A062503 -id:A062503 - cp's OEIS frontend

A320966 Powerful numbers A001694 divisible by a cube > 1.

8, 16, 27, 32, 64, 72, 81, 108, 125, 128, 144, 200, 216, 243, 256, 288, 324, 343, 392, 400, 432, 500, 512, 576, 625, 648, 675, 729, 784, 800, 864, 968, 972, 1000, 1024, 1125, 1152, 1296, 1323, 1331, 1352, 1372, 1568, 1600, 1728, 1800, 1936, 1944, 2000, 2025, 2048, 2187, 2197, 2304, 2312, 2401, 2500

Offset: 1

Hugo Pfoertner, Oct 25 2018

nonn

Powerful numbers that are not squares of squarefree numbers. - Amiram Eldar, Jun 25 2022

Hugo Pfoertner, Table of n, a(n) for n = 1..10000

Cf. A000578, A320965.
Intersection of A001694 and A046099.
A001694 \ A062503.

Select[Range[2500], (m = MinMax[FactorInteger[#][[;; , 2]]])[[1]] > 1 && m[[2]] > 2 &] (* Amiram Eldar, Jun 25 2022 *)

isA001694(n)=n=factor(n)[, 2]; for(i=1, #n, if(n[i]==1, return(0))); 1 \\ from Charles R Greathouse IV
isA046099(n)=n=factor(n)[, 2]; for(i=1, #n, if(n[i]>2, return(1)));0
for (k=1,2500,if(isA001694(k)&&isA046099(k),print1(k,", ")))

from math import isqrt
from sympy import mobius, integer_nthroot
def A320966(n):
    def squarefreepi(n): return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
    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
    def f(x):
        c, l = n+x+squarefreepi(isqrt(x))-squarefreepi(integer_nthroot(x,3)[0]), 0
        j = isqrt(x)
        while j>1:
            k2 = integer_nthroot(x//j**2,3)[0]+1
            w = squarefreepi(k2-1)
            c -= j*(w-l)
            l, j = w, isqrt(x//k2**3)
        return c+l
    return bisection(f,n,n) # Chai Wah Wu, Sep 15 2024

Sum_{n>=1} 1/a(n) = zeta(2)*zeta(3)/zeta(6) - 15/Pi^2 = 0.4237786821... . - Amiram Eldar, Jun 25 2022

A338325 Biquadratefree powerful numbers: numbers whose exponents in their prime factorization are either 2 or 3.

Original entry on oeis.org

1, 4, 8, 9, 25, 27, 36, 49, 72, 100, 108, 121, 125, 169, 196, 200, 216, 225, 289, 343, 361, 392, 441, 484, 500, 529, 675, 676, 841, 900, 961, 968, 1000, 1089, 1125, 1156, 1225, 1323, 1331, 1352, 1369, 1372, 1444, 1521, 1681, 1764, 1800, 1849, 2116, 2197, 2209

Offset: 1

Amiram Eldar, Oct 22 2020

nonn

Equivalently, numbers k such that if a prime p divides k then p^2 divides k but p^4 does not divide k.
Each term has a unique representation as a^2 * b^3, where a and b are coprime squarefree numbers.
Dehkordi (1998) refers to these numbers as "2-full and 4-free numbers".

			4 = 2^2 is a term since the exponent of its only prime factor is 2.
72 = 2^3 * 3^2 is a terms since the exponents of the primes in its prime factorization are 2 and 3.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Massoud H. Dehkordi, Asymptotic formulae for some arithmetic functions in number theory, Ph.D. thesis, Loughborough University, 1998.
Eric Weisstein's World of Mathematics, Biquadratefree.
Eric Weisstein's World of Mathematics, Powerful Number.

Intersection of A001694 and A046100.
Subsequences: A062503, A062838.
Cf. A005117, A090699, A244000, A330595.

Select[Range[2500], # == 1 || AllTrue[FactorInteger[#][[;; , 2]], MemberQ[{2, 3}, #1] &] &]

The number of terms not exceeding x is asymptotically (zeta(3/2)/zeta(3)) * J_2(1/2) * x^(1/2) + (zeta(2/3)/zeta(2)) * J_2(1/3) * x^(1/3), where J_2(s) = Product_{p prime} (1 - p^(-4*s) - p^(-5*s) - p^(-6*s) + p^(-7*s) + p^(-8*s)) (Dehkordi, 1998).

