A007948 -id:A007948 - cp's OEIS frontend

A073183 Sum of divisors of n that are not greater than the cubefree kernel of n.

1, 3, 4, 7, 6, 12, 8, 7, 13, 18, 12, 28, 14, 24, 24, 7, 18, 39, 20, 42, 32, 36, 24, 36, 31, 42, 13, 56, 30, 72, 32, 7, 48, 54, 48, 91, 38, 60, 56, 50, 42, 96, 44, 84, 78, 72, 48, 36, 57, 93, 72, 98, 54, 39, 72, 64, 80, 90, 60, 168, 62, 96, 104, 7, 84, 144, 68, 126, 96, 144, 72

Offset: 1

Reinhard Zumkeller, Jul 19 2002

a(n) >= A073185(n).

			The cubefree kernel of 56 = 7 * 2^3 is 28 = 7 * 2^2 and the divisors <= 28 of 56 are {1, 2, 4, 7, 8, 14, 28}, therefore a(56) = 1 + 2 + 4 + 7 + 8 + 14 + 28 = 64.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000203, A007948, A073182, A073181.

sdcfk[n_]:=Module[{cf=Times@@Flatten[Table[#[[1]],#[[2]]]&/@({#[[1]],If[ #[[2]]>2,2,#[[2]]]}&/@FactorInteger[n])]},Total[Select[Divisors[n],#<= cf&]]]; Array[sdcfk,80] (* Harvey P. Dale, Jul 14 2018 *)

a007948(n) = my(f=factor(n)); for (i=1, #f~, f[i, 2] = min(f[i, 2], 2)); factorback(f);
a(n) = sumdiv(n, d, d*(d<=a007948(n))); \\ Michel Marcus, Feb 07 2015

A097380 Numbers m such that 1+CubeFreeKernel(m) is prime.

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 12, 16, 18, 22, 24, 28, 30, 32, 36, 42, 46, 48, 52, 54, 56, 58, 60, 64, 66, 70, 72, 78, 82, 96, 100, 102, 104, 106, 108, 112, 120, 126, 128, 130, 138, 144, 148, 150, 156, 162, 166, 172, 178, 180, 190, 192, 196, 198, 200, 208, 210, 216, 222, 224, 226

Offset: 1

Reinhard Zumkeller, Aug 11 2004

nonn

			m = 216 = (2*3)^3 -> A097377(216) = 1+(2*3)^2 = 37 = A000040(12), therefore 216 is a term.

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

Cf. A007948, A089189, A097377, A097379, A097381.

f[p_, e_] := p^Min[e, 2]; s[1] = 2; s[n_] := 1 + Times @@ f @@@ FactorInteger[n]; Select[Range[230], PrimeQ[s[#]] &] (* Amiram Eldar, Feb 01 2024 *)

is(n) = {my(f = factor(n)); isprime(1 + prod(i = 1, #f~, f[i, 1]^min(f[i, 2], 2)));} \\ Amiram Eldar, Feb 01 2024

A097377(a(n)) = A007948(a(n))+1 is prime.

A382902 The largest cubefree divisor of the n-th biquadratefree number.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Apr 08 2025

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

Cf. A046100, A007948, A013662.

Similar sequences: A382903, A382904, A382905, A382906.

f[p_, e_] := p^Min[e, 2]; s[n_] := Module[{fct = FactorInteger[n]}, If[AllTrue[fct[[;; , 2]], # < 4 &], Times @@ f @@@ fct, Nothing]]; Array[s, 100]

list(lim) = {my(f); print1(1, ", "); for(k = 2, lim, f = factor(k); if(vecmax(f[, 2]) < 4, print1(prod(i = 1, #f~, f[i, 1]^min(f[i, 2], 2)), ", ")));}

a(n) = A007948(A046100(n)).
a(n) = (A382903(n) * A046100(n)^2)^(1/3).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(4)^2 * Product_{p prime} (1 - 1/p^3 + 1/p^4 - 1/p^5) = 1.01974824991243823979... .

A097381 Numbers m such that 1+SquareFreeKernel(m)*CubeFreeKernel(m) is prime.

