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Decimal expansion of the asymptotic density of the coreful perfect numbers ( A307958) that are generated from even primitives ( A307959).
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9, 3, 6, 1, 0, 4, 7, 4, 5, 9, 0, 6, 8, 1, 6, 5, 6, 3, 8, 4, 5, 1, 6, 3, 0, 4, 5, 7, 8, 4, 4, 1, 1, 8, 5, 6, 1, 5, 5, 2, 8, 4, 2, 8, 7, 8, 2, 9, 8, 4, 3, 5, 3, 5, 6, 9, 4, 4, 2, 2, 0, 9, 1, 8, 9, 5, 8, 1, 1, 8, 4, 1, 5, 4, 6, 2, 4, 9, 0, 8, 6, 4, 7, 8, 1, 5, 7
COMMENTS
Since the coreful perfect numbers are analogous to e-perfect numbers ( A054979), the result of Hagis (see the formula and compare to A318645) can be also applied here. If there is no odd perfect number, then this constant is the asymptotic density of all the coreful perfect numbers.
FORMULA
Equals Sum_{j>=1} beta(c(j))/c(j), where beta(k) = (6/Pi^2)*Product_{p|k}(p/(p+1)) and c(j) is the j-th even term of A307959.
EXAMPLE
0.0093610474590681656384516304578441185615528428782...
MATHEMATICA
f[p_] := 1/(3 * (2^p-1) * 2^(2*p-1)); v = MersennePrimeExponent/@Range[25]; RealDigits[(6/Pi^2)*Total[f/@v], 10, 100][[1]]
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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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
COMMENTS
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).
LINKS
G. E. Hardy and M. V. Subbarao, Highly powerful numbers, Congress. Numer., Vol. 37 (1983), pp. 277-307. (Annotated scanned copy)
EXAMPLE
36 is in the sequence since its coreful divisors are 6, 12, 18, 36, whose sum is 72 = 2 * 36.
MATHEMATICA
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
PROG
(PARI) rad(n) = factorback(factorint(n)[, 1]); \\ A007947s(n) = rad(n)*sigma(n/rad(n)); \\ A057723
Powerful abundant numbers: numbers that are both powerful ( A001694) and abundant ( A005101).
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36, 72, 100, 108, 144, 196, 200, 216, 288, 324, 392, 400, 432, 500, 576, 648, 784, 800, 864, 900, 968, 972, 1000, 1152, 1296, 1352, 1372, 1568, 1600, 1728, 1764, 1800, 1936, 1944, 2000, 2304, 2500, 2592, 2700, 2704, 2744, 2916, 3136, 3200, 3456, 3528, 3600, 3872, 3888, 4000
COMMENTS
The least odd term is a(90) = 11025, and the least term that is coprime to 6 is 1382511906801025.
Are there two consecutive integers in this sequence? There are none below 10^22.
EXAMPLE
36 = 2^2 * 3^2 is a term since it is powerful, and sigma(36) = 91 > 2*36 = 72.
MATHEMATICA
Select[Range[4000], DivisorSigma[-1, #] > 2 && Min[FactorInteger[#][[;; , 2]]] > 1 &]
PROG
(PARI) is(n) = { my(f = factor(n)); n > 1 && vecmin(f[, 2]) > 1 && sigma(f, -1) > 2; }
Primitive coreful abundant numbers (second definition): coreful abundant numbers ( A308053) that are powerful numbers ( A001694).
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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
COMMENTS
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).
EXAMPLE
72 is a term since csigma(72) = 168 > 2 * 72, and 72 = 2^3 * 3^2 is powerful.
MATHEMATICA
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
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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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
COMMENTS
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).
MATHEMATICA
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
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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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
COMMENTS
The terms are ordered according to the their lesser counterparts ( A307962).
MATHEMATICA
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
Primitive coreful abundant numbers: coreful abundant numbers having no coreful abundant aliquot divisor.
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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
COMMENTS
All the coreful abundant numbers ( A308053) are multiples of terms of this sequence.
MATHEMATICA
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]
Coreful harmonic numbers: nonsquarefree numbers k such that the harmonic mean of the coreful divisors of k is an integer.
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12, 18, 36, 56, 60, 75, 84, 90, 126, 132, 150, 156, 168, 180, 198, 204, 228, 234, 240, 252, 276, 280, 306, 342, 348, 351, 372, 392, 396, 414, 420, 444, 450, 468, 492, 504, 516, 522, 525, 558, 564, 588, 612, 616, 630, 636, 660, 666, 684, 702, 708, 720, 726, 728
COMMENTS
A divisor of a number k is coreful if it is divisible by every prime that divides k.
The sequence is restricted to nonsquarefree numbers since the squarefree numbers have a single coreful divisor and thus they trivially have an integer harmonic mean.
EXAMPLE
12 is a term since its coreful divisors are 6 and 12 and their harmonic mean, 8, is an integer.
MATHEMATICA
rad[n_] := Times @@ FactorInteger[n][[;; , 1]]; corHarmQ[n_] := Module[{r = rad[n], d}, d = Select[Divisors[n], rad[#] == r &]; IntegerQ[HarmonicMean[d]]]; Select[Range[10^3], !SquareFreeQ[#] && corHarmQ[#] &]
Primitive coreful Zumkeller numbers: coreful Zumkeller numbers ( A339979) having no coreful Zumkeller aliquot divisor.
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36, 200, 392, 1936, 2704, 4900, 9248, 11552, 16928, 26912, 30752, 60500, 84500, 87616, 99225, 107584, 118336, 141376, 163592, 165375, 179776, 222784, 231525, 238144, 349448, 574592, 645248, 682112, 798848, 881792, 1013888, 1204352, 1225125, 1305728, 1357952
COMMENTS
If m is a coreful Zumkeller number and k is a squarefree number such that gcd(m, k) = 1, then k*m is also a coreful Zumkeller number.
EXAMPLE
a(1) = 36 since it is the least coreful Zumkeller number.
The next coreful Zumkeller numbers, 72, 144 and 180, are not terms since they are multiples of 36.
MATHEMATICA
corZumQ[n_] := 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]; primczQ[n_] := corZumQ[n] && NoneTrue[Most @ Divisors[n], corZumQ]; Select[Range[10^6], primczQ]
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