A013929 -id:A013929 - cp's OEIS frontend

A212177 Number of exponents >= 2 in the canonical prime factorization of the n-th nonsquarefree number (A013929(n)).

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1

Offset: 1

Matthew Vandermast, Jun 03 2012

Length of second signature of A013929(n) (cf. A212172).

			24 = 2^3*3 has 1 exponent of size 2 or greater in its prime factorization. Since 24 = A013929(8), a(8) = 1.

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.

Reinhard Zumkeller, 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. A013929, A212172, A212174.

Cf. A046028, A056170, A085548, A124010, A229099.

a212177 n = a212177_list !! (n-1)
a212177_list = filter (> 0) a056170_list
-- Reinhard Zumkeller, Dec 29 2012

f[n_] := Module[{c = Count[FactorInteger[n][[;; , 2]], ?(# > 1&)]}, If[n > 1 && c > 0, c, Nothing]]; f[1] = 0; Array[f, 300] (* _Amiram Eldar, Oct 01 2023 *)

from math import isqrt
from sympy import mobius, factorint
def A212177(n):
    def f(x): return n+sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return sum(1 for e in factorint(m).values() if e>1) # Chai Wah Wu, Jul 22 2024

a(n) = A056170(A013929(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (Sum_{p prime} 1/p^2)/(1-1/zeta(2)) = A085548 / A229099 = 1.15347789194214704903... . - Amiram Eldar, Oct 01 2023

A192005 Number of non-cyclic abelian groups of finite order. The order is given by A013929.

Original entry on oeis.org

1, 2, 1, 1, 4, 1, 1, 2, 1, 2, 1, 6, 3, 2, 1, 1, 4, 1, 1, 1, 2, 2, 1, 1, 10, 1, 5, 1, 1, 4, 4, 1, 2, 1, 1, 6, 1, 1, 3, 2, 5, 4, 1, 1, 2, 1, 1, 2, 1, 14, 1, 2, 2, 1, 9, 1, 1, 1, 2, 1, 1, 6, 4, 1, 2, 1, 1, 1, 1, 4, 3, 2, 1, 2, 10, 3, 1, 5, 1, 1, 4, 1, 8, 1, 6, 3, 1, 2, 1, 1, 4, 1, 6, 1, 1, 2, 2, 3, 21, 1, 1, 2, 1, 2, 4, 1, 1, 1, 2

Offset: 1

Wolfdieter Lang, Jul 28 2011

Every abelian group of finite order is the direct product of cyclic groups (there may be only one factor). See, e.g., the A. Speiser reference, Satz 43, p. 49, in combination with Satz 42, p. 47, and also Satz 4, p. 17, with the remark on the direct product on page 28.

See the list of abelian groups of small order in the Wikipedia link.

			n=1: there is one abelian group of order 4=A013929(1), which is not the cyclic group Z_4 (in additive notation), namely the Klein 4-group: Z_2 x Z_2 (also denoted by (Z_2)^2).
n=2: there are 2 non-cyclic abelian groups of order 8=A013929(2), namely Z_2 x Z_4 and (Z_2)^3.
n=3: order 9=A013929(3), (Z_3)^2.
n=4: order 12, Z_3 x (Z_2)^2 (note that Z_6 = Z_3 x Z_2 and Z_12 = Z_4 x Z_3, where = means 'is isomorphic to').
n=5: order 16. The four non-cyclic groups are (Z_2)^4, Z_4 x (Z_2)^2, Z_8 x Z_2 and (Z_4)^2.

Andreas Speiser, Die Theorie der Gruppen von endlicher Ordnung, Vierte Auflage, Birkhäuser, 1956.

Wikipedia, List of small groups.

Cf. A000688, A013929, A013661, A021002.

FiniteAbelianGroupCount /@ Select[Range[300], ! SquareFreeQ[#] &] - 1 (* Amiram Eldar, Oct 01 2023 *)

a(n) = A000688(A013929(n)) - 1, n>=1.
See the formula for A000688 using the product of the number of partitions of the exponents in the prime number factorization.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (zeta(2) * c - 1)/(zeta(2) - 1) - 1 = 3.3025914257..., where c = A021002. - Amiram Eldar, Oct 01 2023

A353282 a(n) is the number of solutions (x,y) to the Diophantine equation S(x,y) = (x+y) + (x-y) + (x*y) + (x/y) = A013929(n) when x >= y > 1 and y | x.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 3, 1, 3, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 3, 3, 1, 3, 1, 1, 2, 1, 3, 1, 2, 3, 1

Offset: 1

Bernard Schott, Apr 09 2022

nonn

This is the generalization of a problem proposed by Yakov Perelman for A013929(93) = 243 (references, links and example).
a(n) is the number of squares > 1 dividing A013929(n), so, there is no solution (x,y) for S(x,y) = m when m is a squarefree number (A005117).
Also, number of times where A013929(n) appears in table A351381.
The smallest nonsquare number m such that equation S(x,y) = m has exactly n solutions, for n >= 0, is A130279(n+1).
Integers k for which number of solutions to the equation S(x,y) = k sets a new record are in A046952.

