A025473 -id:A025473 - cp's OEIS frontend

A053812 Exponents occurring in A053810.

2, 3, 2, 2, 3, 5, 2, 2, 3, 7, 2, 5, 2, 3, 2, 2, 2, 2, 3, 2, 2, 2, 11, 7, 3, 2, 2, 5, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 13, 2, 2, 2, 2, 2, 3, 2, 2, 5, 2, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 7, 2, 3, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 17, 2, 2, 2, 2, 3, 2, 2, 2

Offset: 1

Henry Bottomley, Mar 28 2000

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000961, A000961, A001222, A025473, A053810, A053811.

LIM = prime(80)^2; v = vector(400); count = 0; forprime (p = 2, prime(80), x = 2; while (p^x <= LIM, count++; v[count] = p^x; x = nextprime(x + 1))); v = vecsort(vector(count, i, v[i])); vector(count, i, bigomega(v[i])) \\ David Wasserman, Feb 17 2006

from sympy import primepi, integer_nthroot, primerange, factorint
def A053812(n):
    def f(x): return int(n-1+x-sum(primepi(integer_nthroot(x, p)[0]) for p in primerange(x.bit_length())))
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return list(factorint(kmax).values())[0] # Chai Wah Wu, Aug 13 2024

a(n) = A001222(A053810(n)). - David Wasserman, Feb 17 2006

a(n) = log(A053810(n))/log(A053811(n)). - Amiram Eldar, Nov 21 2020

More terms from David Wasserman, Feb 17 2006

Offset corrected by Amiram Eldar, Nov 21 2020

A085818 For n > 1: a(n) = p if n = p^e with p prime and e > 1, otherwise a(n) = (n-m)-th prime, where m = number of nonprime prime powers <= n; a(1)=1.

Original entry on oeis.org

1, 2, 3, 2, 5, 7, 11, 2, 3, 13, 17, 19, 23, 29, 31, 2, 37, 41, 43, 47, 53, 59, 61, 67, 5, 71, 3, 73, 79, 83, 89, 2, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 7, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241

Offset: 1

Reinhard Zumkeller, Jul 04 2003

nonn

a(n) = A025473(n) if n = p^e with p prime and e > 1, otherwise a(n) = A008578(n-A085501(n));
n divides A085819(n) = Product_{k<=n} a(k), as by construction: a(1)=1; if n divides A085819(n-1) then a(n) = smallest prime not occurring earlier; if n does not divide A085819(n-1) then a(n) = greatest prime factor of n (A006530);
A000040 occurs infinitely many times as a subsequence.
a(A085971(n))=A000040(n) and for all k > 1: a(A000040(n)^k)=A000040(n); A085985(n)=A049084(a(n)). - Reinhard Zumkeller, Jul 06 2003

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

Cf. A000040, A008578, A085971.

Cf. A006530, A025473, A085501, A085819, A085985, A049084.

f(n) = 1 + sum(k=2, n, isprimepower(k) && !isprime(k));  \\ A085501
a(n) = {if (n==1, return (1)); my(p); if (isprimepower(n, &p) && !isprime(n), p, prime(n-f(n)));} \\ Michel Marcus, Jan 28 2021

from sympy import primefactors, prime, primepi, integer_nthroot
def A085818(n): return 1 if n==1 else (f[0] if len(f:=primefactors(n))==1 and f[0]Chai Wah Wu, Aug 20 2024

A086455 Sum of divisors of prime powers: sigma(p^e).

Original entry on oeis.org

1, 3, 4, 7, 6, 8, 15, 13, 12, 14, 31, 18, 20, 24, 31, 40, 30, 32, 63, 38, 42, 44, 48, 57, 54, 60, 62, 127, 68, 72, 74, 80, 121, 84, 90, 98, 102, 104, 108, 110, 114, 133, 156, 128, 255, 132, 138, 140, 150, 152, 158, 164, 168, 183, 174, 180, 182, 192, 194, 198

Offset: 1

Reinhard Zumkeller, Jul 20 2003

nonn

R. J. Mathar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Divisor Function.
Eric Weisstein's World of Mathematics, Prime Power.

