A066503 -id:A066503 - cp's OEIS frontend

A073353 Sum of n and its squarefree kernel.

2, 4, 6, 6, 10, 12, 14, 10, 12, 20, 22, 18, 26, 28, 30, 18, 34, 24, 38, 30, 42, 44, 46, 30, 30, 52, 30, 42, 58, 60, 62, 34, 66, 68, 70, 42, 74, 76, 78, 50, 82, 84, 86, 66, 60, 92, 94, 54, 56, 60, 102, 78, 106, 60, 110, 70, 114, 116, 118, 90, 122, 124, 84, 66, 130, 132

Offset: 1

Reinhard Zumkeller, Jul 29 2002

a(n) is even; a(n)=2*n iff n is squarefree.
Least k >n such that n divides k^n. - Benoit Cloitre, Oct 09 2002
a(n) is the smallest integer > n such that the positive integers coprime to a(n) are also coprime to n. - Leroy Quet, Dec 24 2006

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A007947, A005117, A065463, A066503, A064549, A003557, A073354, A078310.

a073353 n = n + a007947 n  -- Reinhard Zumkeller, Jul 23 2013

a[n_] := n + Times @@ FactorInteger[n][[;;, 1]]; Array[a, 100] (* Amiram Eldar, Dec 07 2023 *)

a(n)=vecprod(factor(n)[,1])+n \\ Charles R Greathouse IV, Nov 05 2017

a(n) = n + A007947(n).

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = 1 + A065463 = 1.704442... . - Amiram Eldar, Dec 07 2023

A326128 a(n) = n - A007913(n), where A007913 gives the squarefree part of n.

Original entry on oeis.org

0, 0, 0, 3, 0, 0, 0, 6, 8, 0, 0, 9, 0, 0, 0, 15, 0, 16, 0, 15, 0, 0, 0, 18, 24, 0, 24, 21, 0, 0, 0, 30, 0, 0, 0, 35, 0, 0, 0, 30, 0, 0, 0, 33, 40, 0, 0, 45, 48, 48, 0, 39, 0, 48, 0, 42, 0, 0, 0, 45, 0, 0, 56, 63, 0, 0, 0, 51, 0, 0, 0, 70, 0, 0, 72, 57, 0, 0, 0, 75, 80, 0, 0, 63, 0, 0, 0, 66, 0, 80, 0, 69, 0, 0, 0, 90, 0, 96, 88, 99, 0, 0, 0, 78, 0

Offset: 1

Antti Karttunen, Jun 09 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537.

Cf. A007913, A033879, A326126, A326127, A326129, A336642.

Cf. also A066503.

f[p_, e_] := p^Mod[e, 2]; a[n_] := n - Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Mar 21 2024 *)

PARI
```
A326128(n) = (n-core(n));
```

a(n) = n - A007913(n).
a(n) = A326127(n) + A033879(n).
a(n) >= A066503(n).
a(n) = A007913(n) * A336642(n). - Antti Karttunen, Jul 28 2020
Sum_{k=1..n} a(k) ~ c * n^2, where c = 1/2 - Pi^2/30 = 0.171013... . - Amiram Eldar, Mar 21 2024

A380987 Position of first appearance of n in A290106 (product of prime indices divided by product of distinct prime indices).

Original entry on oeis.org

1, 9, 25, 27, 121, 169, 289, 81, 125, 841, 961, 675, 1681, 1849, 2209, 243, 3481, 1125, 4489, 3267, 5329, 6241, 6889, 2025, 1331, 10201, 625, 7803, 11881, 12769, 16129, 729, 18769, 19321, 22201, 2197, 24649, 26569, 27889, 9801, 32041, 32761, 36481, 25947

Offset: 1

Gus Wiseman, Feb 14 2025

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

All terms are odd.