			For S(x,y) = (x+y) + (x-y) + (x*y) + (x/y) = A013929(2) = 8, the unique solution is (2,1) because (2+1) + (2-1) + (2*1) + (2/1) = 8, hence a(2) = 1.
For S(x,y) = A013929(93) = 243, the two solutions are (24,8) and (54,2) because S(24,8) = S(54,2) = 243, hence a(93) = 2 (problem from Perelman's book).

I. Perelman, L'Algèbre récréative, Deux nombres et quatre opérations, Editions en langues étrangères, Moscou, 1959, pp. 101-102.
Ya. I. Perelman, Algebra can be fun, Two numbers and four operations, Mir Publishers Moscow, 1979, pp. 131-132.

Ya. I. Perelman, Algebra Can Be Fun, Chapter IV, Diophantine Equations, Two numbers and four operations, Mir Publishers Moscow, 1979, pp. 131-132.
Wikipedia, Yakov Perelman.

Cf. A046951, A005117, A013929, A046952, A130279, A351381.

f[p_, e_] := 1 + Floor[e/2]; s[1] = 0; s[n_] := Times @@ (f @@@ FactorInteger[n]) - 1; s /@ Select[Range[250], ! SquareFreeQ[#] &] (* Amiram Eldar, Apr 09 2022 *)

a(n) = A046951(A013929(n)) - 1.

More terms from Amiram Eldar, Apr 09 2022

A362041 a(0) = 1; for n > 0, a(n) is the largest k < A013929(n) such that rad(k) = rad(A013929(n)), where rad(n) = A007947(n).

Original entry on oeis.org

1, 2, 4, 3, 6, 8, 12, 10, 18, 5, 9, 14, 16, 24, 20, 22, 15, 36, 7, 40, 26, 48, 28, 30, 21, 32, 34, 54, 45, 38, 50, 27, 42, 44, 60, 46, 72, 56, 33, 80, 52, 96, 98, 58, 39, 90, 11, 62, 25, 84, 64, 66, 75, 68, 70, 108, 63, 74, 120, 76, 51, 78, 100, 144, 82, 126, 13, 57, 86, 35, 88, 150, 92, 94, 147, 162

Offset: 0

Michael De Vlieger, May 01 2023

nonn

Permutation of natural numbers.

Let m = A013929(n) and let R_m be the sequence of numbers k such that rad(k) = rad(m). a(n) gives the predecessor of m in R_m.

			A013929(1) = 4; the smallest k < 4 such that rad(k) = rad(4) = 2 is a(1) = 2.
A013929(2) = 8; the smallest k < 8 such that rad(k) = rad(8) = 2 is a(2) = 4.
A013929(3) = 9; the smallest k < 9 such that rad(k) = rad(9) = 3 is a(3) = 3.
A013929(4) = 12; the smallest k < 12 such that rad(k) = 6 is a(4) = 6.

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 0..2^16
Michael De Vlieger, Log log scatterplot of a(n), n = 0..2^12, showing primes in red, composite prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue. Numbers k with omega(k) > 1 and all exponents exceeding 1 are highlighted in large light blue dots.

Cf. A007947, A013929, A289280.

rad[x_] := Times @@ FactorInteger[x][[All, 1]]; {1}~Join~Table[Function[r, SelectFirst[Range[m - 1, 1, -1], r == rad[#] &] ][rad[m]], {m, Select[Range[225], Not @* SquareFreeQ]}]

A013929(n) = p^e, a prime power, e > 0, implies a(n) = p^(e-1).

A013929(n) = p^2 implies a(n) = p.

A115228 Nonsquarefree numbers n such that 2n+1 is also nonsquarefree (A013929).

Original entry on oeis.org

4, 12, 24, 40, 49, 60, 76, 84, 112, 121, 144, 148, 162, 171, 175, 180, 184, 212, 220, 256, 264, 292, 312, 328, 364, 387, 400, 412, 416, 420, 423, 436, 472, 480, 490, 508, 512, 544, 580, 612, 616, 625, 637, 652, 684, 688, 712, 722, 724, 760, 796, 808, 812

Offset: 1

Don Reble, Mar 05 2006

For any distinct primes p, q with q odd, contains all n such that n == 0 (mod p^2) and n == -1/2 (mod q^2). - Robert Israel, Oct 21 2016

			24 is in the sequence because 2^2 divides 24 and 7^2 divides 24*2 + 1.

