A056824 -id:A056824 - cp's OEIS frontend

A381312 Numbers whose powerful part (A057521) is a power of a prime with an odd exponent >= 3 (A056824).

8, 24, 27, 32, 40, 54, 56, 88, 96, 104, 120, 125, 128, 135, 136, 152, 160, 168, 184, 189, 224, 232, 243, 248, 250, 264, 270, 280, 296, 297, 312, 328, 343, 344, 351, 352, 375, 376, 378, 384, 408, 416, 424, 440, 456, 459, 472, 480, 486, 488, 512, 513, 520, 536, 544

Offset: 1

Amiram Eldar, Feb 19 2025

Subsequence of A301517 and A374459 and first differs from them at n = 21. A301517(21) = A374459(21) = 216 is not a term of this sequence.
Numbers having exactly one non-unitary prime factor and its multiplicity is odd.
Numbers whose prime signature (A118914) is of the form {1, 1, ..., 2*m+1} with m >= 1, i.e., any number (including zero) of 1's and then a single odd number > 1.
The asymptotic density of this sequence is (1/zeta(2)) * Sum_{p prime} 1/((p-1)*(p+1)^2) = 0.093382464285953613312...

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

Complement of A381311 within A190641.
Cf. A057521, A118914.
Subsequence of A268335, A301517 and A374459.
Subsequences: A030078, A050997, A056824, A065036, A079395, A092759, A138031, A178740, A179664, A179665, A179667, A179670, A179692, A179696, A179704, A189975, A189984, A190378, A190383, A190473.

q[n_] := Module[{e = ReverseSort[FactorInteger[n][[;; , 2]]]}, e[[1]] > 1 && OddQ[e[[1]]] && (Length[e] == 1 || e[[2]] == 1)]; Select[Range[1000], q]

isok(k) = if(k == 1, 0, my(e = vecsort(factor(k)[, 2], , 4)); e[1] % 2 && e[1] > 1 && (#e == 1 || e[2] == 1));

A176509 Composite numbers m for which A064380(m) = A000010(m).

Original entry on oeis.org

8, 27, 125, 128, 343, 1331, 2187, 2197, 4913, 6859, 12167, 24389, 29791, 32768, 50653, 68921, 78125, 79507, 103823, 148877, 205379, 226981, 300763, 357911, 389017, 493039, 571787, 704969, 823543, 912673, 1030301, 1092727, 1225043, 1295029, 1442897, 2048383, 2248091

Offset: 1

Vladimir Shevelev, Apr 19 2010

nonn

Theorem. A064380(m) = A000010(m) iff m has the form m=p^(2^k-1), k>=1, p a prime. Eliminating the primes (k=1), the terms of the sequence have this form for k>1. All terms of A030078 (k=2) and A092759 (k=3) and prime powers of A010803 (k=4) are in the sequence, for example.

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

Cf. A000010, A010803, A030078, A050376, A056824, A064380, A092759, A176472.

seq[max_] := Module[{ps = Select[Range[Floor[Surd[max, 3]]], PrimeQ], e, k, s = {}}, Do[e = Floor[Log[ps[[i]], max]]; k = Floor[Log2[e + 1]]; s = Join[s, ps[[i]]^(2^Range[2, k] - 1)], {i, 1, Length[ps]}]; Sort[s]]; seq[3*10^6] (* Amiram Eldar, Mar 26 2023 *)

is(n)=my(e=isprimepower(n));e>2 && 2^valuation(e+1,2)==e+1 \\ Charles R Greathouse IV, Feb 19 2013

a(n) ~ n^3 log^3 n. - Charles R Greathouse IV, Feb 19 2013

Sum_{n>=1} 1/a(n) = Sum_{k>=2} 1/P(2^k-1) = 0.183077059924063305405..., where P(s) is the prime zeta function. - Amiram Eldar, Jul 11 2024

128 inserted, 1024 deleted, 2187 inserted, 32768 inserted, etc. - R. J. Mathar, Nov 21 2010

More terms from Amiram Eldar, Mar 26 2023

A301517 Numbers whose ratio (sum of nonsquarefree divisors)/(sum of squarefree divisors) is a positive integer.

