A308135 -id:A308135 - cp's OEIS frontend

A336566 a(n) = gcd(A336563(n), A336564(n)) = gcd(A057723(n)-n, n-A308135(n)).

1, 1, 2, 1, 4, 0, 6, 1, 1, 2, 10, 2, 12, 4, 6, 1, 16, 3, 18, 2, 10, 8, 22, 6, 1, 10, 2, 14, 28, 12, 30, 1, 18, 14, 22, 1, 36, 16, 22, 10, 40, 12, 42, 2, 3, 20, 46, 14, 1, 1, 30, 2, 52, 12, 38, 2, 34, 26, 58, 6, 60, 28, 1, 1, 46, 12, 66, 2, 42, 4, 70, 3, 72, 34, 1, 2, 58, 12, 78, 2, 1, 38, 82, 14, 62, 40, 54, 2, 88

Offset: 1

Antti Karttunen, Jul 27 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A057723, A308135, A336563, A336564, A336565.

Differs from A326144 at the positions given by A336555, for the first time at n=45, where a(45) = 3, while A326144(45) = 6.

A007947(n) = factorback(factorint(n)[, 1]);
A057723(n) = { my(r=A007947(n)); (r*sigma(n/r)); };
A308135(n) = (sigma(n)-A057723(n));
A336563(n) = (A057723(n)-n);
A336564(n) = (n - A308135(n));
A336566(n) = gcd(A336563(n), A336564(n));

a(n) = gcd(A336563(n), A336564(n)) = gcd(A057723(n)-n, n-A308135(n));

A336564 a(n) = n - A308135(n), where A308135(n) is the sum of non-coreful divisors of n.

Original entry on oeis.org

1, 1, 2, 3, 4, 0, 6, 7, 8, 2, 10, 2, 12, 4, 6, 15, 16, 3, 18, 8, 10, 8, 22, 6, 24, 10, 26, 14, 28, -12, 30, 31, 18, 14, 22, 17, 36, 16, 22, 20, 40, -12, 42, 26, 27, 20, 46, 14, 48, 17, 30, 32, 52, 12, 38, 34, 34, 26, 58, -18, 60, 28, 43, 63, 46, -12, 66, 44, 42, -4, 70, 45, 72, 34, 41, 50, 58, -12, 78, 44, 80, 38, 82, -14, 62

Offset: 1

Antti Karttunen, Jul 27 2020

sign

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000203, A033879, A057723, A308135, A336563, A336565, A336566.

Cf. A013661, A065487.

f[p_, e_] := (p^(e + 1) - 1)/(p - 1); fc[p_, e_] := f[p, e] - 1; a[1] = 1; a[n_] := n - Times @@ f @@@ (fct = FactorInteger[n]) + Times @@ fc @@@ fct; Array[a, 100] (* Amiram Eldar, Dec 08 2023 *)

A007947(n) = factorback(factorint(n)[, 1]);
A057723(n) = { my(r=A007947(n)); (r*sigma(n/r)); };
A308135(n) = (sigma(n)-A057723(n));
A336564(n) = (n - A308135(n));

a(n) = n - A308135(n) = n - (sigma(n) - A057723(n)).
a(n) = A336563(n) + A033879(n). [Corrected by Georg Fischer, Dec 13 2022]
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = A065487 - A013661 + 1 = 0.586357... . - Amiram Eldar, Dec 08 2023

A308127 Non-coreful abundant numbers: numbers k such that ncsigma(k) > k, where ncsigma(k) is the sum of the non-coreful divisors of k (A308135).

Original entry on oeis.org

30, 42, 60, 66, 70, 78, 84, 90, 102, 114, 120, 126, 132, 138, 150, 156, 168, 174, 180, 186, 198, 210, 222, 240, 246, 258, 270, 282, 294, 300, 318, 330, 336, 354, 366, 378, 390, 402, 420, 426, 438, 450, 462, 474, 480, 498, 510, 534, 546, 570, 582, 606, 618, 630

Offset: 1

Amiram Eldar and Paolo P. Lava, May 14 2019

nonn

Non-coreful divisor d of a number k is a divisor such that rad(d) != rad(k), where rad(k) is the largest squarefree divisor of k (A007947).