Original entry on oeis.org

1, 2, 6, 10, 12, 14, 18, 24, 26, 48, 54, 60, 66, 74, 84, 94, 96, 98, 110, 120, 130, 132, 134, 146, 162, 168, 170, 192, 204, 206, 210, 230, 234, 240, 264, 300, 314, 326, 336, 372, 384, 386, 406, 408, 430, 466, 470, 474, 480, 486, 490, 528, 570, 588, 600, 634, 646

Offset: 1

Reinhard Zumkeller, Aug 11 2004

nonn

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

Cf. A007947, A007948, A097379, A097380.

f[p_, e_] := p^(1 + Min[e, 2]); s[1] = 2; s[n_] := 1 + Times @@ f @@@ FactorInteger[n]; Select[Range[650], PrimeQ[s[#]] &] (* Amiram Eldar, Feb 01 2024 *)

is(n) = {my(f = factor(n)); isprime(1 + prod(i = 1, #f~, f[i, 1]^(1 + min(f[i, 2], 2))));} \\ Amiram Eldar, Feb 01 2024

A097378(a(n)) is prime.

A076998 Difference between cubefree and squarefree components of n.

Original entry on oeis.org

0, 0, 0, 2, 0, 0, 0, 2, 6, 0, 0, 6, 0, 0, 0, 2, 0, 12, 0, 10, 0, 0, 0, 6, 20, 0, 6, 14, 0, 0, 0, 2, 0, 0, 0, 30, 0, 0, 0, 10, 0, 0, 0, 22, 30, 0, 0, 6, 42, 40, 0, 26, 0, 12, 0, 14, 0, 0, 0, 30, 0, 0, 42, 2, 0, 0, 0, 34, 0, 0, 0, 30, 0, 0, 60, 38, 0, 0, 0, 10, 6, 0, 0, 42, 0, 0, 0, 22, 0, 60, 0, 46

Offset: 1

Jon Perry, Nov 28 2002

			a(4)=2 as cubefree(4)=4 and squarefree(4)=2. 4-2=2

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A005117, A007947, A007948.

Cf. A065463, A065465.

a[n_] := Module[{f = FactorInteger[n]}, Times @@ (First[#]^Min[Last[#], 2] & /@ f) - Times @@ (First[#] & /@ f)]; Array[a, 100] (* Amiram Eldar, Sep 24 2023 *)

rad(n)=local(p,i); p=factor(n)[,1]; prod(i=1,length(p),p[i])
rad2(n)=local(p,pn,i); p=factor(n)[,1]; pn=factor(n)[,2]; prod(i=1,length(p),p[i]^min(2,pn[i]))
for (k=1,100,print1(rad2(k)-rad(k)", "))

a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^min(f[i,2],2)) - vecprod(f[,1]);} \\ Amiram Eldar, Sep 24 2023

a(n) = A007948(n) - A007947(n). - Antti Karttunen, Jul 21 2018
From Amiram Eldar, Sep 24 2023: (Start)
a(n) >= 0, with equality if and only if n is squarefree (A005117).
Sum_{k=1..n} a(k) ~ (1/2) * c * n^2, where c = A065465 - A065463 = 0.17707163872600518419... . (End)

A097377 a(n) = CubeFreeKernel(n) + 1.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 5, 10, 11, 12, 13, 14, 15, 16, 5, 18, 19, 20, 21, 22, 23, 24, 13, 26, 27, 10, 29, 30, 31, 32, 5, 34, 35, 36, 37, 38, 39, 40, 21, 42, 43, 44, 45, 46, 47, 48, 13, 50, 51, 52, 53, 54, 19, 56, 29, 58, 59, 60, 61, 62, 63, 64, 5, 66, 67, 68, 69, 70, 71, 72, 37, 74, 75

Offset: 1

Reinhard Zumkeller, Aug 11 2004

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Cubefree.

Cf. A097380, A004709.

f[p_, e_] := p^Min[e, 2]; a[1] = 2; a[n_] := 1 + Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 01 2024 *)

a(n) = {my(f = factor(n)); 1 + prod(i = 1, #f~, f[i, 1]^min(f[i, 2], 2));} \\ Amiram Eldar, Feb 01 2024

a(n) = A007948(n) + 1.

Dirichlet g.f.: zeta(s) * (1 + Product_{p prime} (1 + 1/p^(s-1) - 1/p^s + 1/p^(2*s-2) - 1/p^(2*s-1)). - Amiram Eldar, Feb 01 2024

A097378 a(n) = SquareFreeKernel(n)*CubeFreeKernel(n) + 1.

Original entry on oeis.org

2, 5, 10, 9, 26, 37, 50, 9, 28, 101, 122, 73, 170, 197, 226, 9, 290, 109, 362, 201, 442, 485, 530, 73, 126, 677, 28, 393, 842, 901, 962, 9, 1090, 1157, 1226, 217, 1370, 1445, 1522, 201, 1682, 1765, 1850, 969, 676, 2117, 2210, 73, 344, 501, 2602, 1353, 2810

Offset: 1

Reinhard Zumkeller, Aug 11 2004

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Cubefree.
Eric Weisstein's World of Mathematics, Squarefree.

