A057723 -id:A057723 - cp's OEIS frontend

A307958 Coreful perfect numbers: numbers k such that csigma(k) = 2*k, where csigma(k) is the sum of the coreful divisors of k (A057723).

36, 180, 252, 392, 396, 468, 612, 684, 828, 1044, 1116, 1176, 1260, 1332, 1476, 1548, 1692, 1908, 1960, 1980, 2124, 2196, 2340, 2412, 2556, 2628, 2772, 2844, 2988, 3060, 3204, 3276, 3420, 3492, 3636, 3708, 3852, 3924, 4068, 4140, 4284, 4312, 4572, 4716, 4788

Offset: 1

Amiram Eldar, May 08 2019

nonn

Hardy and Subbarao defined a coreful divisor d of a number k as 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 number of these divisors is A005361(k) and their sum is csigma(k) = A057723(k). Since csigma(k) is multiplicative and csigma(p) = p for prime p, then if k is coreful perfect number, then also m*k is, for any squarefree number m coprime to k, gcd(m, k) = 1. Thus there are infinitely many coreful perfect numbers, and all of them can be generated from the sequence of primitive coreful perfect numbers (A307959), which is the subsequence of powerful terms of this sequence. This sequence and A307959 are analogous to e-perfect numbers (A054979) and primitive e-perfect numbers (A054980).

			36 is in the sequence since its coreful divisors are 6, 12, 18, 36, whose sum is 72 = 2 * 36.

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. A005361, A007947, A054979, A054980, A057723, A307959, A307960, A307961.

f[p_,e_] := (p^(e+1)-1)/(p-1)-1; a[1]=1; a[n_] := Times @@ (f @@@ FactorInteger[n]); s={}; Do[If[a[n] == 2n, AppendTo[s,n]], {n, 1, 10^6}]; s

rad(n) = factorback(factorint(n)[, 1]); \\ A007947
s(n) = rad(n)*sigma(n/rad(n)); \\ A057723
isok(n) = s(n) == 2*n; \\ Michel Marcus, May 14 2019

A308053 Coreful abundant numbers: numbers k such that csigma(k) > 2*k, where csigma(k) is the sum of the coreful divisors of k (A057723).

Original entry on oeis.org

72, 108, 144, 200, 216, 288, 324, 360, 400, 432, 504, 540, 576, 600, 648, 720, 756, 784, 792, 800, 864, 900, 936, 972, 1000, 1008, 1080, 1152, 1188, 1200, 1224, 1296, 1368, 1400, 1404, 1440, 1512, 1568, 1584, 1600, 1620, 1656, 1728, 1764, 1800, 1836, 1872, 1936

Offset: 1

Amiram Eldar, May 10 2019

nonn

Analogous to A005101 as A307958 is analogous to A000396.

The asymptotic density of this sequence is Sum_{n>=1} f(A356871(n)) = 0.0262215..., where f(n) = (6/(Pi^2*n))*Product_{prime p|n}(p/(p+1)). - Amiram Eldar, Sep 02 2022

			72 is in the sequence since its coreful divisors are 6, 12, 18, 24, 36, 72, whose sum is 168 > 2 * 72.

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

Cf. A005101, A057723, A307958.
A339940 and A356871 are subsequences.
Subsequence of A129575.

f[p_, e_] := (p^(e+1)-1)/(p-1)-1; a[1]=1; a[n_] := Times @@ (f @@@ FactorInteger[n]); s={}; Do[If[a[n] > 2n, AppendTo[s, n]], {n, 1, 2000}]; s

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

isok(k) = {my(f=factor(k)); prod(i = 1, #f~, (f[i,1]^(f[i,2]+1)-1) / (f[i,1]-1)-1) > 2*k}; \\ Amiram Eldar, Sep 02 2022

A307959 Primitive coreful perfect numbers: powerful numbers k such that csigma(k) = 2*k, where csigma(k) is the sum of the coreful divisors of k (A057723).

Original entry on oeis.org

36, 392, 30752, 2064512, 549621604352, 2251765454077952, 144114638320566272, 9903520305059670164485701632, 12259964326927110856232952250923146488025299504439754752, 237142198758023568227473376531545064850552499416058362196624559881526665860349952

Offset: 1

Amiram Eldar, May 08 2019

nonn

All the coreful perfect numbers (A307958) can be obtained from a primitive term k by multiplying it by m, if m is squarefree and coprime to k. The primitive terms are powerful. If k = m * r is in the sequence where r = rad(k) is the squarefree kernel of k (A007947), then the sum of the coreful divisors of k is csigma(k) = csigma(m * r) = sigma(m) * r, where sigma(m) is the sum of all the divisors of m (A000203). Thus k is a primitive coreful perfect number, iff sigma(m) * r = 2 * m * r, or m = k/rad(k) is a perfect number. Since k is powerful k = m * rad(m), thus all the primitive coreful perfect numbers can be generated from the perfect numbers by multiplying them by their squarefree kernel. The even terms of these sequence are (2^p) * (2^p-1)^2, were p is a Mersenne exponent (A000043). There is an odd term in this sequence iff there is an odd perfect number.