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + 1/p^2 + 1/p^3) = 1.748932... (A330595).

A366243 Numbers that are products of "Fermi-Dirac primes" (A050376) that are powers of primes with exponents that are not powers of 4.

Original entry on oeis.org

1, 4, 9, 25, 36, 49, 100, 121, 169, 196, 225, 256, 289, 361, 441, 484, 529, 676, 841, 900, 961, 1024, 1089, 1156, 1225, 1369, 1444, 1521, 1681, 1764, 1849, 2116, 2209, 2304, 2601, 2809, 3025, 3249, 3364, 3481, 3721, 3844, 4225, 4356, 4489, 4761, 4900, 5041, 5329

Offset: 1

Amiram Eldar, Oct 05 2023

Equivalently, numbers that are products of "Fermi-Dirac primes" that are powers of primes with exponents that are powers of 2 with odd exponents.
Products of distinct numbers of the form p^(2^(2*k-1)), where p is prime and k >= 1.
Numbers whose prime factorization has exponents that are positive terms of A062880.
Every integer k has a unique representation as a product of 2 numbers: one is in this sequence and the other is in A366242: k = A366245(k) * A366244(k).

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

Cf. A062880, A050376, A366242, A366244, A366245.

A062503 is a subsequence.

mdQ[n_] := AllTrue[IntegerDigits[n, 4], # < 2 &]; q[e_] := EvenQ[e] && mdQ[e/2]; Select[Range[6000], # == 1 || AllTrue[FactorInteger[#][[;; , 2]], q] &]

ismd(n) = {my(d = digits(n, 4)); for(i = 1, #d, if(d[i] > 1, return(0))); 1;}
is(n) = {my(e = factor(n)[ ,2]); for(i = 1, #e, if(e[i]%2 || !ismd(e[i]/2), return(0))); 1;}

Sum_{n>=1} 1/a(n) = Product_{k>=0} zeta(2^(2*k+1))/zeta(2^(2*k+2)) = 1.52599127273749217982... (this is the constant c in A366242).

A328400 Smallest number with the same set of distinct prime exponents as n.

Original entry on oeis.org

1, 2, 2, 4, 2, 2, 2, 8, 4, 2, 2, 12, 2, 2, 2, 16, 2, 12, 2, 12, 2, 2, 2, 24, 4, 2, 8, 12, 2, 2, 2, 32, 2, 2, 2, 4, 2, 2, 2, 24, 2, 2, 2, 12, 12, 2, 2, 48, 4, 12, 2, 12, 2, 24, 2, 24, 2, 2, 2, 12, 2, 2, 12, 64, 2, 2, 2, 12, 2, 2, 2, 72, 2, 2, 12, 12, 2, 2, 2, 48, 16, 2, 2, 12, 2, 2, 2, 24, 2, 12, 2, 12, 2, 2, 2, 96, 2, 12, 12, 4, 2, 2, 2, 24, 2

Offset: 1

Antti Karttunen, Oct 15 2019

nonn

A variant of A046523 which gives the smallest number with the same prime signature as n. However, in this sequence, if any prime exponent occurs multiple times in n, the extra occurrences are removed and the signature is that of one of the numbers where only distinct values of prime exponents occur (A130091).

			90 = 2^1 * 3^2 * 5^1 has prime signature (1,1,2). The smallest number with prime signature (1,2) is 12 = 2^2 * 3, thus a(90) = 12.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from exponents in factorization of n

Cf. A007947, A046523, A181819, A181821, A328401 (rgs-transform).

Cf. A005117 (gives indices of terms <= 2), A062503 (after its initial 1, gives indices of 4's in this sequence).

Array[Times @@ MapIndexed[Prime[#2[[1]]]^#1 &, Reverse[Flatten[Cases[FactorInteger[#], {p_, k_} :> Table[PrimePi[p], {k}]]]]] &[Times @@ FactorInteger[#][[All, 1]]] &@ If[# == 1, 1, Times @@ Prime@ FactorInteger[#][[All, -1]]] &, 105] (* Michael De Vlieger, Oct 17 2019, after Gus Wiseman at A181821 *)

A007947(n) = factorback(factorint(n)[, 1]);
A181819(n) = factorback(apply(e->prime(e),(factor(n)[,2])));
A181821(n) = { my(f=factor(n),p=0,m=1); forstep(i=#f~,1,-1,while(f[i,2], f[i,2]--; m *= (p=nextprime(p+1))^primepi(f[i,1]))); (m); };
A328400(n) = A181821(A007947(A181819(n)));

a(n) = A181821(A007947(A181819(n))).

