A061345 -id:A061345 - cp's OEIS frontend

A280609 Odd prime powers with prime exponents.

9, 25, 27, 49, 121, 125, 169, 243, 289, 343, 361, 529, 841, 961, 1331, 1369, 1681, 1849, 2187, 2197, 2209, 2809, 3125, 3481, 3721, 4489, 4913, 5041, 5329, 6241, 6859, 6889, 7921, 9409, 10201, 10609, 11449, 11881, 12167, 12769, 16129, 16807, 17161, 18769, 19321, 22201, 22801, 24389, 24649, 26569, 27889, 29791, 29929

Offset: 1

Ilya Gutkovskiy, Jan 06 2017

Intersection of A053810 and A061345.

			9 is in the sequence because 9 = 3^2;
25 is in the sequence because 25 = 5^2;
27 is in the sequence because 27 = 3^3, etc.

Eric Weisstein's World of Mathematics, Prime Power.

Cf. A000961, A034785, A051006, A053810, A061345, A078422, A246547, A246551, A246655.

Select[Range[30000], PrimePowerQ[#1] && PrimeQ[PrimeOmega[#1]] && Mod[#1, 2] == 1 & ]

from sympy import primepi, integer_nthroot, primerange
def A280609(n):
    def f(x): return int(n+x-sum(primepi(integer_nthroot(x, p)[0])-1 for p in primerange(x.bit_length())))
    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
    return bisection(f,n,n) # Chai Wah Wu, Sep 12 2024

a(n) = p^q, where p, q are primes and p > 2.

Sum_{n>=1} 1/a(n) = Sum_{p prime} P(p) - A051006 = 0.25699271237062131298..., where P(s) is the prime zeta function. - Amiram Eldar, Sep 13 2024

A284233 Sum of odd prime power divisors of n (not including 1).

Original entry on oeis.org

0, 0, 3, 0, 5, 3, 7, 0, 12, 5, 11, 3, 13, 7, 8, 0, 17, 12, 19, 5, 10, 11, 23, 3, 30, 13, 39, 7, 29, 8, 31, 0, 14, 17, 12, 12, 37, 19, 16, 5, 41, 10, 43, 11, 17, 23, 47, 3, 56, 30, 20, 13, 53, 39, 16, 7, 22, 29, 59, 8, 61, 31, 19, 0, 18, 14, 67, 17, 26, 12, 71, 12, 73, 37, 33, 19, 18, 16, 79, 5

Offset: 1

Ilya Gutkovskiy, Mar 23 2017

			a(15) = 8 because 15 has 4 divisors {1, 3, 5, 15} among which 2 are odd prime powers {3, 5} therefore 3 + 5 = 8.

Eric Weisstein's World of Mathematics, Prime Power.
Index entries for sequences related to sums of divisors.

Cf. A000961, A005069, A023888, A023889, A038712, A061345, A065091 (fixed points), A087436 (number of odd prime power divisors of n), A206787, A246655, A284117.

nmax = 80; Rest[CoefficientList[Series[Sum[Boole[PrimePowerQ[k] && Mod[k, 2] == 1] k x^k/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
Table[Total[Select[Divisors[n], PrimePowerQ[#] && Mod[#, 2] == 1 &]], {n, 80}]
f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - 1; f[2, e_] := 0; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jul 24 2024 *)

G.f.: Sum_{k>=1} A061345(k)*x^A061345(k)/(1 - x^A061345(k)).
a(n) = Sum_{d|n, d = p^k, p prime, p > 2, k > 0} d.
a(p^k) = p*(p^k - 1)/(p - 1) for p is a prime > 2.
a(2^k*p) = p for p is a prime > 2.
a(2^k) = 0.
Additive with a(2^e) = 0, and a(p^e) = (p^(e+1)-1)/(p-1) - 1 for an odd prime p. - Amiram Eldar, Jul 24 2024

A352165 Number of partitions of n into odd prime powers (1 included).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18, 22, 26, 31, 37, 44, 52, 61, 71, 83, 97, 112, 130, 150, 173, 199, 228, 261, 298, 340, 386, 439, 497, 563, 637, 718, 809, 910, 1023, 1147, 1286, 1439, 1608, 1796, 2003, 2231, 2483, 2761, 3065, 3401, 3770, 4175, 4619

Offset: 0

Ilya Gutkovskiy, Mar 06 2022

nonn

Cf. A000961, A023893, A023894, A061345, A099773, A134345, A280151, A352166.

nmax = 55; CoefficientList[Series[Product[1/(1 - Boole[(PrimePowerQ[k] || k == 1) && OddQ[k]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=0} 1 / (1 - x^A061345(k)).

A352166 Number of partitions of n into distinct odd prime powers (1 included).

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 5, 4, 5, 6, 6, 6, 7, 8, 9, 9, 10, 12, 12, 13, 14, 16, 17, 17, 19, 21, 23, 23, 25, 28, 30, 31, 33, 37, 38, 40, 43, 47, 50, 52, 55, 60, 64, 66, 70, 76, 81, 83, 89, 96, 101, 105, 110, 119, 125, 130, 138, 147, 155, 161

Offset: 0

Ilya Gutkovskiy, Mar 06 2022

nonn

Cf. A000961, A024939, A054685, A061345, A106244, A134337, A280152, A352165.

nmax = 70; CoefficientList[Series[Product[(1 + Boole[(PrimePowerQ[k] || k == 1) && OddQ[k]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=0} (1 + x^A061345(k)).