Original entry on oeis.org

8, 24, 27, 32, 40, 54, 56, 88, 96, 104, 120, 125, 128, 135, 136, 152, 160, 168, 184, 189, 216, 224, 232, 243, 248, 250, 264, 270, 280, 296, 297, 312, 328, 343, 344, 351, 352, 375, 376, 378, 384, 408, 416, 424, 440, 456, 459, 472, 480, 486, 488, 512, 513, 520

Offset: 1

Michel Lagneau, Mar 23 2018

nonn

Or numbers m such that r = A162296(m) / A048250(m) is a positive integer.
Conjecture: if r = A162296(a(n)) / A048250(a(n)) is a perfect square, r belongs to A001248.
The corresponding sequence b(n) = {r} begins with {4, 4, 9, 20, 4, 9, 4, 4, 20, 4, 4, 25, 84, 9, 4, 4, 20, 4, 4, 9, 49, 20, 4, 90, 4, 25, ... }. A majority of numbers of b(n) are perfect squares.
The numbers 2^(2n+1) with k > 0 are in the sequence (A004171).
The numbers prime(n)^3 are in the sequence (A030078).
The numbers 8*prime(n) with n > 1 are in the sequence.
Note that "positive integer", in the definition, eliminates squarefree numbers (A005117) from this sequence. - Michel Marcus, Mar 24 2018
From Robert Israel, Mar 29 2018: (Start)
If n is in the sequence, then so is n*p for any prime p coprime to n.
If m and n are in the sequence and are coprime, then m*n is in the sequence. (End)
The exponentially odd numbers (A268335) that are not squarefree are in the sequence. - Amiram Eldar, Jul 04 2020
The numbers of terms that do not exceed 10^k, for k = 1, 2, ..., are 1, 9, 99, 972, 9672, 96630, 966119, 9660732, 96606486, 966062725, ... . Apparently the asymptotic density of this sequence is 0.096606... . Note that most of the terms are in its subsequence A374459 whose asymptotic density is A065463 - A059956 = 0.096515099145... . - Amiram Eldar, Feb 20 2025

			27 is in the sequence because A162296(27) / A048250(27) = 36/4 = 9.

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

Cf. A004171, A005117, A030078, A048250, A162296, A268335, A335989, A374459.
Contains A056824.
Cf. A059956, A065463.

filter:= proc(n) local S,N; uses numtheory;
  S, N:= selectremove(issqrfree, divisors(n));
  N <> {} and type(convert(N,`+`)/convert(S,`+`),integer)
end proc:
select(filter, [$1..1000]); # Robert Israel, Mar 29 2018

lst={};Do[If[DivisorSigma[1,n]-Total[Select[Divisors[n],SquareFreeQ]]>0&&IntegerQ[(DivisorSigma[1,n]-Total[Select[Divisors[n],SquareFreeQ]])/Total[Select[Divisors[n],SquareFreeQ]]],AppendTo[lst,n]],{n,520}];lst
rpiQ[n_]:=Module[{d=Divisors[n],sf,ot,ra},sf=Select[d,SquareFreeQ];ot=Complement[ d, sf];ra= Total[ ot]/Total[sf];ra>0&&IntegerQ[ra]]; Select[Range[600],rpiQ] (* Harvey P. Dale, Mar 19 2019 *)
f[p_, e_] := (p^(e + 1) - 1)/(p^2 - 1); ratio[n_] := Times @@ (f @@@ FactorInteger[n]); Select[Range[2, 520], (r = ratio[#]) > 1 && IntegerQ[r] &] (* Amiram Eldar, Jul 04 2020 *)

isok(n) = my(s = sumdiv(n, d, !issquarefree(d)*d)); s && !(s % (sigma(n) - s)); \\ Michel Marcus, Mar 24 2018