For all n, a(n) = a(A046523(n)).

A369938 Numbers whose maximal exponent in their prime factorization is a power of 2.

Original entry on oeis.org

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, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77

Offset: 1

Amiram Eldar, Feb 06 2024

First differs from its subsequence A138302 \ {1} at n = 378: a(378) = 432 = 2^4 * 3^3 is not a term of A138302.
First differs from A096432, A220218 \ {1}, A335275 \ {1} and A337052 \ {1} at n = 56, and from A270428 \ {1} at n = 113.
Numbers k such that A051903(k) is a power of 2.
The asymptotic density of this sequence is 1/zeta(3) + Sum_{k>=2} (1/zeta(2^k+1) - 1/zeta(2^k)) = 0.87442038669659566330... .

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

Cf. A000079, A002117, A051903.
Subsequences: A001146, A001248, A005117, A030514, A030635, A050376, A062503, A113849, A138302 \ {1}, A179645.
Similar sequences: A368714, A369937, A369939.
Cf. A096432, A220218, A270428, A335275, A337052.

pow2Q[n_] := n == 2^IntegerExponent[n, 2];
Select[Range[2, 100], pow2Q[Max[FactorInteger[#][[;; , 2]]]] &]
Select[Range[2,80],IntegerQ[Log2[Max[FactorInteger[#][[;;,2]]]]]&] (* Harvey P. Dale, Nov 06 2024 *)

ispow2(n) = n >> valuation(n, 2) == 1;
is(n) = n > 1 && ispow2(vecmax(factor(n)[, 2]));

A369939 Numbers whose maximal exponent in their prime factorization is a Fibonacci number.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71

Offset: 1

Amiram Eldar, Feb 06 2024

First differs from its subsequence A115063 at n = 2448. a(2448) = 2592 = 2^5 * 3^4 is not a term of A115063.
First differs from A209061 at n = 62.
Numbers k such that A051903(k) is a Fibonacci number.
The asymptotic density of this sequence is 1/zeta(4) + Sum_{k>=5} (1/zeta(Fibonacci(k)+1) - 1/zeta(Fibonacci(k))) = 0.94462177878047854647... .

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

Cf. A000045, A013662, A051903, A209061.
Subsequences: A005117, A062503, A062838, A113850, A115063.
Similar sequences: A368714, A369937, A369938.

fibQ[n_] := Or @@ IntegerQ /@ Sqrt[5*n^2 + {-4, 4}];
Select[Range[100], fibQ[Max[FactorInteger[#][[;; , 2]]]] &]

isfib(n) = issquare(5*n^2 - 4) || issquare(5*n^2 + 4);
is(n) = n == 1 || isfib(vecmax(factor(n)[, 2]));

A352780 Square array A(n,k), n >= 1, k >= 0, read by descending antidiagonals, such that the row product is n and column k contains only (2^k)-th powers of squarefree numbers.

Original entry on oeis.org

1, 1, 2, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 4, 5, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 9, 10, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 14

Offset: 1

Antti Karttunen and Peter Munn, Apr 02 2022

This is well-defined because positive integers have a unique factorization into powers of nonunit squarefree numbers with distinct exponents that are powers of 2.
Each (infinite) row is the lexicographically earliest with product n and terms that are a (2^k)-th power for all k.
For all k, column k is column k+1 of A060176 conjugated by A225546.

