A057521 -id:A057521 - cp's OEIS frontend

A382890 The square root of the largest square dividing the n-th cubefree number.

1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 5, 1, 2, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 7, 5, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 5, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 1, 7, 3, 10, 1, 1

Offset: 1

Amiram Eldar, Apr 07 2025

The product of the non-unitary prime divisors of the n-th cubefree number.
Also, the square root of the powerful part of the n-th cubefree number.
All the terms are squarefree.

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

Cf. A000188, A004709, A005117, A057521, A371188 (positions of 1's).

Similar sequences: A382888, A382889, A382891.

f[p_, e_] := p^If[e == 1, 0, 1]; s[n_] := Module[{fct = FactorInteger[n]}, If[AllTrue[fct[[;; , 2]], # < 3 &], Times @@ f @@@ fct, Nothing]]; Array[s, 100]

list(lim) = {my(f); print1(1, ", "); for(k = 2, lim, f = factor(k); if(vecmax(f[, 2]) < 3, print1(prod(i = 1, #f~, f[i, 1]^if(f[i, 2] == 1, 0, 1)), ", ")));}

a(n) = A000188(A004709(n)).
a(n) = sqrt(A382889(n)).
a(n) = A004709(n)/A382888(n).
a(n) = sqrt(A004709(n)/A382891(n)).
a(A371188(n)) = 1.

A386426 Odd nondeficient numbers k such that sigma(k) has the same powerful part as k, where sigma is the sum of divisors function.

Original entry on oeis.org

81022725, 891360225, 992106225, 1863765225, 2349967725, 3322372725, 7211992725, 8670600225, 9156802725, 11101612725, 13208490225, 15477435225, 15963637725, 18394650225, 18880852725, 21311865225, 21960135225, 22446337725, 22932540225, 25687687725, 25849755225, 28280767725, 28604902725, 30711780225, 31035915225

Offset: 1

Antti Karttunen, Aug 17 2025

nonn

Sequence by definition contains also any such hypothetical odd terms of A007691 that are mentioned in the comments of A386425. However, if no such terms exist, then this is a subsequence of A386427.
This sequence contains also the intersection of A001694 and A386425, even though it is probably an empty set. See comments in A386428.
The first three terms not divisible by 25 are: a(191) = 283806508293, a(247) = 371184932349, a(328) = 502252568433.

Intersection of A023196 and A386425.
Conjectured to be a subsequence of A386427.
Cf. A000203, A001694, A003557, A007691, A057521, A386428.
Cf. also A005231.

A057521(n)=my(f=factor(n)); prod(i=1, #f~, if(f[i, 2]>1, f[i, 1]^f[i, 2], 1))
isA386426(n) = if(!(n%2), 0, my(s=sigma(n)); ((s>=(2*n)) && (A057521(s)==A057521(n))));

{k | k is odd, A000203(k) >= 2*k and A003557(A000203(k)) = A003557(k)}.

a(8)-a(25) from Giovanni Resta, Aug 18 2025

A212173 First integer with same second signature as n (cf. A212172).

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 1, 8, 4, 1, 1, 4, 1, 1, 1, 16, 1, 4, 1, 4, 1, 1, 1, 8, 4, 1, 8, 4, 1, 1, 1, 32, 1, 1, 1, 36, 1, 1, 1, 8, 1, 1, 1, 4, 4, 1, 1, 16, 4, 4, 1, 4, 1, 8, 1, 8, 1, 1, 1, 4, 1, 1, 4, 64, 1, 1, 1, 4, 1, 1, 1, 72, 1, 1, 4, 4, 1, 1, 1, 16, 16, 1, 1, 4

Offset: 1

Matthew Vandermast, Jun 03 2012

Two integers have the same second signature iff the same exponents >= 2 occur in the canonical prime factorization of each integer, regardless of the order in which they occur in each factorization.