A331593 Numbers k that have the same number of distinct prime factors as A225546(k).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 24, 25, 28, 29, 31, 37, 40, 41, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 61, 63, 67, 68, 71, 72, 73, 75, 76, 79, 80, 81, 83, 88, 89, 92, 96, 97, 98, 99, 101, 103, 104, 107, 108, 109, 112, 113, 116, 117, 121, 124, 127, 131, 135, 136, 137, 139, 144, 147, 148, 149

Offset: 1

Antti Karttunen and Peter Munn, Jan 21 2020

nonn

Numbers k for which A001221(k) = A331591(k).
Numbers k that have the same number of terms in their factorization into powers of distinct primes as in their factorization into powers of squarefree numbers with distinct exponents that are powers of 2. See A329332 for a description of the relationship between the two factorizations and A225546.
If k is included, then all such x that A046523(x) = k are also included, i.e., all numbers with the same prime signature as k. Notably, primes (A000040) are included, but squarefree semiprimes (A006881) are not.
k^2 is included if and only if k is included, for example A001248 is included, but A085986 is not.

			There are 2 terms in the factorization of 36 into powers of distinct primes, which is 36 = 2^2 * 3^2 = 4 * 9; but only 1 term in its factorization into powers of squarefree numbers with distinct exponents that are powers of 2, which is 36 = 6^(2^1). So 36 is not included.
There are 2 terms in the factorization of 40 into powers of distinct primes, which is 40 = 2^3 * 5^1 = 8 * 5; and also 2 terms in its factorization into powers of squarefree numbers with distinct exponents that are powers of 2, which is 40 = 10^(2^0) * 2^(2^1) = 10 * 4. So 40 is included.

Cf. A000120, A046523, A225546, A267116, A329332.
Sequences with related definitions: A001221, A331591, A331592.
Subsequences: A000040, A001248, A050376, A054753, A065036, A162142, A225547.
Subsequences of complement: A006881, A056824, A085986, A120944, A177492.

Select[Range@ 150, Equal @@ PrimeNu@ {#, If[# == 1, 1, Apply[Times, Flatten@ Map[Function[{p, e}, Map[Prime[Log2@ # + 1]^(2^(PrimePi@ p - 1)) &, DeleteCases[NumberExpand[e, 2], 0]]] @@ # &, FactorInteger[#]]]]} &] (* Michael De Vlieger, Jan 26 2020 *)

A331591(n) = if(1==n,0,my(f=factor(n),u=#binary(vecmax(f[, 2])),xs=vector(u),m=1,e); for(i=1,u,for(k=1,#f~, if(bitand(f[k,2],m),xs[i]++)); m<<=1); #select(x -> (x>0),xs));
k=0; n=0; while(k<105, n++; if(omega(n)==A331591(n), k++; print1(n,", ")));

{a(n)} = {k : A001221(k) = A000120(A267116(k))}.

A304291 Composite numbers k such that for all primes p dividing k, p-1 divides k-1 and p+1 divides k+1.

Original entry on oeis.org

8, 27, 32, 125, 128, 243, 343, 512, 1331, 2048, 2187, 2197, 3125, 4913, 6859, 8192, 12167, 16807, 19683, 24389, 29791, 32768, 50653, 68921, 74431, 78125, 79507, 103823, 131072, 148877, 161051, 177147, 205379, 226981, 300763, 357911, 371293, 389017, 493039, 524288