			60 is in the sequence since its non-coreful divisors are 1, 2, 3, 4, 5, 6, 10, 12, 15, and 20 whose sum is 78 > 60.

Amiram Eldar, Table of n, a(n) for n = 1..10000
G. E. Hardy and M. V. Subbarao, Highly powerful numbers, Congress. Numer., Vol. 37 (1983), pp. 277-307. (Annotated scanned copy)

Cf. A000203, A005101, A007947, A057723, A307888, A307958, A308029, A308135.

with(numtheory): P:=proc(k) local a,n; a:=mul(n,n=factorset(k));
if sigma(k)-a*sigma(k/a)>k then k; fi;  end: seq(P(i),i=1..630);

f[p_, e_] := (p^(e + 1) - 1)/(p - 1); fc[p_, e_] := f[p, e] - 1; ncAbQ[n_] := Times @@ (f @@@ FactorInteger[n]) - Times @@ (fc @@@ FactorInteger[n]) > n; Select[Range[2, 1000], ncAbQ]

rad(n) = factorback(factorint(n)[, 1]); \\ A007947
s(n) = my(r=rad(n)); sumdiv(n, d, if (rad(d)!=r, d));
isok(n) = s(n) > n; \\ Michel Marcus, May 14 2019

A336565 Numbers k for which (A057723(k)-k) is equal to gcd(k-A308135(k), A057723(k)-k).

Original entry on oeis.org

6, 28, 234, 496, 588, 600, 1521, 1638, 6552, 8128, 55860, 89376, 33550336, 168836850

Offset: 1

Antti Karttunen, Jul 26 2020

Numbers k for which A336563(k) = A336566(n) [= gcd(A336563(n), A336564(n))].
Numbers k such that either both A336563(k) and A336564(k) are zero (in which case k is squarefree), or A336563(k) divides A336564(k), in which case k is not squarefree.
Also numbers k for which A336647(n) = 2*n - A057723(n).
Question: Are there any other odd terms apart from 1521 = 39^2 ?

Index entries for sequences where odd perfect numbers must occur, if they exist at all

Cf. A057723, A308135, A336563, A336564, A336566, A336647.
Cf. A000396 (a subsequence).
Cf. also A326145.

A007947(n) = factorback(factorint(n)[, 1]);
A057723(n) = { my(r=A007947(n)); (r*sigma(n/r)); };
isA336565(n) = { my(b=A057723(n), c=(sigma(n)-b), d=(b-n)); (gcd(d,(n-c))==d); };

A336647 a(n) = n - A336566(n).

Original entry on oeis.org

0, 1, 1, 3, 1, 6, 1, 7, 8, 8, 1, 10, 1, 10, 9, 15, 1, 15, 1, 18, 11, 14, 1, 18, 24, 16, 25, 14, 1, 18, 1, 31, 15, 20, 13, 35, 1, 22, 17, 30, 1, 30, 1, 42, 42, 26, 1, 34, 48, 49, 21, 50, 1, 42, 17, 54, 23, 32, 1, 54, 1, 34, 62, 63, 19, 54, 1, 66, 27, 66, 1, 69, 1, 40, 74, 74, 19, 66, 1, 78, 80, 44, 1, 70, 23, 46, 33, 86

Offset: 1

Antti Karttunen, Jul 30 2020

sign

Some terms, for example a(600) and a(6552), are negative. - Georg Fischer, Jul 31 2020

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, A057723, A308135, A336563, A336564, A336566.
Cf. A336555 (positions where differs from A336646).
Cf. A336565 (positions where a(n) = 2*n - A057723(n) = n - A336563(n)).
Cf. also A336645.

A007947(n) = factorback(factorint(n)[, 1]);
A057723(n) = { my(r=A007947(n)); (r*sigma(n/r)); };
A308135(n) = (sigma(n)-A057723(n));
A336563(n) = (A057723(n)-n);
A336564(n) = (n - A308135(n));
A336566(n) = gcd(A336563(n), A336564(n));
A336647(n) = (n - A336566(n));

a(n) = n - A336566(n).

A339938 Odd non-coreful abundant numbers: the odd terms of A308127.