			12 = 2^2*3 has 1 exponent >= 2 in its prime factorization, namely, 2. Hence, its second signature is {2}.  The smallest number with second signature {2} is 4; hence, a(12) = 4.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 844.

Jason Kimberley, Table of n, a(n) for n = 1..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Primefan, The First 2500 Integers Factored (1st of 5 pages)

Cf. A212172, A046523. All terms belong to A181800.

A212173 := func; [A212173(n):n in[1..85]]; // Jason Kimberley, Jun 14 2012

f:= proc(n) local E,i;
E:= sort(select(`>`, map(t -> t[2], ifactors(n)[2]), 1),`>`);
mul(ithprime(i)^E[i],i=1..nops(E))
end proc:
map(f, [$1..100]); # Robert Israel, Jul 19 2017

Function[s, Sort[Apply[Join, Map[Function[k, Map[{#, First@ k} &, k]], Values@ s]]][[All, -1]]]@ KeySort@ PositionIndex@ Table[Sort@ DeleteCases[FactorInteger[n][[All, -1]], e_ /; e < 2] /. {} -> {1}, {n, 84}] (* Michael De Vlieger, Jul 19 2017 *)

a(n) = {my(sn = vecsort(select(x->(x>=2), factor(n)[,2]))); for (i=1, n, if (vecsort(select(x->(x>=2),factor(i)[,2])) == sn, return(i)););} \\ Michel Marcus, Jul 19 2017

from functools import reduce
from sympy import factorint
from operator import mul
def P(n): return sorted(factorint(n).values())
def a046523(n):
    x=1
    while True:
        if P(n)==P(x): return x
        else: x+=1
def a057521(n): return 1 if n==1 else reduce(mul, [1 if e==1 else p**e for p, e in factorint(n).items()])
def a(n): return a046523(a057521(n))
print([a(n) for n in range(1, 151)]) # Indranil Ghosh, Jul 19 2017

a(n) = A046523(A057521(n)) = A057521(A046523(n)).

A240370 Positive integers n such that every element in the ring of integers modulo n can be written as the sum of two squares modulo n.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 50, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102, 103, 105, 106, 107, 109, 110, 111, 113, 114, 115, 118, 119, 122, 123, 125, 127, 129, 130, 131, 133, 134, 137, 138, 139, 141, 142, 143, 145, 146, 149, 150, 151, 154, 155, 157, 158, 159, 161, 163, 165, 166, 167, 169

Offset: 1

Charles R Greathouse IV, Apr 04 2014

nonn

Numbers n such that, if p^2 divides n for any prime p, then p = 1 mod 4.
Equivalently, squarefree numbers times A004613.
Thus, numbers k such that A065338(A057521(k)) = 1. - Antti Karttunen, Jun 21 2014
Different from A193304: terms 169, 289, 338, 507, 578, 841, 845, 867, ... are here but not in A193304. - Michel Marcus, Jun 20 2014
The asymptotic density of this sequence is 3/(8*K^2) = (3/4) * A243379 = 0.64208..., where K is the Landau-Ramanujan constant (A064533). - Amiram Eldar, Dec 19 2020

			In Z_7, 0^2 + 0^2 = 0, 1^2 + 0^2 = 1, 1^2 + 1^2 = 2, 3^2 + 1^2 = 3, 2^2 + 0^2 = 4, 2^2 + 1^2 = 5, 3^2 + 2^2 = 6. Therefore 7 is in the sequence.
In Z_8, there is no way to express 3 as a sum of two squares. Therefore 8 is not in the sequence.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Joshua Harrington, Lenny Jones, and Alicia Lamarche, Representing Integers as the Sum of Two Squares in the Ring Z_n, arXiv:1404.0187 [math.NT], Apr 01 2014 and, J. Int. Seq. 17 (2014) # 14.7.4.

