= proc(n) option remember; `if`(n=0, 0,
      a(n-1)+`if`(nops(ifactors(n)[2])=2, 1, 0))
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Aug 23 2021

a[n_] := Count[PrimeNu[Range[n]], 2];
Array[a, 100] (* Jean-François Alcover, Mar 02 2022 *)

PARI
```
a(n)=sum(i=1,n,if(omega(i)-2,0,1))
```

a(n) = my(s = sqrtint(n), p = 2, j = 1, count = 0); while(p <= s, my(r = nextprime(p+1)); my(t = p); while (t <= n, my(w = n\t); if(r > w, break); count += primepi(w) - j; my(r2 = r); while(r2 <= w, my(u = t*r2*r2); if(u > n, break); while (u <= n, count += 1; u *= r2); r2 = nextprime(r2+1)); t *= p); p = r; j += 1); count; \\ Daniel Suteu, Jul 21 2021

from sympy import factorint
from itertools import accumulate
def cond(n): return int(len(factorint(n))==2)
def aupto(nn): return list(accumulate(map(cond, range(1, nn+1))))
print(aupto(77)) # Michael S. Branicky, Jul 21 2021

a(n) ~ (n/log(n))*log(log(n)).

a(A007774(n)) = n. - Daniel Suteu, Jul 21 2021

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

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.

Original entry on oeis.org

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

cp's OEIS Frontend

A375734 Indices of consecutive prime-powers (exclusive) differing by 1. Positions of 1's in A057820.

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A182908 Rank of 2^n when all prime powers (A246655) p^n, for n>=1, are jointly ranked.

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A376341 Position of first appearance of n in A057820, the sequence of first differences of prime-powers, or 0 if n does not appear.

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A302778 Number of "Fermi-Dirac primes" (A050376) <= n.

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A024620 Positions of primes among the powers of primes (A000961).

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A024622 Position of 2^n among the powers of primes (A000961).

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A069637 Number of prime powers <= n with exponents > 1.

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