Cf. A000203, A000961, A025473, A025474, A086454.

A086455 := proc(n)
    numtheory[sigma](A000961(n)) ;
end proc: # R. J. Mathar, Jun 04 2016

DivisorSigma[1, #]& /@ Join[{1}, Select[Range[2, 200], PrimePowerQ]] (* Jean-François Alcover, Feb 10 2018 *)

list(lim) = apply(sigma, select(x -> x == 1 || isprimepower(x), vector(lim, i, i))); \\ Amiram Eldar, May 07 2025

a(n) = A000203(A000961(n)).
a(n) = (p^(e+1)-1)/(p-1), where p^e = A000961(n).
a(n) = (A025473(n)^(A025474(n)+1)-1)/(A025473(n)-1).

A192133 Difference of base and exponent of prime powers (cf. A000961).

Original entry on oeis.org

1, 1, 2, 0, 4, 6, -1, 1, 10, 12, -2, 16, 18, 22, 3, 0, 28, 30, -3, 36, 40, 42, 46, 5, 52, 58, 60, -4, 66, 70, 72, 78, -1, 82, 88, 96, 100, 102, 106, 108, 112, 9, 2, 126, -5, 130, 136, 138, 148, 150, 156, 162, 166, 11, 172, 178, 180, 190, 192, 196, 198, 210

Offset: 1

Reinhard Zumkeller, Jun 26 2011

sign

a(1) = 1 by convention, in accordance with A025473(1) = 1 and A025474(1) = 0.

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

A006093 and A090076 are subsequences.

Cf. A000961, A025473, A025474, A192134.

s[n_] := Module[{f = FactorInteger[n]}, If[Length[f] == 1, f[[1, 1]] - f[[1, 2]], Nothing]]; s[1] = 1; Array[s, 250] (* Amiram Eldar, May 16 2025 *)

a(n) = A025473(n)-A025474(n) = A192134(n)*A025473(n)/A000961(n).

A257572 Prime root of n-th term in A257278.

Original entry on oeis.org

2, 2, 2, 3, 2, 2, 3, 2, 3, 2, 2, 3, 2, 2, 3, 5, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 3, 2, 5, 2, 3, 7, 2, 3, 5, 2, 2, 3, 7, 2, 5, 3, 2, 2, 7, 3, 5, 2, 3, 2, 5, 2, 7, 3, 2, 2, 3, 5, 7, 2, 3, 2, 5, 2, 3, 7, 2, 5, 3, 2, 2, 3, 7, 2, 5, 2, 3, 11, 2, 7, 5, 3, 2, 2, 3

Offset: 1

Reinhard Zumkeller, May 01 2015

nonn

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

Cf. A257278, A020639, A025473, A257573.

Haskell
```
a257572 = a020639 . a257278
```

seq[lim_] := Module[{s = {}, p = 2, r}, While[p^p <= lim, r = Range[p, Log[p, lim]]; AppendTo[s, Transpose[{ConstantArray[p, Length[r]], p^r}]]; p = NextPrime[p]]; SortBy[Flatten[s, 1], Last][[;; , 1]]]; seq[10^13] (* Amiram Eldar, Apr 14 2025 *)

a(n) = A020639(A257278(n)).
A257278(n) = a(n) ^ A257573(n).
a(n) <= A257573(n) by definition of A257278.

A085730 Euler's totient function applied to the sequence of prime powers.

Original entry on oeis.org

1, 1, 2, 2, 4, 6, 4, 6, 10, 12, 8, 16, 18, 22, 20, 18, 28, 30, 16, 36, 40, 42, 46, 42, 52, 58, 60, 32, 66, 70, 72, 78, 54, 82, 88, 96, 100, 102, 106, 108, 112, 110, 100, 126, 64, 130, 136, 138, 148, 150, 156, 162, 166, 156, 172, 178, 180, 190, 192, 196, 198, 210

Offset: 1

Reinhard Zumkeller, Jul 20 2003

nonn

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

Cf. A000010, A000961, A025473, A025474, A207193.

