A057723 -id:A057723 - cp's OEIS frontend

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

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

A335341 Sum of divisors of A003557(n).

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 7, 4, 1, 1, 3, 1, 1, 1, 15, 1, 4, 1, 3, 1, 1, 1, 7, 6, 1, 13, 3, 1, 1, 1, 31, 1, 1, 1, 12, 1, 1, 1, 7, 1, 1, 1, 3, 4, 1, 1, 15, 8, 6, 1, 3, 1, 13, 1, 7, 1, 1, 1, 3, 1, 1, 4, 63, 1, 1, 1, 3, 1, 1, 1, 28, 1, 1, 6, 3, 1, 1, 1

Offset: 1

R. J. Mathar, Jun 02 2020

The sum of the divisors d of n such that n/d is a coreful divisor of n (a coreful divisor of n is a divisor with the same squarefree kernel as n). The number of these divisors is A005361(n). - Amiram Eldar, Jun 30 2023

Antti Karttunen, Table of n, a(n) for n = 1..16383
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A000203, A003557, A005361 (number of divisors of A003557), A336567.

Cf. A047994, A173557.

A335341 := proc(n)
    local a,pe,p,e ;
    a := 1;
    for pe in ifactors(n)[2] do
        p := op(1,pe) ;
        e := op(2,pe) ;
        if e > 1 then
            a := a*(p^e-1)/(p-1) ;
        end if;
    end do:
    a ;
end proc:

f[p_, e_] := (p^e-1)/(p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 26 2020 *)

a(n) = sigma(n/factorback(factor(n)[, 1])); \\ Michel Marcus, Jun 02 2020

a(n) = A000203(A003557(n)).
Multiplicative with a(p^1)=1 and a(p^e) = (p^e-1)/(p-1) if e>1.
A057723(n) = A007947(n)*a(n).
a(n) = 1 iff n in A005117.
a(n) = A336567(n) + A003557(n). - Antti Karttunen, Jul 28 2020
Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (1 - 1/p^(s-1) + 1/p^(2*s-1)). - Amiram Eldar, Sep 09 2023
a(n) = A047994(n)/A173557(n). - Ridouane Oudra, Oct 30 2023

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

A356871 Primitive coreful abundant numbers (second definition): coreful abundant numbers (A308053) that are powerful numbers (A001694).

Original entry on oeis.org

72, 108, 144, 200, 216, 288, 324, 400, 432, 576, 648, 784, 800, 864, 900, 972, 1000, 1152, 1296, 1568, 1600, 1728, 1764, 1800, 1936, 1944, 2000, 2304, 2592, 2700, 2704, 2744, 2916, 3136, 3200, 3456, 3528, 3600, 3872, 3888, 4000, 4356, 4500, 4608, 4900, 5000, 5184

Offset: 1

Amiram Eldar, Sep 02 2022

nonn

For squarefree numbers k, csigma(k) = k, where csigma(k) is the sum of the coreful divisors of k (A057723). Thus, if m is a term (csigma(m) > 2*m) and k is a squarefree number coprime to k, then csigma(k*m) = csigma(k) * csigma(m) = k * csigma(m) > 2*k*m, so k*m is a coreful abundant number. Therefore, the sequence of coreful abundant numbers (A308053) can be generated from this sequence by multiplying with coprime squarefree numbers. The asymptotic density of the coreful abundant numbers can be calculated from this sequence (see comment in A308053).

			72 is a term since csigma(72) = 168 > 2 * 72, and 72 = 2^3 * 3^2 is powerful.

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

Intersection of A001694 and A308053.
A339940 is a subsequence.
Cf. A057723.
Similar sequences: A307959, A328136.

f[p_, e_] := (p^(e+1)-1)/(p-1)-1; s[1] = 1; s[n_] := If[AllTrue[(fct = FactorInteger[n])[[;;, 2]], #>1 &], Times @@ f @@@ fct, 0]; seq={}; Do[If[s[n] > 2*n, AppendTo[seq, n]], {n, 1, 5000}]; seq

A307888 Non-coreful perfect numbers.

Original entry on oeis.org

6, 234, 588, 600, 6552, 89376, 209195610624

Offset: 1

Paolo P. Lava, May 09 2019

A coreful divisor d of a number k is a divisor with the same set of distinct prime factors as k (see LINKS).

Here, only the non-coreful divisors of k are considered.