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

Cf. A013939, A065474, A115170.

select(n -> not numtheory:-issqrfree(n) and not numtheory:-issqrfree(2*n+1), [$1..2000]); # Robert Israel, Oct 21 2016

fQ[n_] := ! SquareFreeQ[n] && ! SquareFreeQ[2 n + 1]; Select[Range[1000], fQ] (* Robert G. Wilson v, Oct 21 2016 *)

isok(n) = !issquarefree(n) && ! issquarefree(2*n+1); \\ Michel Marcus, Oct 22 2016

a(n) ~ n/(1 - 14/Pi^2 + 3*k/2 ) as n -> infinity, where k is the Feller-Tornier constant (A065474). - Robert Israel, Oct 21 2016

Corrected by Zak Seidov, Oct 21 2016

A212174 Row n of table represents second signature of A013929(n): list of exponents >= 2 in canonical prime factorization of A013929(n), in nonincreasing order.

Original entry on oeis.org

2, 3, 2, 2, 4, 2, 2, 3, 2, 3, 2, 5, 2, 2, 3, 2, 2, 4, 2, 2, 2, 3, 3, 2, 2, 6, 2, 3, 2, 2, 2, 4, 4, 2, 3, 2, 2, 5, 2, 2, 2, 2, 3, 3, 2, 4, 2, 2, 3, 2, 2, 3, 2, 7, 2, 3, 3, 2, 4, 2, 2, 2, 2, 3, 2, 2, 5, 4, 2, 3, 2, 2, 2, 2, 4, 2, 2, 3, 2, 3, 6, 2, 2, 2, 3, 2, 2, 2

Offset: 1

Matthew Vandermast, Jun 03 2012

Length of row n equals A212177(n).

			First rows of table read: 2; 3; 2; 2; 4; 2; 2; 3;...
12 = 2^2*3 has positive exponents 2 and 1 in its prime factorization, but only exponents that are 2 or greater appear in a number's second signature. Hence, 12's second signature is {2}. Since 12 = A013929(4), row 4 of the table represents the second signature {2}.

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 i, a(i) for i = 1..4492 (n = 1..3917)
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. A013929, A212172, A212175, A212176, A212177.

&cat[Reverse(Sort([pe[2]:pe in Factorisation(n)|pe[2]gt 1])):n in[1..247]]; // Jason Kimberley, Jun 13 2012

a(n) = A212172(A013929(n)).

This sequence is both the subsequence of A212171 formed by omitting all 1s and the subsequence of A212172 formed by omitting all 0's. - Jason Kimberley, Jun 13 2012

A283919 The smallest square referenced in A013929 (Numbers that are not squarefree).

Original entry on oeis.org

4, 4, 9, 4, 4, 9, 4, 4, 25, 9, 4, 4, 4, 4, 4, 9, 4, 49, 25, 4, 9, 4, 4, 9, 4, 4, 4, 25, 4, 4, 9, 4, 4, 9, 4, 4, 49, 9, 4, 4, 4, 4, 4, 9, 4, 121, 4, 25, 9, 4, 4, 9, 4, 4, 4, 49, 4, 25, 4, 9, 4, 4, 9, 4, 4, 169, 9, 4, 25, 4, 4, 4, 4, 9, 4, 4, 9, 4, 4, 9, 4, 4

Offset: 1

Robert Price, Mar 17 2017

nonn

			A013929(4) = 12, 12 = 2*2*3, so 12 is not squarefree, the square being 2*2 = 4.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Robert Price)

Cf. A013929, A046027.

s[n_] := If[(pos = Position[(f = FactorInteger[n])[[;; , 2]], ?(# >= 2 &)]) == {}, 1, f[[pos[[1, 1]], 1]]]; Select[Array[s, 250], # > 1 &]^2 (* _Amiram Eldar, Nov 14 2020 *)

f(n) = fordiv(n, d, if ((d>1) && issquare(d), return (d))); return (1);
apply(f, select(x->!issquarefree(x), [1..200])) \\ Michel Marcus, Nov 14 2020

A335234 Number of partitions of k_n into two parts (s,t) such that k_n | s*t, where k_n is the n-th nonsquarefree number (A013929).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 2, 3, 2, 1, 1, 1, 1, 1, 4, 1, 3, 2, 1, 2, 4, 1, 1, 1, 1, 2, 3, 1, 5, 1, 3, 2, 1, 1, 1, 5, 1, 2, 1, 4, 1, 1, 1, 1, 6, 3, 1, 2, 1, 1, 1, 2, 4, 1, 1, 6, 1, 1, 2, 2, 3, 1, 1, 1, 4, 7, 1, 5, 1, 1, 2, 1, 3, 1, 2, 7, 1, 1, 1, 1, 2, 5, 4