Original entry on oeis.org

15015, 19635, 21945, 23205, 25935, 26565, 31395, 33495, 33915, 35805, 39585, 41055, 42315, 42735, 45885, 47355, 49665, 50505, 51765, 54285, 55965, 58695, 61215, 64155, 68145, 70455, 72345, 75075, 77385, 80535, 82005, 83265, 84315, 91245, 95865, 102795, 105105

Offset: 1

Amiram Eldar, Dec 23 2020

nonn

First differs from A112643, A129485 and A249263 at n = 28.

			15015 is a term since it is odd and the sum of its non-coreful divisors is A308135(15015) = 17241 > 15015.

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

Intersection of A005408 and A308127.
Cf. A308135.
Similar sequences: A005231, A094889, A112643, A127666, A129485, A249263, A293186, A321147, A333950.

f[p_, e_] := (p^(e + 1) - 1)/(p - 1); fc[p_, e_] := f[p, e] - 1; s[1] = 0; s[n_] := Times @@ (f @@@ FactorInteger[n]) - Times @@ (fc @@@ FactorInteger[n]); Select[Range[1, 10^5, 2], s[#] > # &]

A378689 a(n) = product of divisors d of n that are not coreful.

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 1, 1, 1, 10, 1, 24, 1, 14, 15, 1, 1, 54, 1, 40, 21, 22, 1, 192, 1, 26, 1, 56, 1, 27000, 1, 1, 33, 34, 35, 216, 1, 38, 39, 320, 1, 74088, 1, 88, 135, 46, 1, 3072, 1, 250, 51, 104, 1, 1458, 55, 448, 57, 58, 1, 25920000, 1, 62, 189, 1, 65, 287496

Offset: 1

Michael De Vlieger, Feb 05 2025

			Table of n, a(n), and divisors that are not coreful that produce a(n) for select n:
   n     a(n)
  -----------------------------
   1       1   (empty product)
   2       1 = 1
   3       1 = 1
   4       1 = 1
   5       1 = 1
   6       6 = 1*2*3
  10      10 = 1*2*5
  12      24 = 1*2*3*4
  14      14 = 1*2*7
  15      15 = 1*3*5
  18      54 = 1*2*3*9
  20      40 = 1*2*4*5
  21      21 = 1*3*7
  22      22 = 1*2*11
  24     192 = 1*2*3*4*8
  30   27000 = 1*2*3*5*6*10*15
  36     216 = 1*2*3*4*9

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of log_10(a(n)) for n = 1..2^20.
Michael De Vlieger, Log log scatterplot of log_10(a(n)) for n = 1..2^16 (ignoring a(n) = 1, i.e., n that is a power of a prime), showing a(n) such that n is in A286708 in purple, n in A332785 in blue, n in A120944 in green, highlighting n in A002110 in large green points.

Cf. A007955, A027750, A308135 (sums), A308360 (product of coreful divisors of n).

rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]]; Table[r = rad[n]; Times @@ Select[Divisors[n], rad[#] != r &], {n, 120}]

rad(n) = factorback(factorint(n)[, 1]); \\ A007947
a(n) = my(d=divisors(n), c=rad(n), p=1); for (i=1, #d~, if (rad(d[i]) != c, p *= d[i])); p; \\ Michel Marcus, Feb 07 2025

a(n) = A007955(n) / A308360(n).

a(n) = 1 for powers of primes n (i.e., n in A000961), since d | n such that d > 1 are coreful.

cp's OEIS Frontend

A336566 a(n) = gcd(A336563(n), A336564(n)) = gcd(A057723(n)-n, n-A308135(n)).

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A336564 a(n) = n - A308135(n), where A308135(n) is the sum of non-coreful divisors of n.

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A308127 Non-coreful abundant numbers: numbers k such that ncsigma(k) > k, where ncsigma(k) is the sum of the non-coreful divisors of k (A308135).

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A336565 Numbers k for which (A057723(k)-k) is equal to gcd(k-A308135(k), A057723(k)-k).

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A336647 a(n) = n - A336566(n).

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A339938 Odd non-coreful abundant numbers: the odd terms of A308127.

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A378689 a(n) = product of divisors d of n that are not coreful.

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