			The first coreful perfect numbers are 36, 180 = 36*5, 252 = 36*7, 392, 396 = 36*11, 468 = 36*13, ... thus the primitive ones are 36, 392, ...

Cf. A000043, A000203, A000396, A000668, A007947, A057723, A064549, A307958.

f[p_] := 2^p*(2^p-1)^2; f/@MersennePrimeExponent/@Range[10] (* assuming no odd perfect number exists *)

a(n) = A064549(A000396(n)) = A000396(n) * A007947(A000396(n)).

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

Original entry on oeis.org

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));

A336652 Sum of positive divisors of odd part of n that are divisible by every (odd) prime dividing it: a(n) = A057723(A000265(n)).

Original entry on oeis.org

1, 1, 3, 1, 5, 3, 7, 1, 12, 5, 11, 3, 13, 7, 15, 1, 17, 12, 19, 5, 21, 11, 23, 3, 30, 13, 39, 7, 29, 15, 31, 1, 33, 17, 35, 12, 37, 19, 39, 5, 41, 21, 43, 11, 60, 23, 47, 3, 56, 30, 51, 13, 53, 39, 55, 7, 57, 29, 59, 15, 61, 31, 84, 1, 65, 33, 67, 17, 69, 35, 71, 12, 73, 37, 90, 19, 77, 39, 79, 5, 120, 41, 83, 21, 85, 43

Offset: 1

Antti Karttunen, Jul 30 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. A000265, A000593, A057723, A204455, A330596, A336649, A336651.

f[2, e_] := 1; f[p_, e_] := (p^(e+1) - 1)/(p-1) - 1; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Dec 07 2020 *)

A336652(n) = if(1==n,n,my(f=factor(n)); prod(i=1,#f~,if(2==f[i,1],1,-1+(((f[i,1]^(1+f[i,2]))-1) / (f[i,1]-1)))));

Multiplicative with a(2^e) = 1, and for odd primes p, a(p^e) = (p + p^2 + ... + p^e) = sigma(p^e)-1.
a(n) = A057723(A000265(n)).
a(n) = A204455(n) * A336649(n).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^2/21) * Product_{p prime} (1 - 1/p^2 + 1/p^3) = (Pi^2/21) * A330596 = 0.3517974711... . - Amiram Eldar, Nov 12 2022

A340109 Coreful 3-abundant numbers: numbers k such that csigma(k) > 3*k, where csigma(k) is the sum of the coreful divisors of k (A057723).

Original entry on oeis.org

5400, 7200, 10800, 14400, 16200, 18000, 21168, 21600, 27000, 28800, 32400, 36000, 37800, 42336, 43200, 48600, 50400, 54000, 56448, 57600, 59400, 63504, 64800, 70200, 72000, 75600, 79200, 81000, 84672, 86400, 88200, 90000, 91800, 93600, 97200, 98784, 100800, 102600

Offset: 1

Amiram Eldar, Dec 28 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).
Analogous to A068403 as A308053 is analogous to A005101.
Apparently, the least odd term in this sequence is 3^4 * 5^3 * 7^3 * 11^2 * 13^2 * 17^2 * 19^2 * 23^2 * 29^2 = 3296233276111741840875.
The asymptotic density of this sequence is Sum_{n>=1} f(A364991(n)) = 0.0004006..., where f(n) = (6/(Pi^2*n)) * Product_{prime p|n} (p/(p+1)). - Amiram Eldar, Aug 15 2023

			5400 is a term since csigma(5400) = 16380 > 3 * 5400.

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

Subsequence of A308053.
Cf. A007947, A057723, A364991 (primitive terms).
Similar sequences: A068403, A285615, A293187, A300664, A307112, A328135.

f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - 1; s[1] = 1; s[n_] := Times @@ (f @@@ FactorInteger[n]); Select[Range[10^5], s[#] > 3*# &]

s(n) = {my(f = factor(n)); prod(i = 1, #f~, sigma(f[i, 1]^f[i, 2]) - 1);}
is(n) = s(n) > 3*n; \\ Amiram Eldar, Aug 15 2023

A307962 Lesser of coreful amicable numbers pair: numbers (m, n) such that csigma(m) = csigma(n) = m + n, where csigma(n) is the sum of the coreful divisors of n (A057723).

Original entry on oeis.org

1718200, 4818880, 5154600, 12027400, 14456640, 22336600, 29209400, 32645800, 33732160, 36082200, 39518600, 49827800, 53264200, 62645440, 63573400, 67009800, 70446200, 73882600, 80755400, 81920960, 87628200, 91064600, 91558720, 97937400, 101196480, 101373800

Offset: 1

Amiram Eldar, May 08 2019

nonn

The larger counterparts are in A307963.