			The top left corner of the array:
  n/k |   0   1   2   3   4   5   6
------+------------------------------
    1 |   1,  1,  1,  1,  1,  1,  1,
    2 |   2,  1,  1,  1,  1,  1,  1,
    3 |   3,  1,  1,  1,  1,  1,  1,
    4 |   1,  4,  1,  1,  1,  1,  1,
    5 |   5,  1,  1,  1,  1,  1,  1,
    6 |   6,  1,  1,  1,  1,  1,  1,
    7 |   7,  1,  1,  1,  1,  1,  1,
    8 |   2,  4,  1,  1,  1,  1,  1,
    9 |   1,  9,  1,  1,  1,  1,  1,
   10 |  10,  1,  1,  1,  1,  1,  1,
   11 |  11,  1,  1,  1,  1,  1,  1,
   12 |   3,  4,  1,  1,  1,  1,  1,
   13 |  13,  1,  1,  1,  1,  1,  1,
   14 |  14,  1,  1,  1,  1,  1,  1,
   15 |  15,  1,  1,  1,  1,  1,  1,
   16 |   1,  1, 16,  1,  1,  1,  1,
   17 |  17,  1,  1,  1,  1,  1,  1,
   18 |   2,  9,  1,  1,  1,  1,  1,
   19 |  19,  1,  1,  1,  1,  1,  1,
   20 |   5,  4,  1,  1,  1,  1,  1,

Antti Karttunen, Table of n, a(n) for n = 1..33153; the first 257 antidiagonals

Sequences used in a formula defining this sequence: A000188, A007913, A060176, A225546.
Cf. A007913 (column 0), A335324 (column 1).
Range of values: {1} U A340682 (whole table), A005117 (column 0), A062503 (column 1), {1} U A113849 (column 2).
Row numbers of rows:
- with a 1 in column 0: A000290\{0};
- with a 1 in column 1: A252895;
- with a 1 in column 0, but not in column 1: A030140;
- where every 1 is followed by another 1: A337533;
- with 1's in all even columns: A366243;
- with 1's in all odd columns: A366242;
- where every term has an even number of distinct prime factors: A268390;
- where every term is a power of a prime: A268375;
- where the terms are pairwise coprime: A138302;
- where the last nonunit term is coprime to the earlier terms: A369938;
- where the last nonunit term is a power of 2: A335738.
Number of nonunit terms in row n is A331591(n); their positions are given (in reversed binary) by A267116(n); the first nonunit is in column A352080(n)-1 and the infinite run of 1's starts in column A299090(n).

up_to = 105;
A352780sq(n, k) = if(k==0, core(n), A352780sq(core(n, 1)[2], k-1)^2);
A352780list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, forstep(col=a-1,0,-1, i++; if(i > up_to, return(v)); v[i] = A352780sq(a-col,col))); (v); };
v352780 = A352780list(up_to);
A352780(n) = v352780[n];

A(n,0) = A007913(n); for k > 0, A(n,k) = A(A000188(n), k-1)^2.
A(n,k) = A225546(A060176(A225546(n), k+1)).
A331591(A(n,k)) <= 1.

A030140 The nonsquares squared.

Original entry on oeis.org

4, 9, 25, 36, 49, 64, 100, 121, 144, 169, 196, 225, 289, 324, 361, 400, 441, 484, 529, 576, 676, 729, 784, 841, 900, 961, 1024, 1089, 1156, 1225, 1369, 1444, 1521, 1600, 1681, 1764, 1849, 1936, 2025, 2116, 2209, 2304, 2500, 2601, 2704, 2809, 2916, 3025

Offset: 1

N. J. A. Sloane

The complement of the fourth powers A000583 within the squares A000290. - Peter Munn, Aug 20 2019

			a(1)=2^2, a(2)=3^2, a(3)=5^2, a(4)=6^2, a(5)=7^2, ..., a(n)=(integer which is not a perfect square)^2.

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

Cf. A000037, A000290, A000583, A013661, A013662, A062503.
Positions of 2's in A352080.
Related to A016945 via A225546.

[(n + Floor(1/2 + Sqrt(n)))^2: n in [1..60]]; // Vincenzo Librandi, Apr 06 2020

a:=proc(n) if type(sqrt(n),integer)=false then n^2 else fi end: seq(a(n),n=1..70); # Emeric Deutsch, Apr 11 2007

a[n_] := (n + Floor[1/2 + Sqrt[n]])^2;
Array[a, 50] (* Jean-François Alcover, Apr 05 2020 *)

from math import isqrt
def A030140(n): return (n+(k:=isqrt(n))+int(n>=k*(k+1)+1))**2 # Chai Wah Wu, Jun 17 2024

a(n) = A000037(n)^2.
Sum_{n>=1} 1/a(n) = zeta(2) - zeta(4) = A013661 - A013662 = 0.5626108331... - Amiram Eldar, Nov 14 2020
{a(n) : n >= 1} = {A225546(6m+3) : m >= 0}. - Peter Munn, Nov 17 2022

Edited by N. J. A. Sloane, Jul 02 2008 at the suggestion of R. J. Mathar

A046685 Numbers k such that the sum of cubes of divisors of k and the sum of 4th powers of divisors of k are relatively prime.