Cf. A002117, A004709, A005117, A007947, A007948, A097381.

f[p_, e_] := p^(1 + Min[e, 2]); a[1] = 2; a[n_] := 1 + Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 01 2024 *)

a(n) = {my(f = factor(n)); 1 + prod(i = 1, #f~, f[i, 1]^(1 + min(f[i, 2], 2)));} \\ Amiram Eldar, Feb 01 2024

a(n) = A007947(n)*A007948(n) + 1.
From Amiram Eldar, Feb 01 2024: (Start)
b(n) = a(n) - 1 is multiplicative with b(p^e) = p^(1 + min(e, 2)).
Dirichlet g.f.: zeta(s) * (1 + Product_{p prime} (1 + 1/p^(s-2) - 1/p^s + 1/p^(2*s-3) - 1/p^(2*s-2)).
Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = zeta(3) * Product_{p prime} (1 - 1/p^2 - 1/p^4 + 1/p^5) = 0.69256837284462414024... . (End)

A128972 n^3 - 1 divided by its largest cube divisor.

Original entry on oeis.org

7, 26, 63, 124, 215, 342, 511, 91, 37, 1330, 1727, 2196, 2743, 3374, 4095, 614, 17, 254, 7999, 9260, 10647, 12166, 13823, 1953, 17575, 19682, 813, 24388, 26999, 29790, 32767, 4492, 39303, 42874, 46655, 1876, 54871, 59318, 63999, 8615, 74087, 79506

Offset: 2

Jonathan Vos Post, Apr 28 2007

In other words, cubefree part of n^3-1, or cubefree kernel of n^3-1. Cube analog of A068310.

			a(9) = (9^3-1)/8 = (2^3 * 7 * 13)/(2^3) = 728/8 = 91.
a(10) = (10^3-1)/27 = (3^3 * 37)/(3^3) = 999/27 = 37.
a(18) = (18^3-1)/343 = (7^3 * 17)/(7^3) = 5831/343 = 17.

Robert Israel, Table of n, a(n) for n = 2..10000

Cf. A002350, A004709, A007948, A062378, A067872, A033314, A068310.

a:= n -> mul(f[1]^(f[2] mod 3), f = ifactors(n^3-1)[2]):
seq(a(n),n=2..100); # Robert Israel, Sep 24 2014

a(n) = A062378(A068601(n)) = A062378(n^3-1).

More terms from Carl R. White, Nov 09 2010

A343033 Array T(n, k), n, k > 0, read by antidiagonals; a variant of lunar multiplication (A087062) based on prime exponents of numbers (see Comments section for precise definition).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 2, 5, 2, 1, 1, 5, 3, 3, 5, 1, 1, 6, 7, 4, 7, 6, 1, 1, 7, 15, 5, 5, 15, 7, 1, 1, 2, 11, 6, 11, 6, 11, 2, 1, 1, 3, 3, 7, 35, 35, 7, 3, 3, 1, 1, 10, 5, 4, 13, 30, 13, 4, 5, 10, 1, 1, 11, 21, 9, 5, 77, 77, 5, 9, 21, 11, 1

Offset: 1

Rémy Sigrist and N. J. A. Sloane, Apr 03 2021

To compute T(n, k):
- write the prime exponents of n and of k on two lines, right aligned (these lines correspond to rows of A067255 in reversed order),
- to "multiply" two prime numbers: take the smallest,
- to "add" two prime numbers: take the largest,
- for example, for T(12, 14):
                (11 7 5 3 2)
    12 -->              1 2
    14 -->        x 1 0 0 1
                  ---------
                        1 1
                      0 0
                    0 0
                + 1 1
                -----------
                  1 1 0 1 1  --> 462 = T(12, 14)
This sequence is closely related to lunar multiplication (A087062):
- for any b > 1, let S_b be the set of nonnegative integers m such that A051903(m)< b,
- there is a natural bijection f from S_b to the set of nonnegative integers:
        f(Product_{k >= 0} prime(k)^d(k)) = Sum_{k >= 0} d(k) * b^k,
- let g be the inverse of f,
- then for any numbers n and k in S_b, we have:
        T(n, k) = g(f(n) "*" f(k)) where "*" denotes lunar product in base b,
- the corresponding addition table is A003990.

