A010055 -id:A010055 - cp's OEIS frontend

A119288 a(n) is the second smallest prime factor of n, or 1 if n is a prime power.

1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 1, 3, 1, 7, 5, 1, 1, 3, 1, 5, 7, 11, 1, 3, 1, 13, 1, 7, 1, 3, 1, 1, 11, 17, 7, 3, 1, 19, 13, 5, 1, 3, 1, 11, 5, 23, 1, 3, 1, 5, 17, 13, 1, 3, 11, 7, 19, 29, 1, 3, 1, 31, 7, 1, 13, 3, 1, 17, 23, 5, 1, 3, 1, 37, 5, 19, 11, 3, 1, 5, 1, 41, 1, 3, 17, 43, 29, 11, 1, 3, 13, 23

Offset: 1

Reinhard Zumkeller, May 13 2006

nonn

Least prime factor of {n divided by the maximal power of the least prime factor of n}. - after the original name of the sequence.
a(n) = A020639(A028234(n)).
a(n) = 1 iff n is a prime power: a(A000961(n))=1 and a(A024619(n))>1.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A014673, A020639, A028234, A292269.

Join[{1},Table[Which[PrimePowerQ[n],1,True,FactorInteger[n][[2,1]]],{n,2,100}]] (* Harvey P. Dale, Feb 08 2020 *)

a(n) = if (isprimepower(n) || (n==1), 1, my(f=factor(n)[,1]); f[2]); \\ Michel Marcus, Mar 01 2023

from sympy import primefactors
def A119288(n): return 1 if len(s:=primefactors(n)) <= 1 else sorted(s)[1] # Chai Wah Wu, Mar 31 2023

A010055(n) = 0^(a(n)-1). - Reinhard Zumkeller, May 13 2006

Name changed by Antti Karttunen, Oct 04 2017

A100995 If n is a prime power p^m, m >= 1, then m, otherwise 0.

Original entry on oeis.org

0, 1, 1, 2, 1, 0, 1, 3, 2, 0, 1, 0, 1, 0, 0, 4, 1, 0, 1, 0, 0, 0, 1, 0, 2, 0, 3, 0, 1, 0, 1, 5, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 6, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 4, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0

Offset: 1

Reinhard Zumkeller, Nov 26 2004

nonn

Calculate matrix powers: (A175992^1)/1 - (A175992^2)/2 + (A175992^3)/3 - (A175992^4)/4 + ... Then the nonzero values of a(n) are found as reciprocals in the first column. Compare this to the Taylor series for log(1+x) = (x)/1 - (x^2)/2 + (x^3)/3 - (x^4)/4 + ... Therefore it is natural to write 0, 1/1, 1/1, 1/2, 1/1, 0, 1/1, 1/3, 1/2, 0, 1/1, ... Raising n to a such power gives A014963. - Mats Granvik, Gary W. Adamson, Apr 04 2011
The Dirichlet series that generates the reciprocals of this sequence is the logarithm of the Riemann zeta function. - Mats Granvik, Gary W. Adamson, Apr 04 2011
Number of automorphisms of the finite field with n elements, or 0 if the field does not exist. For n=p^k where p is a prime and k is an integer, the automorphism group of the finite field with n elements is a cyclic group of order k generated by the Frobenius endomorphism. - Yancheng Lu, Jan 11 2021

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

Cf. A028233, A069513, A010055.