a085730 1 = 1
a085730 n = (p - 1) * p ^ (e - 1)
   where p =  a025473 n; e =  a025474 n
-- Reinhard Zumkeller, Feb 16 2012

f[p_, e_] := (p-1)*p^(e-1); s[n_] := If[n == 1, 1, If[PrimePowerQ[n], f @@ (FactorInteger[n][[1]]), Nothing]]; Array[s, 220] (* Amiram Eldar, Apr 05 2025 *)

list(lim)=my(v=List(primes(primepi(lim)))); listput(v,1); for(e=2, log(lim+.5)\log(2),forprime(p=2,(lim+.5)^(1/e),listput(v, p^e))); apply(n->eulerphi(n),vecsort(Vec(v))) \\ Charles R Greathouse IV, Apr 30 2012

a(n) = A000010(A000961(n)).
a(p^e) = (p-1)*p^(e-1).
a(n) = (A025473(n)-1)*A025473(n)^(A025474(n)-1).

A192083 Arithmetic derivative of squares of prime powers: a(n) = A003415(A056798(n)).

Original entry on oeis.org

0, 4, 6, 32, 10, 14, 192, 108, 22, 26, 1024, 34, 38, 46, 500, 1458, 58, 62, 5120, 74, 82, 86, 94, 1372, 106, 118, 122, 24576, 134, 142, 146, 158, 17496, 166, 178, 194, 202, 206, 214, 218, 226, 5324, 18750, 254, 114688, 262, 274, 278, 298, 302, 314, 326, 334

Offset: 1

Reinhard Zumkeller, Jun 26 2011

nonn

A001787 and A024622 give record values and where they occur.

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

Cf. A003415, A024622, A025473, A025474, A056798, A192084.

Subsequences: A001787, A027471, A051674, A079705, A100484.

s[n_] := If[PrimePowerQ[n], f = FactorInteger[n][[1]]; 2*f[[2]]*n^(2 - 1/f[[2]]), Nothing]; s[1] = 0; Array[s, 200] (* Amiram Eldar, Apr 06 2025 *)

a(n) = 2 * A025474(n) * A025473(n)^(2*A025474(n) - 1).

A192084(n) = A003415(a(n)).

A207193 Auxiliary function for computing the Carmichael lambda function (A002322).

Original entry on oeis.org

1, 1, 2, 2, 4, 6, 2, 6, 10, 12, 4, 16, 18, 22, 20, 18, 28, 30, 8, 36, 40, 42, 46, 42, 52, 58, 60, 16, 66, 70, 72, 78, 54, 82, 88, 96, 100, 102, 106, 108, 112, 110, 100, 126, 32, 130, 136, 138, 148, 150, 156, 162, 166, 156, 172, 178, 180, 190, 192, 196, 198

Offset: 1

Reinhard Zumkeller, Feb 16 2012

nonn

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

Cf. A000961, A002322, A025473, A025474, A085730.

a207193 1 = 1
a207193 n | p == 2 && e > 2 = 2 ^ (e - 2)
          | otherwise       = (p - 1) * p ^ (e - 1)
          where p = a025473 n; e = a025474 n

f[p_, e_] := If[p == 2 && e > 2, 2^(e-2), (p-1)*p^(e-1)]; s[n_] := If[n == 1, 1, If[PrimePowerQ[n], f @@ (FactorInteger[n][[1]]), Nothing]]; Array[s, 200] (* Amiram Eldar, Apr 05 2025 *)

a(n) = f(A000961(n)), where f(1) = 1, and f(p^e) = 2^(e-2) if p = 2 and e > 2, and f(p^e) = (p-1)*p^(e-1) otherwise.

A063872 Let m be the n-th positive integer such that phi(m) is divisible by m - phi(m). Then a(n) = phi(m)/(m - phi(m)).

