A074583 -id:A074583 - cp's OEIS frontend

A384915 The number of unordered factorizations of n into powers of primes of the form p^e where p is prime and 0 <= e <= p (A074583).

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 3, 2, 1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 3, 2, 2, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 2, 2, 1, 1, 1, 3, 4, 1, 1, 2, 1, 1, 1

Offset: 1

Amiram Eldar, Jun 12 2025

			a(4) = 2 since 4 has 2 factorizations: 2^1 * 2^1 and 2^2, with exponents 1 and 2 that are <= 2.

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

Cf. A026820, A048103, A074583.

Similar sequences: A000688, A046951, A050361, A050377, A188581, A188585, A322885, A370256, A384912, A384913, A384914, A384916.

f[p_, e_] := Length[IntegerPartitions[e, p]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

T(n, k)=my(s); forpart(v=n, s++, , k); s \\ Charles R Greathouse IV at A026820
a(n) = {my(f = factor(n)); prod(i = 1, #f~, T(f[i,2], f[i,1]));}

Multiplicative with a(p^e) = A026820(e, p).

a(n) >= A384916(n), with equality if and only if n is in A048103.

A344029 Numbers included in A343983 but not in A074583.

Original entry on oeis.org

72, 2025, 78447, 5922181, 84238825, 1141011175

Offset: 1

Seiichi Manyama, May 07 2021

			If n is in A074583, n can be expressed as n = p^e (p>=e) using the prime p.
On the other hand, the terms of this sequence are factorized as follows.
72 = 2^3 * 3^2.
2025 = 3^4 * 5^2.
78447 = 3 * 79 * 331.
5922181 = 71 * 239 * 349.
84238825 = 5^2 * 11 * 17 * 37 * 487.

Cf. A074583, A343983.

isok(n) = my(f=factor(n)); sumdiv(n, d, Mod(d, n)^d)==1 && n>1 && !(#f~==1 && f[1, 1]>=f[1, 2]);

a(6) from Seiichi Manyama, Aug 01 2023

A192135 Prime powers p^e with p < e.

Original entry on oeis.org

8, 16, 32, 64, 81, 128, 243, 256, 512, 729, 1024, 2048, 2187, 4096, 6561, 8192, 15625, 16384, 19683, 32768, 59049, 65536, 78125, 131072, 177147, 262144, 390625, 524288, 531441, 1048576, 1594323, 1953125, 2097152, 4194304, 4782969, 5764801, 8388608, 9765625, 14348907

Offset: 1

Reinhard Zumkeller, Jun 26 2011

nonn

Donovan Johnson, Table of n, a(n) for n = 1..10000

Complement to A074583 with respect to A000961.

Cf. A095874, A192187, A257278.

A192135 := proc(nmax) local s ,i,p,e ; s := {} ; for i from 1 do p := ithprime(i) ; if p^(p+1) > nmax then break; end if; for e from p+1 do if p^e > nmax then break; end if; s := s union {p^e} ; end do: end do: sort(s) ; end proc:
A192135(20000000) ; # R. J. Mathar, Jul 09 2011

seq[lim_] := Module[{s = {}, p = 2}, While[p^p <= lim, AppendTo[s, p^Range[p+1, Log[p, lim]]]; p = NextPrime[p]]; Sort[Flatten[s]]]; seq[10^7] (* Amiram Eldar, Apr 14 2025 *)

a(n) = A000961(A192187(n)).
A095874(a(n)) =  A192187(n).
Sum_{n>=1} 1/a(n) = Sum_{p prime} 1/(p^p*(p-1)) = 0.26859872089648243789... . - Amiram Eldar, Apr 14 2025

A192188 Positions of prime powers p^e with e <= p within A000961.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25, 26, 27, 29, 30, 31, 32, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 70, 72, 73, 74, 75

Offset: 1

Reinhard Zumkeller, Jun 26 2011

nonn

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

Complement of A192187.
a(n) = A095874(A074583(n)).
A000961(a(n)) = A074583(n).

A343983 Numbers k such that Sum_{j|k} j^j == 1 (mod k).

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, 72, 73, 79, 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, 233, 239, 241, 251, 257

Offset: 1

Seiichi Manyama, May 06 2021

nonn

This sequence is different from A074583.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A062796, A188776.

q[n_] := Divisible[DivisorSum[n, #^# &] - 1, n]; Select[Range[260], q] (* Amiram Eldar, May 06 2021 *)

isok(n) = sumdiv(n, d, Mod(d, n)^d)==1;

from itertools import count, islice
from sympy import divisors
def A343983_gen(): # generator of terms
    yield 1
    for k in count(1):
        if sum(pow(j,j,k) for j in divisors(k,generator=True)) % k == 1:
            yield k
A343983_list = list(islice(A343983_gen(),30)) # Chai Wah Wu, Jun 19 2022

A073045 Nonprime solutions to sopfr(n) = S(n), where sopfr(n) = A001414 and S(n) = A002034.

Original entry on oeis.org

4, 9, 25, 27, 49, 121, 125, 169, 289, 343, 361, 529, 625, 841, 961, 1331, 1369, 1681, 1849, 2197, 2209, 2401, 2809, 3125, 3481, 3721, 4489, 4913, 5041, 5329, 6241, 6859, 6889, 7921, 9409, 10201, 10609, 11449, 11881, 12167, 12769, 14641, 16129, 16807

Offset: 1

Jason Earls, Aug 24 2002

nonn

			sopfr(9) = S(9) = 6 and 6 is composite, so 9 is a term.

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

Cf. A001414, A002034, A074583.

{for(n=1,17000,if(!isprime(n),s=0; fac=factor(n); for(i=1,matsize(fac)[1],s=s+fac[i,1]*fac[i,2]); m=1; p=1; while(p%n>0,m++; p=p*m); if(s==m,print1(n,","))))}

Edited and extended by Klaus Brockhaus, Aug 26 2002

cp's OEIS Frontend

A384915 The number of unordered factorizations of n into powers of primes of the form p^e where p is prime and 0 <= e <= p (A074583).

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A344029 Numbers included in A343983 but not in A074583.

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A192135 Prime powers p^e with p < e.

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A192188 Positions of prime powers p^e with e <= p within A000961.

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A343983 Numbers k such that Sum_{j|k} j^j == 1 (mod k).

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A073045 Nonprime solutions to sopfr(n) = S(n), where sopfr(n) = A001414 and S(n) = A002034.

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