a100995 n = f 0 n where
   f e 1 = e
   f e x = if r > 0 then 0 else f (e + 1) x'
           where (x', r) = divMod x p
   p = a020639 n
-- Reinhard Zumkeller, Mar 19 2013

f:= proc(n) local F;
    F:= ifactors(n)[2];
    if nops(F) = 1 then F[1][2]
    else 0
    fi
end proc:
map(f, [$1..100]); # Robert Israel, Jun 09 2015

ppm[n_]:=If[PrimePowerQ[n],FactorInteger[n][[1,2]],0]; Array[ppm,110] (* Harvey P. Dale, Mar 03 2014 *)
a=Table[Limit[Sum[If[Mod[n, k] == 0, MoebiusMu[n/k]/(n/k)^(s - 1)/(1 - 1/n^(s - 1)), 0], {k, 1, n}], s -> 1], {n, 1, 105}];
Numerator[a]*Denominator[a] (* Mats Granvik, Jun 09 2015 *)
a = FullSimplify[Table[MangoldtLambda[n]/Log[n], {n, 1, 105}]]
Numerator[a]*Denominator[a] (* Mats Granvik, Jun 09 2015 *)

{a(n) = my(t); if( n<1, 0, t = factor(n); if( [1,2] == matsize(t), t[1,2], 0))} /* Michael Somos, Aug 15 2012 */

{a(n) = my(t); if( n<1, 0, if( t = isprimepower(n), t))} /* Michael Somos, Aug 15 2012 */

A100994(n) = A014963(n)^a(n);
a(A000961(n)) = A025474(n).
a(n) = Sum_{d|n, gcd(d, n/d) = 1} (-1)^omega(n/d) * bigomega(d). - Ilya Gutkovskiy, Apr 15 2021

Edited by Daniel Forgues and N. J. A. Sloane, Aug 18 2009

A069513 Characteristic function of the prime powers p^k, k >= 1.

Original entry on oeis.org

0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0

Offset: 1

Benoit Cloitre, Apr 15 2002

Also, number of Galois fields of order n. - Charles R Greathouse IV, Mar 12 2008

Also, number of abelian indecomposable groups of order n. - Kevin Lamoreau, Mar 13 2023

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

Cf. A246655, A010055, A001221, A001222, A008683, A090751, A000688, A361414.
The partial sums of this sequence give A025528. - Daniel Forgues, Mar 02 2009
Cf. A014963, A003418. - Enrique Pérez Herrero, Jun 01 2011

a069513 1 = 0
a069513 n = a010055 n  -- Reinhard Zumkeller, Mar 19 2013

A069513 := proc(n)
    if n = 1 then
        0 ;
    elif A001221(n) > 1 then
        0;
    else
        1 ;
    end if ;
end proc:
seq(A069513(n),n=1..80) ; # R. J. Mathar, Nov 02 2016

A069513[n_]:=Boole[PrimeNu[n]==1]; A069513/@Range[20] (* Enrique Pérez Herrero, May 30 2011 *)

PARI
```
for(n=1,120,print1(omega(n)==1,","))
```

from sympy import primefactors
def A069513(n): return int(len(primefactors(n)) == 1) # Chai Wah Wu, Mar 31 2023

If n >= 2, a(n) = A010055(n).
a(n) = Sum_{d|n} bigomega(d)*mu(n/d); equivalently, Sum_{d|n} a(d) = bigomega(n); equivalently, Möbius transform of bigomega(n).
Dirichlet g.f.: ppzeta(s). Here ppzeta(s) = Sum_{p prime} Sum_{k>=1} 1/(p^k)^s. Note that ppzeta(s) = Sum_{p prime} 1/(p^s - 1) = Sum_{k>=1} primezeta(k*s). - Franklin T. Adams-Watters, Sep 11 2005
a(n) = floor(1/A001221(n)), for n > 1. - Enrique Pérez Herrero, Jun 01 2011
a(n) = - Sum_{d|n} mu(d)*bigomega(d), where bigomega = A001222. - Ridouane Oudra, Oct 29 2024
a(n) = - Sum_{d|n} mu(d)*omega(d), where omega = A001221. - Ridouane Oudra, Jul 30 2025

Moved original definition to formula line. Used comment (that I previously added) as definition. - Daniel Forgues, Mar 08 2009

Edited by Franklin T. Adams-Watters, Nov 02 2009

A065515 Number of prime powers <= n.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Nov 27 2001

a(n) > pi(n) = A000720(n).
From Chayim Lowen, Aug 05 2015: (Start)
a(n) <= pi(n) + A069623(n).
Conjecture: a(n) >= pi(A069623(n)) + pi(n) + 1.
Each term m is repeated A057820(m) times. (End)

			There are 9 prime powers <= 12: 1=2^0, 2, 3, 4=2^2, 5, 7, 8=2^3, 9=3^2 and 11, so a(12) = 9.

