A025528 -id:A025528 - cp's OEIS frontend

A322981 If n is the k-th prime power > 1, a(n) = k, otherwise a(n) = 0.

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

Offset: 1

Antti Karttunen, Jan 01 2019

nonn

This is a ("corrected") variant of A095874, which uses the list of "powers of primes" A000961 instead of prime powers A246655. - M. F. Hasler, Jun 16 2021

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000961 (powers of primes, including 1), A010055, A025528, A069513, A095874 (analog based on A000961), A246655 (prime powers > 1).

Cf. A049084.

up_to = 16384;
A322981list(up_to) = { my(v=vector(up_to), k=0); for(n=1,up_to,if(isprimepower(n),k++; v[n] = k, v[n] = 0)); (v); };
v322981 = A322981list(up_to);
A322981(n) = v322981[n];

A322981(n)=if(isprimepower(n),sum(i=1,exponent(n),primepi(sqrtnint(n,i)))) \\ M. F. Hasler, Jun 16 2021

a(n) = A010055(n) * A025528(n) = A069513(n) * A025528(n).
a(n) = A025528(A069513(n)*n), when assuming that A025528(0) = 0.
a(A000961(1+n)) = n for all n >= 1.

A333235 a(n) is the product of indices of unitary prime power divisors of n.

Original entry on oeis.org

1, 1, 2, 3, 4, 2, 5, 6, 7, 4, 8, 6, 9, 5, 8, 10, 11, 7, 12, 12, 10, 8, 13, 12, 14, 9, 15, 15, 16, 8, 17, 18, 16, 11, 20, 21, 19, 12, 18, 24, 20, 10, 21, 24, 28, 13, 22, 20, 23, 14, 22, 27, 24, 15, 32, 30, 24, 16, 25, 24, 26, 17, 35, 27, 36, 16, 28, 33, 26, 20

Offset: 1

Ilya Gutkovskiy, Mar 12 2020

Equivalently: replace each prime power p^e in the prime factorization of n by its index in A246655. - M. F. Hasler, Jun 16 2021

			a(600) = a(2^3 * 3 * 5^2) = a(A246655(6) * A246655(2) * A246655(14)) = 6 * 2 * 14 = 168.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A003963, A025528, A027883, A141128, A141809, A156061, A182908, A246655.

Cf. A322981 (the index of n = p^e in A246655).

N:= 1000: # for a(1)..a(N)
R:= NULL: p:= 2:
while p < N do
  R:= R,  seq(p^k,k=1..ilog[p](N));
  p:= nextprime(p);
od:
L:= sort([R]):
f:= proc(n) local F, t;
  F:= ifactors(n)[2];
  mul(ListTools:-BinarySearch(L,t[1]^t[2]),t=F)
end proc:
map(f, [$1..N]); # Robert Israel, Feb 11 2021

PrimePowerPi[n_] := Sum[Boole[PrimePowerQ[k]], {k, 1, n}]; a[1] = 1; a[n_] := Times @@ (PrimePowerPi[#[[1]]^#[[2]]] & /@ FactorInteger[n]); Table[a[n], {n, 1, 70}]

apply( {A333235(n)=vecprod([A322981(f[1]^f[2])|f<-factor(n)~])}, [1..99]) \\ M. F. Hasler, Jun 16 2021

If n = Product (p_j^k_j) then a(n) = Product (A025528(p_j^k_j)).
a(prime(n)) = A027883(n).
a(2^n) = A182908(n).
a(A246655(n)) = n.

A378615 Number of non prime powers <= prime(n).

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 6, 7, 10, 13, 14, 18, 21, 22, 25, 29, 34, 35, 39, 42, 43, 48, 50, 55, 62, 65, 66, 69, 70, 73, 84, 86, 91, 92, 101, 102, 107, 112, 115, 119, 124, 125, 134, 135, 138, 139, 150, 161, 164, 165, 168, 173, 174, 182, 186, 191, 196, 197, 202, 205

Offset: 1

Gus Wiseman, Dec 06 2024

nonn

			The non prime powers counted under each term:
  n=1  n=2  n=3  n=4  n=5  n=6  n=7  n=8  n=9  n=10
  -------------------------------------------------
   1    1    1    6   10   12   15   18   22   28
                  1    6   10   14   15   21   26
                       1    6   12   14   20   24
                            1   10   12   18   22
                                 6   10   15   21
                                 1    6   14   20
                                      1   12   18
                                          10   15
                                           6   14
                                           1   12
                                               10
                                                6
                                                1

Restriction of A356068 (first-differences A143731).
First-differences are A368748.
Maxima are A378616.
Other classes of numbers (instead of non prime powers):
- prime: A000027 (diffs A000012), restriction of A000720 (diffs A010051)
- squarefree: A071403 (diffs A373198), restriction of A013928 (diffs A008966)
- nonsquarefree: A378086 (diffs A061399), restriction of A057627 (diffs A107078)
- prime power: A027883 (diffs A366833), restriction of A025528 (diffs A010055)
- composite: A065890 (diffs A046933), restriction of A065855 (diffs  A005171)
A000040 lists the primes, differences A001223
A000961 and A246655 list the prime powers, differences A057820.
A024619 lists the non prime powers, differences A375735, seconds A376599.
A080101 counts prime powers between primes (exclusive), inclusive A366833.
A361102 lists the non powers of primes, differences A375708.
Cf. A027883, A053607, A304521, A343249, A345531, A377057, A377281, A377286, A377289, A377703, A377781, A378032.