If (m, n) is an amicable pair (A259180), then the pair (m*k, n*k) with k=rad(m*n) is a coreful amicable pair (rad(i)=A007947(i) is the squarefree kernel of i), and so are all the pairs (m*k*s, n*k*s) where s is a squarefree number with gcd(s, k) = 1. Proof: k = rad(m*n) = rad(m)*rad(n)/rad(gcd(m,n)), csigma(m*k) = csigma(m*rad(m)*j) where j = rad(n)/rad(gcd(m,n)) is squarefree and coprime to m*rad(m), so csigma(m*k) = j * csigma(m*rad(m)) = j * rad(m)* sigma(m) = rad(m)*rad(n)/rad(gcd(m,n)) * sigma(m) = rad(m)*rad(n)/rad(gcd(m,n)) * (n+m) = k *(n+m) = csigma(n*k).

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

Cf. A007947, A057723, A063990, A259180, A307958, A307959, A307963.

f[p_,e_] := (p^(e+1)-1)/(p-1)-1; csigma[1]=1; csigma[n_] := Times @@ (f @@@ FactorInteger[n]); s={}; Do[m = csigma[n] - n; If[m > n && csigma[m] - m == n, AppendTo[s, n]], {n, 1, 10^8}]; s

Wrong terms corrected by Amiram Eldar, Dec 02 2019

A307963 Larger of coreful amicable numbers pair: numbers (m, n) such that csigma(m) = csigma(n) = m + n, where csigma(n) is the sum of the coreful divisors of n (A057723).

Original entry on oeis.org

2218040, 4924700, 6654120, 15526280, 14774100, 28834520, 37706680, 42142760, 34472900, 46578840, 51014920, 64323160, 68759240, 64021100, 82067480, 86503560, 90939640, 95375720, 104247880, 83719900, 113120040, 117556120, 93569300, 126428280, 103418700, 130864360

Offset: 1

Amiram Eldar, May 08 2019

nonn

The terms are ordered according to the their lesser counterparts (A307962).

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

Cf. A057723, A307958, A307959, A307962.

f[p_,e_] := (p^(e+1)-1)/(p-1)-1; csigma[1]=1; csigma[n_] := Times @@ (f @@@ FactorInteger[n]); s={}; Do[m = csigma[n] - n; If[m > n && csigma[m] - m == n, AppendTo[s, m]], {n, 1, 10^8}]; s

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); };

A336465 Recursive variant of A057723: If n is cubefree, then a(n) = A057723(n), otherwise a(n) = A007947(n) * a(n/A007947(n)).

Original entry on oeis.org

1, 2, 3, 6, 5, 6, 7, 12, 12, 10, 11, 18, 13, 14, 15, 24, 17, 24, 19, 30, 21, 22, 23, 36, 30, 26, 36, 42, 29, 30, 31, 48, 33, 34, 35, 72, 37, 38, 39, 60, 41, 42, 43, 66, 60, 46, 47, 72, 56, 60, 51, 78, 53, 72, 55, 84, 57, 58, 59, 90, 61, 62, 84, 96, 65, 66, 67, 102, 69, 70, 71, 108, 73, 74, 90, 114, 77, 78, 79, 120

Offset: 1

Antti Karttunen, Jul 31 2020

nonn

Not multiplicative. The least counterexample is 72 = 8*9: a(72) = 108, while a(8) * a(9) = 12 * 12 = 144. - Amiram Eldar, Sep 09 2023

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences related to sigma(n).

Cf. A000203, A007947, A057723.

f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - 1; a[1] = 1; a[n_] := a[n] = Module[{fct = FactorInteger[n]}, If[Max[fct[[;; , 2]]] < 3, Times @@ f @@@ fct, r = Times @@ fct[[;; , 1]]; r*a[n/r]]]; Array[a, 100] (* Amiram Eldar, Sep 09 2023 *)

A336465(n) = if(issquarefree(n), n, my(f=factor(n), r=factorback(f[,1])); if(issquarefree(n/r),r*sigma(n/r),r*A336465(n/r)));

a(n) = Sum_{k=1..n} (1 - ceiling(n/k^2) + floor(n/k^2)) * mu(k)^2 * (n/k), where mu is the Möbius function (A008683). - Wesley Ivan Hurt, Jan 29 2021

Keyword mult removed by Amiram Eldar, Sep 09 2023

cp's OEIS Frontend

A307958 Coreful perfect numbers: numbers k such that csigma(k) = 2*k, where csigma(k) is the sum of the coreful divisors of k (A057723).

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A308053 Coreful abundant numbers: numbers k such that csigma(k) > 2*k, where csigma(k) is the sum of the coreful divisors of k (A057723).

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A307959 Primitive coreful perfect numbers: powerful numbers k such that csigma(k) = 2*k, where csigma(k) is the sum of the coreful divisors of k (A057723).

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

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A336652 Sum of positive divisors of odd part of n that are divisible by every (odd) prime dividing it: a(n) = A057723(A000265(n)).

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A340109 Coreful 3-abundant numbers: numbers k such that csigma(k) > 3*k, where csigma(k) is the sum of the coreful divisors of k (A057723).

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A307962 Lesser of coreful amicable numbers pair: numbers (m, n) such that csigma(m) = csigma(n) = m + n, where csigma(n) is the sum of the coreful divisors of n (A057723).

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A307963 Larger of coreful amicable numbers pair: numbers (m, n) such that csigma(m) = csigma(n) = m + n, where csigma(n) is the sum of the coreful divisors of n (A057723).

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