			Divisors of 234 are 1, 2, 3, 6, 9, 13, 18, 26, 39, 78, 117, 234 and its prime factors are 2, 3, 13. Among the divisors, 78 and 234 are divided by all the prime factors and 1 + 2 + 3 + 6 + 9 + 13 + 18 + 26 + 39 + 117 = 234.

G. E. Hardy and M. V. Subbarao, Highly powerful numbers, Congress. Numer. 37 (1983), 277-307. (Annotated scanned copy)

Cf. A000203, A007947, A057723, A307958, A307986, A308029.

with(numtheory): P:=proc(q) local a,k,n; for n from 1 to q do
a:=mul(k,k=factorset(n)); if n=sigma(n)-a*sigma(n/a) then print(n); fi;
od; end: P(10^7);

f[p_, e_] := (p^(e + 1) - 1)/(p - 1); fc[p_, e_] := f[p, e] - 1; ncQ[n_] := Times @@ (f @@@ FactorInteger[n]) - Times @@ (fc @@@ FactorInteger[n]) == n; Select[Range[2, 10^5], ncQ] (* Amiram Eldar, May 11 2019 *)

rad(n) = factorback(factorint(n)[, 1]); \\ A007947
s(n) = rad(n)*sigma(n/rad(n)); \\ A057723
isok(n) = sigma(n) - s(n) == n; \\ Michel Marcus, May 11 2019

Solutions of k = A000203(k) - A057723(k).

a(7) from Giovanni Resta, May 09 2019

A308029 Numbers whose sum of coreful divisors is equal to the sum of non-coreful divisors.

Original entry on oeis.org

6, 1638, 55860, 168836850, 12854283750

Offset: 1

Paolo P. Lava, May 10 2019

A coreful divisor d of a number k is a divisor with the same set of distinct prime factors as k (see LINKS).
Sequence is a subset of A083207.
Tested up to 10^12. - Giovanni Resta, May 10 2019

			Divisors of 1638 are 1, 2, 3, 6, 7, 9, 13, 14, 18, 21, 26, 39, 42, 63, 78, 91, 117, 126, 182, 234, 273, 546, 819, 1638. The coreful ones are 546, 1638 and 1 + 2 + 3 + 6 + 7 + 9 + 13 + 14 + 18 + 21 + 26 + 39 + 42 + 63 + 78 + 91 + 117 + 126 + 182 + 234 + 273 + 819 = 546 + 1638 = 2184.

G. E. Hardy and M. V. Subbarao, Highly powerful numbers, Congress. Numer. 37 (1983), 277-307. (Annotated scanned copy)

Cf. A000203, A007947, A057723, A083207, A307888, A307958, A307962, A307963, A307986.

with(numtheory): P:=proc(q) local a, k, n; for n from 1 to q do
a:=mul(k, k=factorset(n)); if sigma(n)=2*a*sigma(n/a)
then print(n); fi; od; end: P(10^7);

f[p_, e_] := (p^(e + 1) - 1)/(p - 1); fc[p_, e_] := f[p, e] - 1; csigmaQ[n_] := Times @@ (fc @@@ FactorInteger[n]) == Times @@ (f @@@ FactorInteger[n])/2; Select[Range[2, 10^5], csigmaQ] (* Amiram Eldar, May 11 2019 *)

rad(n) = factorback(factorint(n)[, 1]); \\ A007947
s(n) = my(rn=rad(n)); rn*sigma(n/rn); \\ A057723
isok(n) = 2*s(n) == sigma(n); \\ Michel Marcus, May 11 2019

Solutions of A000203(k) = 2*A057723(k).

a(4)-a(5) from Giovanni Resta, May 10 2019

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

A339940 Primitive coreful abundant numbers: coreful abundant numbers having no coreful abundant aliquot divisor.

Original entry on oeis.org

72, 108, 200, 784, 900, 1764, 1936, 2704, 2744, 4356, 4900, 6084, 9248, 10404, 11552, 12996, 16928, 19044, 26912, 30276, 34596, 47432, 49284, 60500, 60516, 61504, 66248, 66564, 79524, 84500, 87616, 99225, 101124, 107584, 113288, 118336, 125316, 133956, 141376

Offset: 1

Amiram Eldar, Dec 23 2020

nonn

Analogous to A091191 as A057723 is analogous to A000203.