Original entry on oeis.org

1, 2, 1, 4, 6, 1, 2, 10, 12, 1, 16, 18, 22, 4, 2, 28, 30, 1, 36, 40, 42, 46, 6, 52, 58, 60, 1, 66, 70, 72, 78, 2, 82, 88, 96, 100, 102, 106, 108, 112, 10, 4, 126, 1, 130, 136, 138, 148, 150, 156, 162, 166, 12, 172, 178, 180, 190, 192, 196, 198, 210, 222, 226, 228, 232

Offset: 1

Labos Elemer, Aug 27 2001

m is the n-th prime power larger than 1; i.e., m = A000961(n+1). Proof: If phi(m) is divisible by m-phi(m), then m is divisible by m-phi(m). Let k be the product of the distinct prime factors of m. Then phi(m)/m = phi(k)/k, so k/(k-phi(k)) = m/(m-phi(m)) is an integer. Thus k is divisible by k-phi(k) and k is squarefree. Let k-phi(k) = d and k/(k-phi(k)) = e; note that e>1 and GCD(d,e)=1. Thus d = k - phi(k) = d e - phi(d e) = d e - phi(d) phi(e) so d (e-1) = d e - d = phi(d) phi(e) <= phi(d) (e-1) and d <= phi(d). But this implies that d=1, so phi(k)=k-1 and k is prime. Hence m is a prime power. - Dean Hickerson, Aug 28 2001
For primes, quotient = (p - 1) / 1 = p - 1; for prime powers, p^a, a > 1: quotient = p^(a - 1)(p - 1) / p^(a - 1) = p - 1, so each p - 1 values occur infinitely often: a(n) + 1 = root of n-th prime power with positive exponent, i.e., A025473(n+1). - [Edited by] Daniel Forgues, May 08 2014
"LCM numeral system": a(n+1) is maximum digit for index n, n >= 0; a(-n) is maximum digit for index n, n < 0. - Daniel Forgues, May 03 2014

Harvey P. Dale, Table of n, a(n) for n = 1..1000
OEIS Wiki, LCM numeral system

Cf. A000010, A051953, A007694, A000961, A054740, A049237, A025473.

epd[n_]:=Module[{ep=EulerPhi[n]},If[Divisible[ep,n-ep],ep/(n-ep),Nothing]]; Array[epd,300,2] (* Harvey P. Dale, Dec 27 2020 *)

M(n) = ispower(n, , &n); if (isprime(n), n, 1); \\ A014963
apply(x->x-1, select(isprime, apply(x->M(x+1), [1..260]))) \\ Michel Marcus, Sep 14 2021

a(n) = A025473(n + 1) - 1. - Bill McEachen, Sep 11 2021

A085729 Sum of prime factors of prime powers.

Original entry on oeis.org

0, 2, 3, 4, 5, 7, 6, 6, 11, 13, 8, 17, 19, 23, 10, 9, 29, 31, 10, 37, 41, 43, 47, 14, 53, 59, 61, 12, 67, 71, 73, 79, 12, 83, 89, 97, 101, 103, 107, 109, 113, 22, 15, 127, 14, 131, 137, 139, 149, 151, 157, 163, 167, 26, 173, 179, 181, 191, 193, 197, 199, 211, 223

Offset: 1

Reinhard Zumkeller, Jul 20 2003

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime Power.
Eric Weisstein's World of Mathematics, Sum of Prime Factors.

Cf. A000961, A001414, A025473, A025474.

s[n_] := Module[{f = FactorInteger[n]}, If[Length[f] == 1, Times @@ f[[1]], Nothing]]; s[1] = 0; Array[s, 225] (* Amiram Eldar, May 14 2025 *)

a(n) = A001414(A000961(n)).

a(n) = e*p when n = p^e: a(n) = A025474(n)*A025473(n).

cp's OEIS Frontend

A053812 Exponents occurring in A053810.

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A085818 For n > 1: a(n) = p if n = p^e with p prime and e > 1, otherwise a(n) = (n-m)-th prime, where m = number of nonprime prime powers <= n; a(1)=1.

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A086455 Sum of divisors of prime powers: sigma(p^e).

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Maple

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A192133 Difference of base and exponent of prime powers (cf. A000961).

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Mathematica

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A257572 Prime root of n-th term in A257278.

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Haskell

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A085730 Euler's totient function applied to the sequence of prime powers.

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Haskell

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PARI

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A192083 Arithmetic derivative of squares of prime powers: a(n) = A003415(A056798(n)).

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Mathematica

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A207193 Auxiliary function for computing the Carmichael lambda function (A002322).

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