Original entry on oeis.org

1, 2, 4, 8, 9, 18, 25, 100, 121, 225, 289, 484, 529, 841, 1089, 1156, 1681, 2116, 2209, 2601, 2809, 3364, 3481, 4761, 5041, 6724, 6889, 7225, 7569, 7921, 8836, 10201, 11236, 11449, 12769, 13225, 13924, 15129, 17161, 18769, 19881, 20164, 21025

Offset: 1

Labos Elemer

nonn

It can be shown that this is a subsequence of A028982.
From Robert Israel, Jul 09 2018: (Start)
The only terms that are not in A062503 are 2, 8 and 18.
No term is divisible by a term of A002476.
p^2 is a term for every p in A003627. (End)

Robert Israel, Table of n, a(n) for n = 1..10000
Mathematics StackExchange, Are 1+p^3+p^6 and 1+p^4+p^8 coprime?

Cf. A002476, A003627, A028982, A046678, A046679, A046680, A046681, A046683, A062503.

N:= 10^6: # to get all terms <= N
sort(select(filter, [seq(t^2,t=1..isqrt(N)),seq(2*t^2,t=1..isqrt(N/2))])); # Robert Israel, Jul 09 2018

Select[Range[25000], CoprimeQ[DivisorSigma[3, #], DivisorSigma[4, #]] &] (* Michael De Vlieger, Aug 10 2023 *)

isok(n) = gcd(sigma(n, 3), sigma(n, 4)) == 1; \\ Michel Marcus, Sep 24 2019

A322527 Number of integer partitions of n whose product of parts is a power of a squarefree number (A072774).

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 13, 18, 21, 31, 34, 45, 51, 63, 72, 88, 97, 120, 128, 158, 174, 201, 222, 264, 287, 333, 359, 416, 441, 518, 557, 631, 684, 770, 833, 954, 1017, 1141, 1222, 1378, 1475, 1643, 1755, 1939, 2097, 2327, 2471, 2758, 2928, 3233, 3470, 3813, 4085

Offset: 0

Gus Wiseman, Dec 14 2018

nonn

			The a(1) = 1 through a(8) = 18 integer partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (52)       (44)
             (111)  (31)    (41)     (42)      (61)       (53)
                    (211)   (221)    (51)      (331)      (71)
                    (1111)  (311)    (222)     (421)      (422)
                            (2111)   (321)     (511)      (521)
                            (11111)  (411)     (2221)     (611)
                                     (2211)    (3211)     (2222)
                                     (3111)    (4111)     (3311)
                                     (21111)   (22111)    (4211)
                                     (111111)  (31111)    (5111)
                                               (211111)   (22211)
                                               (1111111)  (32111)
                                                          (41111)
                                                          (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)
Missing from the list for n = 7 through 9:
  (43)   (62)    (54)
  (322)  (332)   (63)
         (431)   (432)
         (3221)  (522)
                 (621)
                 (3222)
                 (3321)
                 (4311)
                 (32211)

Cf. A003963, A005117, A038041, A062503, A064573, A072774, A295193, A302505, A306021, A319169, A320322, A322526, A322528, A322530.

Table[Length[Select[IntegerPartitions[n],SameQ@@Last/@FactorInteger[Times@@#]&]],{n,30}]

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A320966 Powerful numbers A001694 divisible by a cube > 1.

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A338325 Biquadratefree powerful numbers: numbers whose exponents in their prime factorization are either 2 or 3.

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A366243 Numbers that are products of "Fermi-Dirac primes" (A050376) that are powers of primes with exponents that are not powers of 4.

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A328400 Smallest number with the same set of distinct prime exponents as n.

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A369938 Numbers whose maximal exponent in their prime factorization is a power of 2.

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A369939 Numbers whose maximal exponent in their prime factorization is a Fibonacci number.

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A352780 Square array A(n,k), n >= 1, k >= 0, read by descending antidiagonals, such that the row product is n and column k contains only (2^k)-th powers of squarefree numbers.

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A030140 The nonsquares squared.

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