The subsequence A240109 is a version not allowing 0.
Different from A193304.
Complement of A053443. Subsequence of A192450.
Cf. A004613, A057521, A064533, A065338, A243379.

rad[n_] := Times @@ First /@ FactorInteger[n];
a57521[n_] := n/Denominator[n/rad[n]^2];
a65338[n_] := a65338[n] = If[n==1, 1, Mod[p = FactorInteger[n][[1, 1]], 4]* a65338[n/p]];
Select[Range[200], a65338[a57521[#]] == 1&] (* Jean-François Alcover, Sep 22 2018, after Antti Karttunen *)
Select[Range[200], AllTrue[FactorInteger[#], Mod[First[#1], 4] == 1 || Last[#1] == 1 &] &] (* Amiram Eldar, Dec 19 2020 *)

is(n)=my(f=factor(n)); for(i=1,#f~,if(f[i,2]>1 && f[i,1]%4>1, return(0))); 1

isok(n) = { if (n < 2, return (0)); if ((n % 4) == 0, return (0)); forprime(q = 2, n, if (((q % 4) == 3) && ((n % q) == 0) && ((n % q^2) == 0), return (0)); ); return (1); } \\ Michel Marcus, Jun 08 2014

;; With Antti Karttunen's IntSeq-library.
(define A240370 (MATCHING-POS 1 1 (lambda (k) (= 1 (A065338 (A057521 k))))))
;; Antti Karttunen, Jun 21 2014

A336253 Exponential barely deficient numbers: exponential deficient numbers whose exponential abundancy is closer to 2 than that of any smaller exponential deficient number.

Original entry on oeis.org

1, 4, 72, 100, 144, 3528, 12100, 15876, 24336, 441000, 1334025, 2205000, 5664400, 24206400, 71267364, 151880976, 3252372552, 9346201200, 13319078472, 26828235000, 347372082000, 1851803856100, 2260121356900, 3198696480100, 5202286387272, 10330374528100, 16316106062400

Offset: 1

Amiram Eldar, Jul 14 2020

nonn

The exponential abundancy of a number k is esigma(k)/k, where esigma is the sum of exponential divisors of k (A051377).
Exponential deficient numbers are numbers k with esigma(k)/k < 2. These are numbers that are neither e-perfect (A054979) nor exponential abundant (A129575).
The corresponding values of the exponential abundancy are 1, 1.5, 1.666..., 1.8..., 1.833..., ...
All the terms are powerful numbers (A001694) because esigma(k)/k depends only on the powerful part of k (A057521). - Amiram Eldar, May 06 2025

			4 is a term since it is exponential deficient, and esigma(4)/4 = 3/2 is higher than esigma(k)/k for all the exponential deficient numbers k < 4.

Amiram Eldar, Table of n, a(n) for n = 1..36

Cf. A051377, A054979, A057521, A129575.
Subsequence of A001694.
Similar sequences: A302572, A228450, A262228, A307122, A336252, A336254.

fun[p_, e_] := DivisorSum[e, p^# &]; esigma[1] = 1; esigma[n_] := Times @@ fun @@@ FactorInteger[n]; rm = 0; s={}; Do[r = esigma[n]/n; If[r >= 2, Continue[]]; If[r > rm, rm = r; AppendTo[s, n]], {n, 1, 10^6}]; s

a(21)-a(27) from Amiram Eldar, May 06 2025

A336254 Exponential barely abundant numbers: exponential abundant numbers whose exponential abundancy is closer to 2 than that of any smaller exponential abundant number.

Original entry on oeis.org

900, 1764, 3600, 4356, 4500, 4900, 12348, 47916, 79092, 112500, 605052, 2812500, 13366548, 29647548, 89139564, 231708348, 701538156, 1757812500, 14772192228, 32179382604, 43945312500, 71183762748, 620995547124, 990454107996, 3417547576788, 3488004374652, 10271220141996

Offset: 1

Amiram Eldar, Jul 14 2020

nonn

The exponential abundancy of a number k is esigma(k)/k, where esigma is the sum of exponential divisors of k (A051377).