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier-North Holland, 1978, Chapter 4.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime Power

Cf. A000040, A000961, A000720, A276781 (ordinal transform).
A025528(n) = a(n) - 1.
Cf. A139555. - Reinhard Zumkeller, Oct 27 2010

a065515 n = length $ takeWhile (<= n) a000961_list
-- Reinhard Zumkeller, Apr 25 2011

N:= 100: # to get a(1) to a(N)
L:= Vector(N):
L[1]:= 1:
p:= 1:
while p < N do
  p:= nextprime(p);
  for k from 1 to floor(log[p](N)) do
    L[p^k] := 1;
  od
od:
ListTools:-PartialSums(convert(L,list)); # Robert Israel, May 03 2015

a[n_] := 1 + Count[ Range[2, n], p_ /; Length[ FactorInteger[p]] == 1]; Table[a[n], {n, 1, 73}] (* Jean-François Alcover, Oct 12 2011 *)
Accumulate[Table[If[Length[FactorInteger[n]]==1,1,0],{n,80}]] (* Harvey P. Dale, Aug 06 2016 *)
Accumulate[Table[If[PrimePowerQ[n],1,0],{n,120}]]+1 (* Harvey P. Dale, Sep 29 2016 *)

a(n)=n+=.5;1+sum(k=1,log(n)\log(2),primepi(n^(1/k))) \\ Charles R Greathouse IV, Apr 26 2012

from sympy import primepi
from sympy.ntheory.primetest import integer_nthroot
def A065515(n): return 1+sum(primepi(integer_nthroot(n,k)[0]) for k in range(1,n.bit_length())) # Chai Wah Wu, Jul 23 2024

Partial sums of A010055. - Reinhard Zumkeller, Nov 22 2009
a(n) = 1 + Sum_{k=1..log_2(n)} pi(floor(n^(1/k))). - Chayim Lowen, Aug 05 2015
a(n) = 1 + Sum_{k=2..n} floor(2*A001222(k)/(tau(k^2)-1)) where tau is A000005(n). - Anthony Browne, May 17 2016

A276781 a(n) = 1+n-(nearest power of prime <= n); for n > 1, a(n) = minimal b such that the numbers binomial(n,k) for b <= k <= n-b have a common divisor greater than 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 3, 1, 1, 2, 1, 2, 3, 4, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2, 3, 4, 5, 1, 2, 3, 4, 1, 2, 1, 2, 3, 4, 1, 2, 1, 2, 3, 4, 1, 2, 3, 4, 5, 6, 1, 2, 1, 2, 3, 1, 2, 3, 1, 2, 3, 4, 1, 2, 1, 2, 3, 4, 5, 6, 1, 2, 1, 2, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 7, 8, 1, 2, 3, 4, 1, 2

Offset: 1

N. J. A. Sloane, Sep 29 2016, following a suggestion from Eric Desbiaux

The definition in the video has "b < k < n-b" rather than "b <= k <= n-b", but that appears to be a typographical error.
From Antti Karttunen, Jan 21 2020: (Start)
a(n) = 1 if n is a power of prime (term of A000961), otherwise a(n) is one more than the distance to the nearest preceding prime power.
For n > 1, a(n) indicates the maximum region on the row n of Pascal's triangle (A007318) such that binomial terms C(n,a(n)) .. C(n,n-a(n)) all share a common prime factor. Because for all prime powers, p^k, the binomial terms C(p^k,1) .. C(p^k,p^k-1) have p as their prime factor, we have a(A000961(n)) = 1 for all n, while for each successive n that is not a prime power, the region of shared prime factor shrinks one step more towards the center of the triangle. From this follows that this is the ordinal transform of A025528 (equally, of A065515, or of A003418(n) from n >= 1 onward), equivalent to the simple definition given above.
(End)

			Row 6 of Pascal's triangle is 1,6,15,20,15,6,1 and [15,20,15] have a common divisor of 5. Since 15 = binomial(6,2), a(6)=2.