Offset: 1

Wesley Ivan Hurt, Jun 09 2020

nonn

a(n) >= 1.

			a(4) = 1; The 4th nonsquarefree number, A013929(4) = 12 has 6 partitions into two parts: (11,1), (10,2), (9,3), (8,4), (7,5) and (6,6) with corresponding products 11, 20, 27, 32, 35, 36. A013929(4) = 12 only divides the product 36, so a(4) = 1.
a(5) = 2; The 5th nonsquarefree number, A013929(5) = 16 has 8 partitions into two parts: (15,1), (14,2), (13,3), (12,4), (11,5), (10,6), (9,7) and (8,8) with corresponding products 15, 28, 39, 48, 55, 60, 63 and 64. A013929(5) = 16 divides two of these products, 48 and 64, so a(5) = 2.

Index entries for sequences related to partitions

Cf. A013929.

Table[If[Sum[(1 - Ceiling[(i*(n - i))/n] + Floor[(i*(n - i))/n]), {i, Floor[n/2]}] > 0, Sum[(1 - Ceiling[(i*(n - i))/n] + Floor[(i*(n - i))/n]), {i, Floor[n/2]}], {}], {n, 300}] // Flatten

A383263 Union of prime powers (A246655) and numbers that are not squarefree (A013929).

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 24, 25, 27, 28, 29, 31, 32, 36, 37, 40, 41, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 60, 61, 63, 64, 67, 68, 71, 72, 73, 75, 76, 79, 80, 81, 83, 84, 88, 89, 90, 92, 96, 97, 98, 99, 100, 101, 103

Offset: 1

Peter Luschny, Apr 27 2025

nonn

Union of A013929 and A000040. - Chai Wah Wu, Apr 27 2025

Cf. A000040, A013929, A246655.

Essentially the same as A363597.

with(NumberTheory):
IsPrimePower := n -> nops(PrimeFactors(n)) = 1:
IsA383263 := n -> IsPrimePower(n) or not IsSquareFree(n):
select(IsA383263, [seq(1..104)]);

Select[Range[120], Or[PrimePowerQ[#], ! SquareFreeQ[#]] &] (* Michael De Vlieger, Apr 27 2025 *)

isok(k) = isprimepower(k) || !issquarefree(k);

from math import isqrt
from sympy import mobius, primepi
def A383263(n):
    def f(x): return int(n+x+sum(mobius(k)*(x//k**2) for k in range(2, isqrt(x)+1))-primepi(x))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Apr 27 2025

def isA383263(n: int) -> bool: return is_prime_power(n) or not is_squarefree(n)

A062322 Factorials of nonsquarefree numbers, or A013929(n)!, (including 1).

Original entry on oeis.org

1, 24, 40320, 362880, 479001600, 20922789888000, 6402373705728000, 2432902008176640000, 620448401733239439360000, 15511210043330985984000000, 10888869450418352160768000000

Offset: 0

Jason Earls, Jul 05 2001

Harry J. Smith, Table of n, a(n) for n = 0..100

Cf. A013929.

Select[Range[0,30],!SquareFreeQ[#]&]! (* Harvey P. Dale, Aug 27 2016 *)

for(n=0,38, if(issquarefree(n), n+1,print(n!)))

{ n=-1; for (m=0, 10^9, if (m>0, f*=m, f=1); if (!issquarefree(m), write("b062322.txt", n++, " ", f); if (n==100, break)); ) } \\ Harry J. Smith, Aug 04 2009

Better name from Jon E. Schoenfield, Aug 09 2015

cp's OEIS Frontend

A212177 Number of exponents >= 2 in the canonical prime factorization of the n-th nonsquarefree number (A013929(n)).

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A192005 Number of non-cyclic abelian groups of finite order. The order is given by A013929.

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A353282 a(n) is the number of solutions (x,y) to the Diophantine equation S(x,y) = (x+y) + (x-y) + (x*y) + (x/y) = A013929(n) when x >= y > 1 and y | x.

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A362041 a(0) = 1; for n > 0, a(n) is the largest k < A013929(n) such that rad(k) = rad(A013929(n)), where rad(n) = A007947(n).

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A115228 Nonsquarefree numbers n such that 2n+1 is also nonsquarefree (A013929).

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A212174 Row n of table represents second signature of A013929(n): list of exponents >= 2 in canonical prime factorization of A013929(n), in nonincreasing order.

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A283919 The smallest square referenced in A013929 (Numbers that are not squarefree).

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