All the terms are powerful numbers (A001694) because esigma(k)/k depends only on the powerful part of k (A057521). - Amiram Eldar, May 06 2025

			The first 6 exponential abundant numbers, 900, 1764, 3600, 4356, 4500 and 4900, have decreasing values of exponential abundancy: 2.4, 2.285..., 2.2, 2.181..., 2.08, 2.057... and therefore they are in this sequence. The next exponential abundant number with a lower exponential abundancy is 12348 with eisgma(12348)/12348 = 2.040...

Amiram Eldar, Table of n, a(n) for n = 1..36

The exponential version of A071927.
Subsequence of A001694 and A328136.
Cf. A051377, A057521, A129575, A336253.
Similar sequences: A188263, A302570, A302571, A335054.

fun[p_, e_] := DivisorSum[e, p^# &]; esigma[1] = 1; esigma[n_] := Times @@ fun @@@ FactorInteger[n]; rm = 3; s={}; Do[r = esigma[n]/n; If[r <= 2, Continue[]]; If[r < rm, rm = r; AppendTo[s, n]], {n, 1, 10^6}]; s

a(23)-a(27) from Amiram Eldar, May 06 2025

A337052 Numbers k such that the powerful part of k has an even number of prime divisors counted with multiplicity.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76

Offset: 1

Amiram Eldar, Aug 12 2020

nonn

Numbers k such that A001222(A057521(k)) == 0 (mod 2).
Numbers k such that A057521(k) is in A028260.
Differs from A096432 by having the additional terms 1 and 216, 256, 768, 864, ... and not having the terms 432, 648, ...
First differs from both A220218 and A335275 at n = 193: a(193) = 216 is not a term of these two sequences.
Cohen (1964) proved that this sequence has an asymptotic density, and gave the value 1/2 + (1/5) * Product_{p prime} (1 + (p^2 + p + 1)/(p^3 * (p + 1))) = 0.8172707179... But the numbers of terms not exceeding 10^k for k = 1, 2, ... are 9, 90, 885, 8849, 88499, 884993, 8849889, 88498711, 884987643, 8849876178, ... indicating that the asymptotic density is about 0.88498...

			2 is a term since the powerful part of 2 is 1, which has 0 prime divisors, and 0 is even.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eckford Cohen, Some asymptotic formulas in the theory of numbers, Trans. Amer. Math. Soc., Vol. 112, No. 2 (1964), pp. 214-227. See corollary 4.2.2, pp. 226-227.

Cf. A001222, A028260, A057521, A096432.

Subsequences: A000290, A004709, A005117, A138302, A220218, A335275 and A336615.

Select[Range[100], EvenQ @ Total @ Select[FactorInteger[#][[;; , 2]], #1 > 1 &] &]

A351449 a(n) = A064989(A295294(A003961(n))).

Original entry on oeis.org

1, 1, 1, 11, 1, 1, 1, 3, 29, 1, 1, 11, 1, 1, 1, 49, 1, 29, 1, 11, 1, 1, 1, 3, 34, 1, 22, 11, 1, 1, 1, 55, 1, 1, 1, 319, 1, 1, 1, 3, 1, 1, 1, 11, 29, 1, 1, 49, 85, 34, 1, 11, 1, 22, 1, 3, 1, 1, 1, 11, 1, 1, 29, 1091, 1, 1, 1, 11, 1, 1, 1, 87, 1, 1, 34, 11, 1, 1, 1, 49, 469, 1, 1, 11, 1, 1, 1, 3, 1

Offset: 1

Antti Karttunen, Feb 11 2022

Antti Karttunen, Table of n, a(n) for n = 1..20000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A003961, A048250, A057521, A064989, A151800, A295294, A326042, A351450, A351451.