Antti Karttunen, Table of n, a(n) for n = 1..16384 (terms for n = 2..1685 from R. J. Mathar, terms 1686..10000 from Chai Wah Wu).
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Christophe Soulé, Le triangle de Pascal et ses propriétés, Lecture, Soc. Math. de France, Feb 13 2008; SMF link (in French).

Cf. A007318, A010055, A276782 (positions of records), A000961 (positions of ones), A024619 (positions of terms > 1).

Cf. A002944, A003418, A025528, A065515, A175851.

mygcd:=proc(lis) local i,g,m;
m:=nops(lis); g:=lis[1];
for i from 2 to m do g:=gcd(g,lis[i]); od:
g; end;
f:=proc(n) local b,lis; global mygcd;
for b from floor(n/2) by -1 to 1 do
lis:=[seq(binomial(n,i),i=b..n-b)];
if mygcd(lis)=1 then break; fi; od:
b+1;
end;
[seq(f(n),n=2..120)];

Table[b = 1; While[GCD @@ Map[Binomial[n, #] &, Range[b, n - b]] == 1, b++]; b, {n, 92}] (* Michael De Vlieger, Oct 03 2016 *)

A276781(n) = if(1==n,1,forstep(k=n,1,-1,if(isprimepower(k),return(1+n-k)))); \\ Antti Karttunen, Jan 21 2020

from sympy import factorint
def A276781(n): return 1+n-next(filter(lambda m:len(factorint(m))<=1, range(n,0,-1))) # Chai Wah Wu, Oct 25 2024

If A010055(n) == 1, a(n) = 1, otherwise a(n) = 1 + a(n-1). - Antti Karttunen, Jan 21 2020

Term a(1) = 1 prepended and alternative simpler definition added to the name by Antti Karttunen, Jan 20 2020

A071330 Number of decompositions of n into sum of two prime powers.

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 2, 3, 3, 4, 3, 4, 3, 4, 3, 4, 3, 5, 3, 5, 4, 4, 2, 5, 3, 5, 4, 5, 3, 6, 3, 7, 5, 7, 4, 7, 2, 6, 4, 6, 3, 6, 3, 6, 5, 6, 2, 8, 3, 8, 4, 6, 2, 9, 3, 7, 4, 6, 2, 8, 3, 7, 4, 7, 3, 9, 2, 8, 5, 7, 2, 10, 3, 8, 6, 7, 3, 9, 2, 9, 4, 7, 4, 11, 3, 9, 4, 7, 3, 12, 4, 8, 3, 7, 2

Offset: 1

Reinhard Zumkeller, May 19 2002

a(2*n) > 0 (Goldbach's conjecture).

a(A071331(n)) = 0; A095840(n) = a(A000961(n)).

			10 = 1 + 3^2 = 2 + 2^3 = 3 + 7 = 5 + 5, therefore a(10) = 4;
11 = 2 + 3^2 = 3 + 2^3 = 4 + 7, therefore a(11) = 3;
12 = 1 + 11 = 3 + 3^2 = 2^2 + 2^3 = 5 + 7, therefore a(12) = 4;
a(149)=0, as for all x<149: if x is a prime power then 149-x is not.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A000961, A002375, A010055, A061358, A071331, A109829.