Table[Length[Select[Range[Prime[n]],Not@*PrimePowerQ]],{n,100}]

from sympy import prime, primepi, integer_nthroot
def A378615(n): return int((p:=prime(n))-n-sum(primepi(integer_nthroot(p,k)[0]) for k in range(2,p.bit_length()))) # Chai Wah Wu, Dec 07 2024

a(n) = prime(n) - A027883(n). - Chai Wah Wu, Dec 08 2024

A082998 a(n) = card{ x <= n : omega(x) = 3 }.

Original entry on oeis.org

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

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, A025528, A033992, A072000, A072114, A082997.

PARI
```
a(n)=sum(i=1,n,if(omega(i)-3,0,1))
```

a(n, k = 3, m = 1, p = 2, s = sqrtnint(n\m, k), j = 1) = my(count = 0); if (k==2, while(p <= s, my(r = nextprime(p+1)); my(t = m*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); return(count)); while(p <= s, my(r = nextprime(p+1)); my(t = m*p); while(t <= n, my(s = sqrtnint(n\t, k-1)); if(r > s, break); count += a(n, k-1, t, r, s, j+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))==3)
def aupto(nn): return list(accumulate(map(cond, range(1, nn+1))))
print(aupto(105)) # Michael S. Branicky, Jul 21 2021

a(n) ~ (1/2)*(n/log(n))*log(log(n))^2.

a(A033992(n)) = n. - Daniel Suteu, Jul 21 2021

A025532 a(n) is the sum of exponents in the prime factorization of lcm{C(n,0), C(n,1), ..., C(n,n)}.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..2000

{0, 0}~Join~Table[Total@ FactorInteger[LCM @@ Array[Binomial[n, #] &, n]][[All, -1]], {n, 2, 74}] (* Michael De Vlieger, Jan 13 2018 *)

for(n=0, 100, l=1; for(k=0, n, l=lcm(l,binomial(n,k))); v=factor(l); s=0; for(k=1, matsize(v)[1], s=s+v[k,2]); print1(s","))

a(n) = bigomega(lcm(vector(n+1, k, binomial(n, k-1)))); \\ Michel Marcus, Jan 06 2018

a(n) = A025528(n + 1) - A001222(n + 1). - Luc Rousseau, Jan 04 2018

a(n) = A001222(A002944(n+1)). - Michel Marcus, Jan 05 2018

More terms from Ralf Stephan, Mar 28 2003

A330292 a(n) = number of integers 1 <= k < n such that omega(k) <= omega(n), where omega = A001221.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 5, 6, 7, 9, 8, 11, 9, 13, 14, 10, 11, 17, 12, 19, 20, 21, 13, 23, 14, 25, 15, 27, 16, 29, 17, 18, 31, 32, 33, 34, 19, 36, 37, 38, 20, 41, 21, 41, 42, 43, 22, 45, 23, 47, 48, 49, 24, 51, 52, 53, 54, 55, 25, 59, 26, 58, 59, 27, 61, 65, 28, 63, 64

Offset: 1

Dilshod Urazov, Dec 10 2019

Any natural number n can be represented as n = (k_1)^p_1 * (k_2)^p_2 * ... * (k_h)^p_h, where k_i is prime for any i from 1 to h. Let us consider the function omega(n) = h, which represents the number of distinct prime factors of n. Then a(k) is the number of positive integers j less than k for which the value of function omega(j) is <= omega(k).
a(P) = A025528(P) for P a prime power in A246655.
a(Q) = Q - 1 for Q a primorial number in A002110.
Let us consider n > k such that omega(n) = omega(k) = omega and there is no w such that n > w > k and omega(w) > omega. Hence a(n) - a(k) = n - k.

			a(1) = 0: 1 has no predecessor, omega(1) = 0 by convention;
a(2) = 1 because omega(2) = 1, 1 >= omega(0);
a(3) = 2 because omega(3) = 1 and none of omega(1), omega(2) >= 1;
a(4) = 3 because omega(4) = 1 and none of omega(1), omega(2), omega(3) >= 1.