Offset: 1

Paolo P. Lava, May 17 2018

nonn

Intersection of A080062 and A056729.
Mainly odd powers of a prime: A056824 is a subset of this sequence.
If the additional limitations p-2|n-2 and p+2|n+2 should be added, only 243, 19683, 78125, 1594323 would be terms of the sequence for n <= 10^7.
Terms that are not perfect powers are 31*7^4, 31^3*7^4, 71*11^6, .... - Altug Alkan, May 17 2018
It appears that this is the intersection of A002808 and A171561. - Michel Marcus, May 19 2018
From Robert Israel, May 25 2018: (Start)
  If i is odd and 4|j, then 31^i*7^j is a member.
  If i is odd and 6|j, then 71^i*11^j is a member.
  If i is odd and 12|j, then 17^i*5^j is a member.
  If i is odd and 36|j, then 53^i*5^j is a member.
  If i == 9 (mod 18) and 6|j, then 13^i*37^j is a member.
  If i == 9 (mod 18) and 12|j, then 29^i*53^j is a member.
  If i == 18 (mod 36), j == 3 (mod 6) and k == 2 (mod 4), then 5^i*17^j*53^k is a member.
(End)
Composite numbers k such that for all primes p dividing k, p+1 divides k-1 and p-1 divides k+1 are the union of 2^2j and 3^2j, with j>0. - Paolo P. Lava, May 16 2019

			Prime factors of 74431 are 7 and 31 and (74431-1)/(7-1) = 12405, (74431-1)/(31-1) = 2481, (74431+1)/(7+1) = 9304, (74431+1)/(31+1) = 2326.

Giovanni Resta, Table of n, a(n) for n = 1..1000 (first 192 terms from Robert Israel)

Cf. A056729, A056824, A080062.

sol:=[]; m:=1; p:=[]; for u in [1..600000] do if not IsPrime(u) then p:=PrimeDivisors(u);  s:=0; for i in [1..#p] do if IsIntegral((u-1)/(p[i]-1)) and  IsIntegral((u+1)/(p[i]+1)) then  s:=s+1; end if; if s eq #p then sol[m]:=u; m:=m+1; end if; end for; end if; end for; sol; // Marius A. Burtea, May 16 2019

with(numtheory): P:=proc(q) local a,b,k,n,ok;
for n from 2 to q do if not isprime(n) then a:=factorset(n); ok:=1;
for k from 1 to nops(a) do if frac((n-1)/(a[k]-1))>0 or frac((n+1)/(a[k]+1))>0 then ok:=0; break; fi; od;
if ok=1 then print(n); fi; fi; od; end: P(10^6);

Select[Range[4, 2^19], Function[k, And[CompositeQ@ k, AllTrue[FactorInteger[k][[All, 1]], And[Mod[k - 1, # - 1] == 0, Mod[k + 1, # + 1] == 0] &]]]] (* Michael De Vlieger, May 22 2018 *)

lista(nn) = {forcomposite(c=1, nn, my(f = factor(c)); ok = 1; for (k=1, #f~, my(p = f[k,1]); if (((c-1) % (p-1)) || ((c+1) % (p+1)), ok = 0; break);); if (ok, print1(c, ", ")););} \\ Michel Marcus, May 19 2018

A301482 Composite numbers whose sum of aliquot parts divide the sum of the squares of their aliquot parts.

Original entry on oeis.org

8, 22, 27, 32, 77, 125, 128, 243, 343, 494, 512, 611, 660, 1073, 1281, 1331, 1425, 2033, 2048, 2187, 2197, 2332, 3125, 4172, 4565, 4913, 5293, 6031, 6859, 8192, 9983, 12167, 13969, 15818, 15947, 16807, 17485, 19683, 23489, 23840, 24389, 25241, 25389, 29791, 32768

Offset: 1

Paolo P. Lava, Mar 22 2018

Semiprimes in the sequence: 22, 77, 611, 1073, 2033, 5293, 6031, 9983, 13969, 15947, 23489, 25241, 40301, 49901, 50249, 51101, 56759, 65017, 71677, 85079, 97217, 98099, 99101, .... - Robert Israel, Mar 29 2018

2^k is a term for all odd k > 1. - Michael S. Branicky, Aug 22 2021

			Aliquot parts of 77 are 1, 7, 11. Then (1^2 + 7^2 + 11^2)/(1 + 7 + 11) = 171/19 = 9.