All the coreful abundant numbers (A308053) are multiples of terms of this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A057723, A091191, A302574, A307959, A308053.

f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - 1; s[1] = 1; s[n_] := Times @@ (f @@@ FactorInteger[n]); cabQ[n_] := s[n] > 2*n; pricabQ[n_] := cabQ[n] && AllTrue[Most @ Divisors[n], !cabQ[#] &]; Select[Range[10^5], pricabQ]

A339979 Coreful Zumkeller numbers: numbers whose set of coreful divisors can be partitioned into two disjoint sets of equal sum.

Original entry on oeis.org

36, 72, 144, 180, 200, 252, 288, 324, 360, 392, 396, 400, 468, 504, 576, 600, 612, 648, 684, 720, 784, 792, 800, 828, 900, 936, 1008, 1044, 1116, 1152, 1176, 1200, 1224, 1260, 1296, 1332, 1368, 1400, 1440, 1476, 1548, 1568, 1584, 1600, 1620, 1656, 1692, 1764

Offset: 1

Amiram Eldar, Dec 25 2020

nonn

A coreful divisor d of a number k is a divisor with the same set of distinct prime factors as k, or rad(d) = rad(k), where rad(k) is the largest squarefree divisor of k (A007947).

The coreful perfect numbers (A307958) are a subsequence.

			36 is a term since its set of coreful divisors, {6, 12, 18, 36}, can be partitioned into the two disjoint sets, {6, 12, 18} and {36}, whose sums are equal: 6 + 12 + 18 = 36.

A307958 is a subsequence.
Subsequence of A308053.
Cf. A007947, A057723.
Similar sequences: A083207, A290466, A335197, A335142, A335215, A335218.

corZumQ[n_] := Module[{r = Times @@ FactorInteger[n][[;; , 1]], d, sum, x}, d = r * Divisors[n/r]; (sum = Plus @@ d) >= 2*n && EvenQ[sum] && CoefficientList[Product[1 + x^i, {i, d}], x][[1 + sum/2]] > 0]; Select[Range[1800], corZumQ]

from itertools import count, islice
from sympy import primefactors, divisors
def A339979_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue,1)):
        f = primefactors(n)
        d = [x for x in divisors(n) if primefactors(x)==f]
        s = sum(d)
        if s&1^1 and n<<1<=s:
            d = d[:-1]
            s2, ld = (s>>1)-n, len(d)
            z = [[0 for  in range(s2+1)] for  in range(ld+1)]
            for i in range(1, ld+1):
                y = min(d[i-1], s2+1)
                z[i][:y] = z[i-1][:y]
                for j in range(y,s2+1):
                    z[i][j] = max(z[i-1][j],z[i-1][j-y]+y)
                if z[i][s2] == s2:
                    yield n
                    break
A339979_list = list(islice(A339979_gen(),20)) # Chai Wah Wu, Feb 14 2023

A364988 a(n) is the sum of coreful divisors d of n such that n/d is also a coreful divisor.

Original entry on oeis.org

1, 0, 0, 2, 0, 0, 0, 6, 3, 0, 0, 0, 0, 0, 0, 14, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 12, 0, 0, 0, 0, 30, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 62, 0, 0, 0, 0, 0, 0, 0, 18, 0, 0, 0, 0, 0, 0, 0, 0, 39, 0, 0, 0, 0

Offset: 1

Amiram Eldar, Aug 15 2023

A coreful divisor d of a number n is a divisor with the same set of distinct prime factors as n (see A307958).

The number of these divisors is A361430(n).

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

Cf. A001694, A307958, A361430.

Similar sequences: A000203, A057723 (sum of coreful divisors).

f[p_, e_] := (p^e - 1)/(p-1) - 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^f[i,2] - 1)/(f[i,1] - 1) - 1);}

Multiplicative with a(p^e) = (p^e - 1)/(p-1) - 1.
Dirichlet g.f.: zeta(s) * zeta(s-1) * Product_{p prime} (1 + (2*p - p^s*(p+1))/p^(2*s)).
a(n) > 0 if and only if n is powerful (A001694).
a(n) <= n with equality only when n = 1.
a(p^2) = p for a prime p.

cp's OEIS Frontend

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

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A335341 Sum of divisors of A003557(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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A356871 Primitive coreful abundant numbers (second definition): coreful abundant numbers (A308053) that are powerful numbers (A001694).

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A307888 Non-coreful perfect numbers.

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A308029 Numbers whose sum of coreful divisors is equal to the sum of non-coreful divisors.

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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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