			Array T(n, k) begins:
  n\k|  1   2   3   4   5    6    7   8   9   10   11   12   13   14
  ----  -  --  --  --  --  ---  ---  --  --  ---  ---  ---  ---  ---
    1|  1   1   1   1   1    1    1   1   1    1    1    1    1    1
    2|  1   2   3   2   5    6    7   2   3   10   11    6   13   14  --> A007947
    3|  1   3   5   3   7   15   11   3   5   21   13   15   17   33  --> A328915
    4|  1   2   3   4   5    6    7   4   9   10   11   12   13   14  --> A007948
    5|  1   5   7   5  11   35   13   5   7   55   17   35   19   65
    6|  1   6  15   6  35   30   77   6  15  210  143   30  221  462
    7|  1   7  11   7  13   77   17   7  11   91   19   77   23  119
    8|  1   2   3   4   5    6    7   8   9   10   11   12   13   14
    9|  1   3   5   9   7   15   11   9  25   21   13   45   17   33
   10|  1  10  21  10  55  210   91  10  21  110  187  210  247  910
   11|  1  11  13  11  17  143   19  11  13  187   23  143   29  209
   12|  1   6  15  12  35   30   77  12  45  210  143   60  221  462
   13|  1  13  17  13  19  221   23  13  17  247   29  221   31  299
   14|  1  14  33  14  65  462  119  14  33  910  209  462  299  238

Cf. A003990, A007947, A007948, A051903, A067255, A087062, A328915, A342767, A343035.

T(n,k) = { my (r=1, pp=factor(n)[,1]~, qq=factor(k)[,1]~); for (i=1, #pp, for (j=1, #qq, my (p=prime(primepi(pp[i])+primepi(qq[j])-1), v=valuation(r, p), w=min(valuation(n, pp[i]), valuation(k, qq[j]))); if (w>v, r*=p^(w-v)))); r }

T(n, k) = T(k, n).
T(n, 1) = 1.
T(n, 2) = A007947(n).
T(n, 3) = A328915(n).
T(n, 4) = A007948(n).
T(n, n) = A343035(n).
A051903(T(n, k)) = min(A051903(n), A051903(k)).

A360906 Numbers with the same number of cubefree divisors and 3-full divisors.

Original entry on oeis.org

1, 16, 81, 384, 625, 640, 896, 1296, 1408, 1664, 2176, 2401, 2432, 2944, 3712, 3968, 4374, 4736, 5248, 5504, 6016, 6784, 7552, 7808, 8576, 9088, 9216, 9344, 10000, 10112, 10624, 10935, 11392, 12416, 12928, 13184, 13696, 13952, 14464, 14641, 15309, 16256, 16768

Offset: 1

Amiram Eldar, Feb 25 2023

nonn

Numbers k such that A073184(k) = A190867(k).
Numbers whose largest cubefree divisor (A007948) and cubefull part (A360540) have the same number of divisors (A000005).
If k and m are coprime terms, then k*m is also a term.
The characteristic function of this sequence depends only on prime signature.
1 is the only cubefree (A004709) term.
Includes the 4th powers of squarefree numbers (1 and A113849).
The 4th powers of primes (A030514) are the only terms that are prime powers (A246655).
Numbers of the for m*p^(3*2^k+1), where m is squarefree, p is prime, gcd(m, p) = 1 and omega(m) = k, are all terms. In particular, this sequence includes numbers of the form p^7*q, where p != q are primes (A179664), and numbers of the form p^13*q*r where p, q, and r are distinct primes.
The corresponding numbers of cubefree (or 3-full) divisors are 1, 3, 3, 6, 3, 6, 6, 9, 6, 6, 6, 3, 6, 6, ... .

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

Cf. A000005, A004709, A007948, A073184, A190867, A246655, A360540, A360902.

Subsequences: A030514, A113849, A179664, A360907.

q[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, Times @@ (Min[#, 3] & /@ (e + 1)) == Times @@ (Max[#, 1] & /@ (e - 1))]; q[1] = True; Select[Range[10^4], q]

is(k) = {my(e = factor(k)[,2]); prod(i = 1, #e, min(e[i] + 1, 3)) == prod(i = 1, #e, max(e[i] - 1, 1)); }

cp's OEIS Frontend

A073183 Sum of divisors of n that are not greater than the cubefree kernel of n.

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A097380 Numbers m such that 1+CubeFreeKernel(m) is prime.

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A382902 The largest cubefree divisor of the n-th biquadratefree number.

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A097381 Numbers m such that 1+SquareFreeKernel(m)*CubeFreeKernel(m) is prime.

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A076998 Difference between cubefree and squarefree components of n.

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A097377 a(n) = CubeFreeKernel(n) + 1.

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A097378 a(n) = SquareFreeKernel(n)*CubeFreeKernel(n) + 1.

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A128972 n^3 - 1 divided by its largest cube divisor.

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