Amiram Eldar, Table of n, a(n) for n = 1..1009 (terms below 10^9, terms 1..100 from Paolo P. Lava)

Cf. A001065, A001157, A020487, A067558.

Contains A056824.

with(numtheory): P:=proc(n)
if not isprime(n) and frac((add(p^2,p=divisors(n))-n^2)/(sigma(n)-n))=0
then n; fi; end: seq(P(i),i=2..35*10^3);

aQ[n_] := CompositeQ[n] && Divisible[DivisorSigma[2, n] - n^2, DivisorSigma[1, n] - n]; Select[Range[33000], aQ] (* Amiram Eldar, Aug 17 2019 *)

isok(n) = (n!=1) && !isprime(n) && (((sigma(n,2) - n^2) % (sigma(n) - n)) == 0); \\ Michel Marcus, Mar 23 2018

from sympy import divisors
def ok(n):
    divs = divisors(n)[:-1]
    return len(divs) > 1 and sum(d**2 for d in divs)%sum(divs) == 0
print(list(filter(ok, range(4, 32769)))) # Michael S. Branicky, Aug 22 2021

A381316 Numbers whose powerful part (A057521) is a power of a prime with an exponent >= 3 (A246549).

Original entry on oeis.org

8, 16, 24, 27, 32, 40, 48, 54, 56, 64, 80, 81, 88, 96, 104, 112, 120, 125, 128, 135, 136, 152, 160, 162, 168, 176, 184, 189, 192, 208, 224, 232, 240, 243, 248, 250, 256, 264, 270, 272, 280, 296, 297, 304, 312, 320, 328, 336, 343, 344, 351, 352, 368, 375, 376, 378

Offset: 1

Amiram Eldar, Feb 19 2025

First differs from A344653 and A345193 at n = 17: a(17) = 120 is not a term of these sequences.
Numbers whose prime signature (A118914) is of the form {1, 1, ..., m} with m >= 3, i.e., any number (including zero) of 1's and then a single number >= 3.
The asymptotic density of this sequence is (1/zeta(2)) * Sum_{p prime} 1/(p*(p^2-1)) = A369632 / A013661 = 0.13463358553764438661... .

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

Complement of A060687 within A190641.
Cf. A013661, A057521, A246549, A369632.
Cf. A344653, A345193.
Subsequences: A030078, A030514, A030516, A030629, A030631, A050997, A056824, A065036, A079395, A092759, A138031, A178739, A178740, A179644, A179645, A179664, A179665, A179667, A179668, A179670, A179672, A179692, A179693, A179696, A179704, A179747, A189975, A189984, A189987, A190110, A190292, A190378, A190383, A190388, A190473, A381312.

q[n_] := Module[{e = ReverseSort[FactorInteger[n][[;; , 2]]]}, e[[1]] > 2 && (Length[e] == 1 || e[[2]] == 1)]; Select[Range[1000], q]

isok(k) = if(k == 1, 0, my(e = vecsort(factor(k)[, 2], , 4)); e[1] > 2 && (#e == 1 || e[2] == 1));

cp's OEIS Frontend

A381312 Numbers whose powerful part (A057521) is a power of a prime with an odd exponent >= 3 (A056824).

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A176509 Composite numbers m for which A064380(m) = A000010(m).

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A301517 Numbers whose ratio (sum of nonsquarefree divisors)/(sum of squarefree divisors) is a positive integer.

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A331593 Numbers k that have the same number of distinct prime factors as A225546(k).

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A304291 Composite numbers k such that for all primes p dividing k, p-1 divides k-1 and p+1 divides k+1.

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A301482 Composite numbers whose sum of aliquot parts divide the sum of the squares of their aliquot parts.

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A381316 Numbers whose powerful part (A057521) is a power of a prime with an exponent >= 3 (A246549).

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