A369690 a(n) = max(A119288(n), A053669(n)).

Original entry on oeis.org

2, 3, 2, 3, 2, 5, 2, 3, 2, 5, 2, 5, 2, 7, 5, 3, 2, 5, 2, 5, 7, 11, 2, 5, 2, 13, 2, 7, 2, 7, 2, 3, 11, 17, 7, 5, 2, 19, 13, 5, 2, 5, 2, 11, 5, 23, 2, 5, 2, 5, 17, 13, 2, 5, 11, 7, 19, 29, 2, 7, 2, 31, 7, 3, 13, 5, 2, 17, 23, 5, 2, 5, 2, 37, 5, 19, 11, 5, 2, 5, 2

Offset: 1

Peter Munn and Michael De Vlieger, Feb 18 2024

Equivalently, a(n) is the largest p such that p is the 2nd smallest prime dividing n or the smallest prime not dividing n.

If squarefree n is such that a(n) = p, then a(k) = p for k in the infinite sequence { k = m*n : rad(m) | n }. Consequence of the fact that both A119288(n) and A053669(n) do not depend on multiplicity of prime divisors p | n.

			Let p be the second least prime factor of n or 1 if n is a prime power, and let q be the smallest prime that does not divide n.
a(1) = 2 since max(p, q) = max(1, 2) = 2.
a(2) = 3 since max(p, q) = max(1, 3) = 3.
a(4) = 3 since max(p, q) = max(1, 3) = 3.
a(6) = 5 since max(p, q) = max(3, 5) = 5.
a(9) = 2 since max(p, q) = max(1, 2) = 2.
a(15) = 5 since max(p, q) = max(5, 2) = 5.
a(36) = 5 since max(p, q) = max(3, 5) = 5.
Generally,
a(n) = 2 for n in A061345 = union of {1} and sequences { m*p : prime p > 2, rad(m) | p }.
a(n) = 3 for n in A000079 = { 2*m : rad(m) | 2 }.
a(n) = 5 for k in { k = m*d : rad(m) | d, d in {6, 10, 15} }.
a(n) = 7 for k in { k = m*d : rad(m) | d, d in {14, 21, 30, 35} }.
a(n) = 11 for k in { k = m*d : rad(m) | d, d in {22, 33, 55, 77, 210} }, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000079, A002110, A003557, A007947, A024619, A053669, A061345, A096015 (smallest instead of 2nd smallest), A100484, A119288, A246547, A361098.

{2}~Join~Array[If[PrimePowerQ[#],
  q = 2; While[Divisible[#, q], q = NextPrime[q]]; q,
  q = 2; While[Divisible[#, q], q = NextPrime[q]];
    Max[FactorInteger[#][[2, 1]], q]] &, 120, 2]

a(n) <= A003557(n) for n > 4 in A246547 and for n in A361098.

Numbers n that set records include 1, 2, and squarefree semiprimes, i.e., (A100484 \ {4}) U {1, 2}.

A380757 Powers of primes that have a primitive root.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 11, 13, 17, 19, 23, 25, 27, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229

Offset: 1

Michael De Vlieger, Feb 01 2025

Proper subset of A033948.

A046022 is a proper subset of this sequence.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000079, A000961, A033948, A046022, A061345, A278568.

With[{nn = 2^8},
  Complement[#, Array[2^# &, Floor@ Log2[#[[-1]]] + 2, 3]] &@
  Union[{1}, Prime@ Range@ PrimePi[#[[-1]] ], #] &@
  Select[Union@ Flatten@
    Table[a^2*b^3, {b, Surd[#, 3]}, {a, Sqrt[nn/b^3]}],
    PrimePowerQ] ]

from sympy import primepi, integer_nthroot
def A380757(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 n if x<6 else int(n+x-3-sum(primepi(integer_nthroot(x,k)[0])-1 for k in range(1,x.bit_length())))
    return bisection(f,n,n) # Chai Wah Wu, Feb 03 2025

Union of {1, 2, 4} and A061345.
This sequence is A000961 without A000079(k) for k > 2.
A033948 = union of {a(n)} and {2*a(n)} without 8 = union of {a(n)} and A278568, where {a(n)} represents this sequence.
Intersection of A000961 and A033948.
Define c(m) to be the number of terms that do not exceed m. Then for m >= 4, c(m) = 3 + (Sum_{k = 1..floor(log_2(m))} pi(floor(m^(1/k)))) - floor(log_2(m)) = 3 + A065515(m) - A113473(m).

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).

A360204 Primitive prime powers. p is a primitive prime power iff it is an odd prime power that exceeds the preceding odd prime power by more than any smaller odd prime power does. ('Prime power' defined in the sense of A246655.)