a071330 n = sum $
   map (a010055 . (n -)) $ takeWhile (<= n `div` 2) a000961_list
-- Reinhard Zumkeller, Jan 11 2013

primePowerQ[n_] := Length[ FactorInteger[n]] == 1; a[n_] := (r = 0; Do[ If[ primePowerQ[k] && primePowerQ[n-k], r++], {k, 1, Floor[n/2]}]; r); Table[a[n], {n, 1, 95}](* Jean-François Alcover, Nov 17 2011, after Michael B. Porter *)

ispp(n) = (omega(n)==1 || n==1)
A071330(n) = {local(r);r=0;for(i=1,floor(n/2),if(ispp(i) && ispp(n-i),r++));r} \\ Michael B. Porter, Dec 04 2009

a(n)=my(s); forprime(p=2,n\2,if(isprimepower(n-p), s++)); for(e=2,log(n)\log(2), forprime(p=2, sqrtnint(n\2,e), if(isprimepower(n-p^e), s++))); s+(!!isprimepower(n-1))+(n==2) \\ Charles R Greathouse IV, Nov 21 2014

A195943 Zeroless prime powers: Intersection of A000961 and A052382.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 243, 251, 256, 257, 263, 269, 271, 277, 281, 283, 289, 293, 311

Offset: 1

M. F. Hasler, Sep 25 2011

In contrast to A195942, we also allow for primes (p^n with n=1) in this sequence.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
C. Rivera, Puzzle 607. A zeroless Prime power, on primepuzzles.net, Sept. 24, 2011.

Cf. A195942, A195944, A195945, A195946, A195908, A195948, A007377, A008839, A030700, A030701, A030702, A030703, A030704, A030705, A030706.

Cf. A038618 (subsequence).

a195943 n = a195943_list !! (n-1)
a195943_list = filter ((== 1) . a010055) a052382_list
-- Reinhard Zumkeller, Sep 27 2011

for( n=1,9999, is_A000961(n) && is_A052382(n) && print1(n","))

A195943 = A000961 intersect A052382 = A000961 \ A011540.

A010055(a(n)) * A168046(a(n)) = 1. - Reinhard Zumkeller, Sep 27 2011

A009087 Numbers whose number of divisors is prime (i.e., numbers of the form p^(q-1) for primes p,q).

Original entry on oeis.org

2, 3, 4, 5, 7, 9, 11, 13, 16, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239

Offset: 1

Simon Colton (simonco(AT)cs.york.ac.uk)

Invented by the HR Automatic Concept Formation Program. If the sum of divisors is prime, then the number of divisors is prime, i.e., this is a supersequence of A023194.

A010055(a(n)) * A010051(A100995(a(n))+1) = 1. - Reinhard Zumkeller, Jun 06 2013

			tau(16)=5 and 5 is prime.

S. Colton, Automated Theory Formation in Pure Mathematics. New York: Springer (2002)

Indranil Ghosh, Table of n, a(n) for n = 1..12546 (terms 1..1000 from T. D. Noe)
S. Colton, Refactorable Numbers - A Machine Invention, J. Integer Sequences, Vol. 2, 1999, #2.

Subsequence of A000961.

Cf. A023194, A036454, A203967.

a009087 n = a009087_list !! (n-1)
a009087_list = filter ((== 1) . a010051 . (+ 1) . a100995) a000961_list
-- Reinhard Zumkeller, Jun 05 2013

Select[Range[250],PrimeQ[DivisorSigma[0,#]]&] (* Harvey P. Dale, Sep 28 2011 *)

is(n)=isprime(isprimepower(n)+1) \\ Charles R Greathouse IV, Sep 16 2015

from sympy import primepi, integer_nthroot, primerange
def A009087(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = 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-sum(primepi(integer_nthroot(x,k-1)[0]) for k in primerange(x.bit_length()+1)))
    return bisection(f,n,n) # Chai Wah Wu, Feb 22 2025

p^(q-1), p, q primes.