			The first position of 12 in A290106 is 675, with prime indices {2,2,2,3,3}, so a(12) = 675.
The terms together with their prime indices begin:
      1: {}
      9: {2,2}
     25: {3,3}
     27: {2,2,2}
    121: {5,5}
    169: {6,6}
    289: {7,7}
     81: {2,2,2,2}
    125: {3,3,3}
    841: {10,10}
    961: {11,11}
    675: {2,2,2,3,3}
   1681: {13,13}
   1849: {14,14}
   2209: {15,15}
    243: {2,2,2,2,2}
   3481: {17,17}
   1125: {2,2,3,3,3}

For factors instead of indices we have A064549 (sorted A001694), firsts of A003557.
The additive version for factors is A280286 (sorted A381075), firsts of A280292.
Position of first appearance of n in A290106.
The additive version is A380956 (sorted A380957), firsts of A380955.
For difference instead of quotient see A380986.
The sorted version is A380988.
A000040 lists the primes, differences A001223.
A003963 gives product of prime indices, distinct A156061.
A005117 lists squarefree numbers, complement A013929.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, length A001222.
A304038 lists distinct prime indices, sum A066328, length A001221.
Cf. A000720, A007947, A046660, A066503, A071625, A136565, A178503, A175508, A325034, A379681.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
q=Table[Times@@prix[n]/Times@@Union[prix[n]],{n,10000}];
Table[Position[q,k][[1,1]],{k,mnrm[q]}]

A380988 Sorted positions of first appearances in A290106 (product of prime indices divided by product of distinct prime indices).

Original entry on oeis.org

1, 9, 25, 27, 81, 121, 125, 169, 243, 289, 625, 675, 729, 841, 961, 1125, 1331, 1681, 1849, 2025, 2187, 2197, 2209, 3125, 3267, 3481, 4489, 4913, 5329, 5625, 6075, 6241, 6561, 6889, 7803, 9801, 10125, 10201, 11881, 11979, 12769, 14641, 15125, 15625, 16129

Offset: 1

Gus Wiseman, Feb 18 2025

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

All terms are odd.

			The prime indices of 225 are {2,2,3,3}, with image A290106(225) = 6. The prime indices of 169 are {6,6}, also with image 6. Since the latter is the first with image 6, 169 is in the sequence, and 225 is not.
The terms together with their prime indices begin:
     1: {}
     9: {2,2}
    25: {3,3}
    27: {2,2,2}
    81: {2,2,2,2}
   121: {5,5}
   125: {3,3,3}
   169: {6,6}
   243: {2,2,2,2,2}
   289: {7,7}
   625: {3,3,3,3}
   675: {2,2,2,3,3}
   729: {2,2,2,2,2,2}
   841: {10,10}
   961: {11,11}
  1125: {2,2,3,3,3}
  1331: {5,5,5}
  1681: {13,13}
  1849: {14,14}
  2025: {2,2,2,2,3,3}

For factors instead of indices we have A001694 (unsorted A064549), firsts of A003557.
Sorted firsts of A290106.
The additive version is A380957 (sorted A380956), firsts of A380955.
For difference instead of quotient see A380986.
The unsorted version is A380987.
The additive version for factors is A381075 (unsorted A280286), firsts of A280292.
A000040 lists the primes, differences A001223.
A003963 gives product of prime indices, distinct A156061.
A005117 lists squarefree numbers, complement A013929.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, length A001222.
A304038 lists distinct prime indices, sum A066328, length A001221.
Cf. A000720, A007947, A046660, A066503, A071625, A136565, A178503, A175508, A325034, A379681.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
q=Table[Times@@prix[n]/Times@@Union[prix[n]],{n,1000}];
Select[Range[Length[q]],FreeQ[Take[q,#-1],q[[#]]]&]

A284318 Triangle read by rows in which row n lists divisors d of n such that n divides d^n.

Original entry on oeis.org

1, 2, 3, 2, 4, 5, 6, 7, 2, 4, 8, 3, 9, 10, 11, 6, 12, 13, 14, 15, 2, 4, 8, 16, 17, 6, 18, 19, 10, 20, 21, 22, 23, 6, 12, 24, 5, 25, 26, 3, 9, 27, 14, 28, 29, 30, 31, 2, 4, 8, 16, 32, 33, 34, 35, 6, 12, 18, 36, 37, 38, 39, 10, 20, 40, 41, 42, 43, 22, 44, 15, 45, 46, 47, 6, 12, 24, 48, 7, 49, 10, 50

Offset: 1

Juri-Stepan Gerasimov, Mar 25 2017

Row n lists divisors of n that are divisible by A007947(n). - Robert Israel, Apr 27 2017

			Triangle begins:
    1;
    2;
    3;
    2, 4;
    5;
    6;
    7;
    2, 4, 8;
    3, 9;
    10;
    11;
    6, 12;
    13;
    14;
    15;
    2, 4, 8, 16.