Original entry on oeis.org

5, 17, 37, 97, 149, 211, 307, 907, 1151, 1361, 5623, 8501, 9587, 15727, 19661, 31469, 156007, 360749, 370373, 492227, 1349651, 1357333, 2010881, 4652507, 17051887, 20831533, 47326913, 122164969, 189695893, 191913031, 387096383, 436273291, 1294268779

Offset: 1

Peter Luschny, Feb 01 2023

nonn

Conjecture: Primitive prime powers are primes.

			The first few terms of the sequence are the minuends in the following differences. The differences are strictly increasing by definition.
  5 -   3 = 2;
 17 -  13 = 4;
 37 -  31 = 6;
 97 -  89 = 8;
149 - 139 = 10;
211 - 199 = 12;
307 - 293 = 14;
907 - 887 = 20.
The example above might suggest the subtrahends are also prime. In general they are not, as the example a(10) shows, where 1361 - 1331 = 30, but 1331 is not prime. - _Ivan N. Ianakiev_, Feb 02 2023

Cf. A246655, A061345.

a[1] = 5; candidates[n_] := Select[Range[NextPrime[n, -1], n], OddQ[#] && PrimePowerQ[#]&];
difference[n_] := candidates[n][[-1]] - candidates[n][[-2]];
a[n_] := a[n] = Module[{k = a[n-1]+2}, While[OddQ[k] && !PrimePowerQ[k] || difference[k] <= difference[a[n-1]], k = k+2]; k];
a/@Range[16] (* Ivan N. Ianakiev, Feb 02 2023 *)

def A360204_list(n):
    R = []; L = 3; MAX = 1
    for C in xsrange(3, n, 2):
        if C.is_prime_power():
            if C - L > MAX:
                MAX = C - L
                R.append(C)
            L = C
    return R
print(A360204_list(6000))

A386894 Markoff numbers that are powers of one odd prime or twice powers of one odd prime.

Original entry on oeis.org

1, 2, 5, 13, 29, 34, 89, 169, 194, 233, 433, 1597, 2897, 5741, 7561, 28657, 33461, 43261, 96557, 426389, 514229, 646018, 1686049, 2012674, 2922509, 3276509, 11485154, 21531778, 94418953, 253191266, 321534781, 433494437, 780291637, 1405695061, 1475706146, 2971215073, 6684339842, 19577194573

Offset: 1

Wolfdieter Lang, Aug 07 2025

A subsequence of A000961 without numbers divisible by 4.
The powers of odd primes are given in A061345 (with offset 0).
These Markoff numbers (see A002559) have been proved to obey the Frobenius-Markoff uniqueness conjecture. See Aigner, Corollary 3.20, p. 59, and there the references [4] A. Baragar, [18] J. O. Button, and [119] Ying Zhang.

			26 = 2*13 is not a Markoff number, hence not in the present sequence.
610 = 2*5*61 is a Markoff number but not a prime power nor is 305 a prime power.

Martin Aigner, Markov's theorem and 100 years of the uniqueness conjecture. A mathematical journey from irrational numbers to perfect matchings. Springer, 2013.

Mong Lung Lang and Ser Peow Tan, A simple proof of the Markoff conjecture for prime powers, arXiv:math/0508443 [math.NT], 2005.
Paul Schmutz, Systoles of arithmetic surfaces and the Markoff spectrum, Math. Ann. 305 (1996), no. 1, 191-203.
Ying Zhang, An elementary proof of uniqueness of Markoff numbers which are prime powers, arXiv:math/0606283 [math.NT], 2007.

Cf. A000040, A000961, A002559, A061345, A256395.

MAX=10^11; data=NestWhile[Select[Union[Sort/@Flatten[Table[{a, b, 3a b -c}/.MapThread[Rule, {{a, b, c}, #}]&/@Map[RotateLeft[ii, #]&, Range[3]], {ii, #}], 1]], Max[#]James C. McMahon, Aug 12 2025 *)

def A386894List(len: int = 50, MAX: int = 10**10) -> list[int]:
    # Using function 'MarkovNumbers' from A002559.
    M = MarkovNumbers(len, MAX)
    U = set([1])
    for m in M:  # if m is a Markov number and ...
        z = ZZ(m)
        if is_prime_power(z) or (is_even(z) and is_prime_power(z//2)):
            U.add(m)
    return sorted(U)
# Balance required sequence length and search depth.
print(A386894List(len=120, MAX=10**12))  # Peter Luschny, Aug 12 2025

Markoff numbers of the form 2^j*p^k, with an odd prime p, j = 0 or 1, and k >= 0, ordered strictly increasing.

cp's OEIS Frontend

A280609 Odd prime powers with prime exponents.

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A284233 Sum of odd prime power divisors of n (not including 1).

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A352165 Number of partitions of n into odd prime powers (1 included).

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A352166 Number of partitions of n into distinct odd prime powers (1 included).

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A369690 a(n) = max(A119288(n), A053669(n)).

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A380757 Powers of primes that have a primitive root.

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

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A360204 Primitive prime powers. p is a primitive prime power iff it is an odd prime power that exceeds the preceding odd prime power by more than any smaller odd prime power does. ('Prime power' defined in the sense of A246655.)

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