A095874 a(n) = k if n = A000961(k) (powers of primes), a(n) = 0 if n is not in A000961.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Jun 10 2004

nonn

The name has been edited to clarify that the indices k refer to A000961 ("powers of primes" = {1} U A246655) and not to the list A246655 of proper prime powers. - M. F. Hasler, Jun 16 2021

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

Cf. A000961 (right inverse), A049084, A097621.

a095874 n | y == n    = length xs + 1
          | otherwise = 0
          where (xs, y:ys) = span (< n) a000961_list
-- Reinhard Zumkeller, Feb 16 2012, Jun 26 2011

Join[{1},Module[{k=2},Table[If[PrimePowerQ[n],k;k++,0],{n,2,100}]]] (* Harvey P. Dale, Aug 15 2020 *)

a(n)=if(isprimepower(n), sum(i=1,logint(n,2), primepi(sqrtnint(n,i)))+1, n==1) \\ Charles R Greathouse IV, Apr 29 2015

{M95874=Map(); A095874(n,k)=if(mapisdefined(M95874,n,&k),k, isprimepower(n), mapput(M95874,n, k=sum(i=1,exponent(n), primepi(sqrtnint(n,i)))+1); k,n==1)} \\ Variant with memoization, possibly useful to compute A097621, A344826 and related. One may omit "isprimepower(n)," (possibly requiring factorization) and ",n==1" if n is known to be a power of a prime, i.e., to get a left inverse for A000961. - M. F. Hasler, Jun 15 2021

from sympy import primepi, integer_nthroot, primefactors
def A095874(n): return 1+int(primepi(n)+sum(primepi(integer_nthroot(n,k)[0]) for k in range(2,n.bit_length()))) if n==1 or len(primefactors(n))==1 else 0 # Chai Wah Wu, Jan 19 2025

a(n) = Sum_{1 <= k <= n} A010055(k); [corrected by M. F. Hasler, Jun 15 2021]
a(n) = A065515(n)*(A065515(n)-A065515(n-1)).
a(n) = A065515(n)*A069513(n). - M. F. Hasler, Jun 16 2021

Edited by M. F. Hasler, Jun 15 2021

A195942 Zeroless prime powers (excluding primes): Intersection of A025475 and A052382.

Original entry on oeis.org

1, 4, 8, 9, 16, 25, 27, 32, 49, 64, 81, 121, 125, 128, 169, 243, 256, 289, 343, 361, 512, 529, 625, 729, 841, 961, 1331, 1369, 1681, 1849, 2187, 2197, 3125, 3481, 3721, 4489, 4913, 5329, 6241, 6561, 6859, 6889, 7921, 8192

Offset: 1

M. F. Hasler, Sep 25 2011

Reinhard Zumkeller and T. D. Noe, Table of n, a(n) for n = 1..4094 (terms < 10^10)
C. Rivera, Puzzle 607. A zeroless Prime power, on primepuzzles.net, Sept. 24, 2011.

Cf. A195985, A195943, A195944, A195945, A195946, A195908, A195948, A007377, A008839, A030700, A030701, A030702, A030703, A030704, A030705, A030706.

a195942 n = a195942_list !! (n-1)
a195942_list = filter (\x -> a010051 x == 0 && a010055 x == 1) a052382_list
-- Reinhard Zumkeller, Sep 27 2011

mx = 10^10; t = {1}; p = 2; While[pw = 2; While[n = p^pw; n <= mx, If[Union[IntegerDigits[n]][[1]] > 0, AppendTo[t, n]]; pw++]; pw > 2, p = NextPrime[p]]; t = Sort[t] (* T. D. Noe, Sep 27 2011 *)

for( n=1,9999, is_A025475(n) && is_A052382(n) && print1(n","))

A195942 = A025475 intersect A052382.

A010055(a(n)) * (1 - A010051(a(n))) * A168046(a(n)) = 1. - Reinhard Zumkeller, Sep 27 2011

cp's OEIS Frontend

A119288 a(n) is the second smallest prime factor of n, or 1 if n is a prime power.

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A100995 If n is a prime power p^m, m >= 1, then m, otherwise 0.

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A069513 Characteristic function of the prime powers p^k, k >= 1.

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A065515 Number of prime powers <= n.

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A071330 Number of decompositions of n into sum of two prime powers.

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