Felix Fröhlich, Table of n, a(n) for n = 1..10000

Cf. A001221 (omega), A025528, A246655, A002110.

a[n_] := Block[{t = PrimeNu[n]}, Length@ Select[Range[n - 1], PrimeNu[#] <= t &]]; Array[a, 70] (* Giovanni Resta, Dec 10 2019 *)

for(n=1,70,my(omn=omega(n),m=0);for(k=1,n-1,if(omega(k)<=omn,m++));print1(m,", ")) \\ Hugo Pfoertner, Dec 10 2019

a(n) = my(omn=omega(n)); sum(k=1, n-1, omega(k) <= omn); \\ Michel Marcus, Dec 11 2019

def primes(n):
    divisors = [ d for d in range(2,n//2+1) if n % d == 0 ]
    return [ d for d in divisors if \
             all( d % od != 0 for od in divisors if od != d ) ]
pprimes = {}
for i in range(1, 10000):
    res = len(primes(i))
    if res == 0:
        res = 1
    pprimes[i] = res
for k in range(1, 10000):
    s = 0
    for i in range(1, k):
        if pprimes[i] <= pprimes[k]:
            s+=1
    print(s)

More terms from Giovanni Resta, Dec 10 2019

A341416 a(n) is the least k such that the product of indices of unitary prime power divisors of k is n.

Original entry on oeis.org

1, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 35, 36, 47, 49, 40, 59, 61, 52, 45, 71, 56, 79, 55, 68, 89, 63, 65, 103, 107, 92, 77, 121, 72, 127, 85, 91, 137, 139, 88, 151, 112, 124, 115, 169, 104, 119, 99, 148, 193, 197, 133, 211, 223, 117, 145, 161, 136, 241, 155, 196

Offset: 1

Robert Israel, Feb 11 2021

a(n) is the least k such that A333235(k) = n.

a(p) = A246655(p) for prime p.

			a(3) = 4 because A333235(4) = 3 and this is the first occurrence of 3 in A333235.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A246655, A333235.

N:= 1000: # for a(1) to a(A025528(N))
R:= NULL: p:= 2:
while p < N do
  R:= R,  seq(p^k,k=1..ilog[p](N));
  p:= nextprime(p);
od:
L:= sort([R]):
M:= nops(L):
f:= proc(n) local F, t;
  F:= ifactors(n)[2];
  mul(ListTools:-BinarySearch(L,t[1]^t[2]),t=F)
end proc:
V:= Vector(M): count:= 0:
for n from 1 while count < M do
  v:= f(n);
if v <= M and V[v] = 0 then count:= count+1; V[v]:= n fi;
od:
convert(V,list);

A333235(a(n)) = n.

A025556 a(n) = sum of the exponents in the prime factorization of LCM{1,3,6,...,C(n+1,2)}.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling

nonn

Index entries for sequences related to lcm's

Cf. A001222, A025528, A025555.

a025555(n) = my(s=1); for(k=1, n, s=lcm(s, k*(k+1)/2)); s \\ after Edward Jiang in A025555
a(n) = bigomega(a025555(n)) \\ Felix Fröhlich, Sep 06 2019

a(n) = A001222(A025555(n)). - Sean A. Irvine, Sep 06 2019

a(n) = A025528(n+1)-1. - Pontus von Brömssen, Sep 28 2024

More terms from Sean A. Irvine, Sep 06 2019

A143039 a(n) = m-th prime power (not counting 1), where m = 10^n.

Original entry on oeis.org

16, 419, 7517, 103511, 1295953, 15474787, 179390821, 2037968761, 22801415981, 252096675073, 2760723662941, 29996212395727, 323780470283789, 3475385632514321, 37124507635789309, 394906912575042053, 4185296577158764117, 44211790220874464021

Offset: 1

Lekraj Beedassy, Jul 18 2008

nonn

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 455, pp 84, Ellipses, Paris 2008.

Cf. A000961.

a(n)=A000961(10^n + 1).

A025528(a(n)) = 10^n.

Edited and extended by Max Alekseyev, May 13 2009

A327247 Number of odd prime powers <= n (with exponents > 0).

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Sep 14 2019

nonn

Cf. A000523, A025528, A033270, A061345, A065515, A069637, A085501, A174275 (first differences), A246655, A285879.

Table[Sum[Boole[OddQ[k] && PrimePowerQ[k]], {k, 1, n}], {n, 1, 75}]

a(n) = {sum(k=2, primepi(n), logint(n, prime(k)))} \\ Andrew Howroyd, Sep 14 2019

a(n) = A025528(n) - A000523(n).

cp's OEIS Frontend

A322981 If n is the k-th prime power > 1, a(n) = k, otherwise a(n) = 0.

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A333235 a(n) is the product of indices of unitary prime power divisors of n.

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A378615 Number of non prime powers <= prime(n).

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A082998 a(n) = card{ x <= n : omega(x) = 3 }.

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A025532 a(n) is the sum of exponents in the prime factorization of lcm{C(n,0), C(n,1), ..., C(n,n)}.

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A330292 a(n) = number of integers 1 <= k < n such that omega(k) <= omega(n), where omega = A001221.

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A341416 a(n) is the least k such that the product of indices of unitary prime power divisors of k is n.

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