Robert Israel, Table of n, a(n) for n = 1..10002 (rows 1 to 5250, flattened)

Cf. A000961 (1 together with k such that k divides p^k for some prime divisor p of k), A005361 (row length), A007774 (m such that m divides s^m for some semiprime divisor s of m), A007947 (smallest u such that u^n|n and n|u, or divisor k such that A000005(k) = 2^A001221(n)), A057723 (row sums), A066503 (difference between largest x and smallest y such that x^n|n, n|x, y^n|n and n|y).

Cf. A003557, A027750.

[[u: u in [1..n] | Denominator(n/u) eq 1 and Denominator(u^n/n) eq 1]: n in [1..50]];

f:= proc(n) local r;
    r:= convert(numtheory:-factorset(n),`*`);
    op(sort(convert(map(`*`, numtheory:-divisors(n/r),r),list)))
end proc:
map(f, [$1..100]); # Robert Israel, Apr 27 2017

Flatten[Table[Select[Range[n], Divisible[n, #] && Divisible[#^n, n] &], {n, 50}]] (* Indranil Ghosh, Mar 25 2017 *)

for(n=1, 50, for(i=1, n, if(n%i==0 & (i^n)%n==0, print1(i,", "););); print();); \\ Indranil Ghosh, Mar 25 2017

for n in range(1, 51):
    print([i for i in range(1, n + 1) if n%i==0 and (i**n)%n==0]) # Indranil Ghosh, Mar 25 2017

T(n,k) = A007947(n) * A027750(A003557(n), k). - Robert Israel, Apr 27 2017

A326142 Sum of all other divisors of n except its largest squarefree divisor: a(n) = sigma(n) - A007947(n).

Original entry on oeis.org

0, 1, 1, 5, 1, 6, 1, 13, 10, 8, 1, 22, 1, 10, 9, 29, 1, 33, 1, 32, 11, 14, 1, 54, 26, 16, 37, 42, 1, 42, 1, 61, 15, 20, 13, 85, 1, 22, 17, 80, 1, 54, 1, 62, 63, 26, 1, 118, 50, 83, 21, 72, 1, 114, 17, 106, 23, 32, 1, 138, 1, 34, 83, 125, 19, 78, 1, 92, 27, 74, 1, 189, 1, 40, 109, 102, 19, 90, 1, 176, 118, 44, 1, 182, 23, 46, 33

Offset: 1

Antti Karttunen, Jun 09 2019

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A000203, A007947, A066503, A326143.
Cf. also A326053, A326061, A326126.
Cf. A013661, A065463.

rad[n_] := Times @@ FactorInteger[n][[;; , 1]]; a[n_] := DivisorSigma[1, n] - rad[n]; Array[a, 100] (* Amiram Eldar, Dec 05 2023 *)

A007947(n) = factorback(factorint(n)[, 1]);
A326142(n) = (sigma(n)-A007947(n));

a(n) = A000203(n) - A007947(n).
a(n) = n + A326143(n).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = A013661 - A065463 = 0.940491... . - Amiram Eldar, Dec 05 2023

A336551 a(n) = A003557(n) - 1.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 3, 2, 0, 0, 1, 0, 0, 0, 7, 0, 2, 0, 1, 0, 0, 0, 3, 4, 0, 8, 1, 0, 0, 0, 15, 0, 0, 0, 5, 0, 0, 0, 3, 0, 0, 0, 1, 2, 0, 0, 7, 6, 4, 0, 1, 0, 8, 0, 3, 0, 0, 0, 1, 0, 0, 2, 31, 0, 0, 0, 1, 0, 0, 0, 11, 0, 0, 4, 1, 0, 0, 0, 7, 26, 0, 0, 1, 0, 0, 0, 3, 0, 2, 0, 1, 0, 0, 0, 15, 0, 6, 2, 9, 0, 0, 0, 3, 0

Offset: 1

Antti Karttunen, Jul 28 2020

nonn

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

Cf. A003557, A007947, A066503.

A336551(n) = { my(f=factor(n)); for(i=1, #f~, f[i, 2] = f[i, 2]-1); (factorback(f)-1); };

a(n) = A066503(n) / A007947(n).

A326145 Numbers n for which n - A007947(n) is equal to gcd(n - A007947(n), sigma(n) - A007947(n) - n).