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
A057521(n) = { my(f=factor(n)); prod(i=1, #f~, if(f[i, 2]>1, f[i, 1]^f[i, 2], 1)); }; \\ From A057521
A064989(n) = { my(f = factor(n>>valuation(n,2))); for(i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
A295294(n) = sigma(A057521(n));
A351449(n) = A064989(A295294(A003961(n)));

Multiplicative with a(p) = 1 and for e > 1, a(p^e) = A064989((q^(e+1)-1)/(q-1)), where q = nextPrime(p) = A151800(p).
a(n) = A064989(A295294(A003961(n))) = A326042(A057521(n)).
a(n) = A326042(n) / A351451(n).

A366765 The largest divisor of n that have no exponent 2 in their prime factorization.

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 8, 3, 10, 11, 6, 13, 14, 15, 16, 17, 6, 19, 10, 21, 22, 23, 24, 5, 26, 27, 14, 29, 30, 31, 32, 33, 34, 35, 6, 37, 38, 39, 40, 41, 42, 43, 22, 15, 46, 47, 48, 7, 10, 51, 26, 53, 54, 55, 56, 57, 58, 59, 30, 61, 62, 21, 64, 65, 66, 67, 34, 69

Offset: 1

Amiram Eldar, Oct 21 2023

The largest term of A337050 that divides n.

The number of these divisors is A366763(n), and their sum is A366764(n).

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

Cf. A337050, A366763, A366764.

Similar sequences: A055231, A057521, A008833, A350390.

f[p_, e_] := p^If[e < 3, 1, e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i, 1] ^ if(f[i, 2] < 3, 1, f[i, 2]));}

Multiplicative with a(p^e) = p if e <= 2 and p^e otherwise.
a(n) <= n, with equality if and only if n is in A337050.
Dirichlet g.f.: zeta(s-1) * Product_{p prime} (1 - 1/p^(2*s-2) + 1/p^(2*s-1) + 1/p^(3*s-3) - 1/p^(3*s-2)).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 - 1/p^2 + 2/p^3 - 1/p^4) = 0.83234421330425224469... .

A368697 Numbers that are divisible by the squares of two distinct primes and whose arithmetic derivative (A003415) is a squarefree number of the form 4k+2.

Original entry on oeis.org

11025, 17325, 27225, 28665, 29925, 36225, 37485, 38025, 40425, 47025, 48825, 49725, 53361, 56925, 63525, 63945, 65025, 69825, 70785, 74025, 74529, 76725, 81225, 81585, 84525, 84825, 88725, 90405, 92169, 92565, 92925, 98325, 105525, 106425, 108225, 110925, 111573, 111825, 113925, 116325, 116865, 117117, 119025, 119925

Offset: 1

Antti Karttunen, Jan 09 2024

nonn

Intersection of A036785 and A368696, i.e., of A036785, A327862 and A328393.

Cf. A003415, A057521.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A057521(n) = { my(f=factor(n)); prod(i=1, #f~, if(f[i, 2]>1, f[i, 1]^f[i, 2], 1)); };
isA368697(n) = if(omega(A057521(n))<2, 0, my(d=A003415(n)); ((2==(d%4))&&issquarefree(d)));

cp's OEIS Frontend

A382890 The square root of the largest square dividing the n-th cubefree number.

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A386426 Odd nondeficient numbers k such that sigma(k) has the same powerful part as k, where sigma is the sum of divisors function.

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A212173 First integer with same second signature as n (cf. A212172).

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A240370 Positive integers n such that every element in the ring of integers modulo n can be written as the sum of two squares modulo n.

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A336253 Exponential barely deficient numbers: exponential deficient numbers whose exponential abundancy is closer to 2 than that of any smaller exponential deficient number.

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A336254 Exponential barely abundant numbers: exponential abundant numbers whose exponential abundancy is closer to 2 than that of any smaller exponential abundant number.

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A337052 Numbers k such that the powerful part of k has an even number of prime divisors counted with multiplicity.

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