Original entry on oeis.org

6, 28, 496, 936, 1638, 8128, 33550336

Offset: 1

Antti Karttunen, Jun 09 2019

Numbers n such that either A066503(n) and A326143(n) are both zero or A066503(n) is not zero and divides A326143(n).
Question: Are there any odd terms?
No other terms < 2^31.

Index entries for sequences where any odd perfect numbers must occur

Cf. A000203, A000396 (a subsequence), A007947, A066503, A228058, A326142, A326143, A326144.

A007947(n) = factorback(factorint(n)[, 1]);
isA326145(n) = { my(b=A007947(n), t=n-b, u = (sigma(n)-b)-n); (gcd(t,u)==t); };
\\ Or alternatively as:
isA326145(n) = { my(t=A326143(n), u=A066503(n)); ((!u && !t)||(u && !(t%u))); };

A336644 a(n) = (n-rad(n)) / core(n), where rad(n) and core(n) give the squarefree kernel and squarefree part of n, respectively.

Original entry on oeis.org

0, 0, 0, 2, 0, 0, 0, 3, 6, 0, 0, 2, 0, 0, 0, 14, 0, 6, 0, 2, 0, 0, 0, 3, 20, 0, 8, 2, 0, 0, 0, 15, 0, 0, 0, 30, 0, 0, 0, 3, 0, 0, 0, 2, 6, 0, 0, 14, 42, 20, 0, 2, 0, 8, 0, 3, 0, 0, 0, 2, 0, 0, 6, 62, 0, 0, 0, 2, 0, 0, 0, 33, 0, 0, 20, 2, 0, 0, 0, 14, 78, 0, 0, 2, 0, 0, 0, 3, 0, 6, 0, 2, 0, 0, 0, 15, 0, 42, 6, 90, 0, 0, 0, 3, 0

Offset: 1

Antti Karttunen, Jul 28 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A007913, A007947, A008833, A066503, A336642, A336643.

A336644(n) = ((n-factorback(factorint(n)[, 1])) / core(n));

from math import prod
from sympy.ntheory.factor_ import primefactors, core
def A336644(n): return (n-prod(primefactors(n)))//core(n) # Chai Wah Wu, Dec 30 2021

a(n) = A066503(n) / A007913(n) = (n-A007947(n)) / A007913(n).

a(n) = A008833(n) - A336643(n).

A073354 Binomial coefficient ( n, squarefree kernel(n) ).

Original entry on oeis.org

1, 1, 1, 6, 1, 1, 1, 28, 84, 1, 1, 924, 1, 1, 1, 120, 1, 18564, 1, 184756, 1, 1, 1, 134596, 53130, 1, 2925, 40116600, 1, 1, 1, 496, 1, 1, 1, 1947792, 1, 1, 1, 847660528, 1, 1, 1, 2104098963720, 344867425584, 1, 1, 12271512, 85900584, 10272278170, 1

Offset: 1

Reinhard Zumkeller, Jul 29 2002

a(n)=1 iff n is squarefree.

Robert Israel, Table of n, a(n) for n = 1..3315

Cf. A007947, A005117, A073353, A066503, A064549, A003557.

f:= proc(n) local k;
  k:= convert(numtheory:-factorset(n),`*`);
  binomial(n,k)
end proc:
map(f, [$1..60]); # Robert Israel, May 07 2021

a[n_] := Binomial[n, Times @@ FactorInteger[n][[All, 1]]];
Table[a[n], {n, 1, 60}] (* Jean-François Alcover, May 11 2023 *)

a(n) = binomial(n, A007947(n)).

cp's OEIS Frontend

A073353 Sum of n and its squarefree kernel.

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A326128 a(n) = n - A007913(n), where A007913 gives the squarefree part of n.

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A380987 Position of first appearance of n in A290106 (product of prime indices divided by product of distinct prime indices).

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A380988 Sorted positions of first appearances in A290106 (product of prime indices divided by product of distinct prime indices).

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A284318 Triangle read by rows in which row n lists divisors d of n such that n divides d^n.

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A326142 Sum of all other divisors of n except its largest squarefree divisor: a(n) = sigma(n) - A007947(n).

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A336551 a(n) = A003557(n) - 1.

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A326145 Numbers n for which n - A007947(n) is equal to gcd(n - A007947(n), sigma(n) - A007947(n) - n).

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