| Displaying 1-8 of 8 results found.
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10854718875, 12131744625, 13408770375, 18091198125, 19538493975, 20219574375, 21070924875, 22347950625, 22915517625, 23880381525, 24902002125, 25327677375, 28307404125, 28733079375, 29462808375, 32564156625, 35118208125, 36395233875, 39800635875, 40226311125
COMMENTS
Odd numbers whose divisors have a mean abundancy index that is larger than 2.
The odd terms in A245214 are relatively rare: a(1) = A245214(276918705). The least term that is not divisible by 3 is 26115176669245401228259189019322202117310546875.
EXAMPLE
10854718875 is a term since it is odd and A374777(10854718875) / A374778(10854718875) = 11975203 / 5955950 = 2.0106... > 2.
MATHEMATICA
f[p_, e_] := ((e+1)*p^2 - (e+2)*p + p^(-e))/((e+1)*(p-1)^2); s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; seq[kmax_] := Module[{v = {}}, Do[If[s[k] > 2, AppendTo[v, k]], {k, 1, kmax, 2}]; v]; seq[2*10^10]
PROG
(PARI) is(k) = if(!(k % 2), 0, my(f = factor(k)); prod(i = 1, #f~, p=f[i, 1]; e=f[i, 2]; (-2*p - e*p + p^2 + e*p^2 + p^(-e))/((e + 1)*(p - 1)^2)) > 2);
Numerator of the mean abundancy index of the divisors of n.
+10
9
1, 5, 7, 17, 11, 35, 15, 49, 34, 11, 23, 119, 27, 75, 77, 129, 35, 85, 39, 187, 5, 115, 47, 343, 86, 135, 71, 85, 59, 77, 63, 107, 161, 175, 33, 289, 75, 195, 63, 539, 83, 25, 87, 391, 187, 235, 95, 301, 54, 43, 245, 153, 107, 355, 23, 105, 91, 295, 119, 1309, 123, 315
COMMENTS
First differs from A318491 at n = 27. The abundancy index of a number k is sigma(k)/k = A017665(k)/ A017666(k).
FORMULA
f(n) = (Sum_{d|n} sigma(d)/d) / tau(n), where sigma(n) is the sum of divisors of n ( A000203), and tau(n) is their number ( A000005). f(n) is multiplicative with f(p^e) = ((e+1)*p^2 - (e+2)*p + p^(-e))/((e+1)*(p-1)^2).
Dirichlet g.f. of f(n): zeta(s) * Product_{p prime} ((p/(p-1)^2) * ((p^s-1)*log((1-1/p^s)/(1-1/p^(s+1))) + p-1)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} f(k) = Product_{p prime} ((p/(p-1)) * (1 - log(1 + 1/p))) = 1.3334768464... . For comparison, the asymptotic mean of the abundancy index over all the positive integers is zeta(2) = 1.644934... ( A013661). Lim sup_{n->oo} f(n) = oo (i.e., f(n) is unbounded).
EXAMPLE
For n = 2, n has 2 divisors, 1 and 2. Their abundancy indices are sigma(1)/1 = 1 and sigma(2)/2 = 3/2, and their mean abundancy index is (1 + 3/2)/2 = 5/4. Therefore a(2) = numerator(5/4) = 5.
MATHEMATICA
f[p_, e_] := ((e+1)*p^2 - (e+2)*p + p^(-e))/((e+1)*(p-1)^2); a[1] = 1; a[n_] := Numerator[Times @@ f @@@ FactorInteger[n]]; Array[a, 100]
PROG
(PARI) a(n) = {my(f = factor(n), p, e); numerator(prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2]; ((e+1)*p^2 - (e+2)*p + p^(-e))/((e+1)*(p-1)^2))); }
CROSSREFS
Cf. A000005, A000203, A013661, A017665, A017666, A038040, A060640, A245214, A318491, A318492, A374778 (denominators), A374779, A374780, A374781.
a(n) = Sum_{(d<n) | n} (d * tau(d)).
+10
5
0, 1, 1, 5, 1, 11, 1, 17, 7, 15, 1, 47, 1, 19, 17, 49, 1, 62, 1, 67, 21, 27, 1, 151, 11, 31, 34, 87, 1, 145, 1, 129, 29, 39, 25, 254, 1, 43, 33, 219, 1, 189, 1, 127, 104, 51, 1, 423, 15, 130, 41, 147, 1, 278, 33, 287, 45, 63, 1, 589, 1, 67, 132, 321, 37, 277
COMMENTS
If q are proper divisors of n then values of sequence a(n) are the bending moments at point 0 of static forces of sizes tau(q) operating in places q on the cantilever as the nonnegative number axis of length n with support at point 0 by the schema: a(n) = Sum_{q | n} (q * tau(q)).
Number n = 144 is the smallest number n such that a(n) > n * tau(n) (see A245212 and A245214). Conjecture: 21 is only number such that a(n) = n.
FORMULA
a(n) = 1 for n = primes.
a(n) = n + 5 for even semiprimes q = 2p > 4 (see A100484) where p = odd prime.
EXAMPLE
For n = 21 with proper divisors [1, 3, 7] we have: a(21) = 7 * tau(7) + 3 * tau(3) + 1 * tau(1) = 7*2 + 3*2 + 1*1 = 21.
PROG
(Magma) [(&+[d*#([e: e in Divisors(d)]): d in Divisors(n)])-(n*(#[d: d in Divisors(n)])): n in [1..1000]];
a(n) = n * tau(n) - Sum_{(d<n) | n} (d * tau(d)).
+10
4
1, 3, 5, 7, 9, 13, 13, 15, 20, 25, 21, 25, 25, 37, 43, 31, 33, 46, 37, 53, 63, 61, 45, 41, 64, 73, 74, 81, 57, 95, 61, 63, 103, 97, 115, 70, 73, 109, 123, 101, 81, 147, 85, 137, 166, 133, 93, 57, 132, 170, 163, 165, 105, 154, 187, 161, 183, 169, 117, 131, 121
COMMENTS
If d are divisors of n then values of sequence a(n) are the bending moments at point 0 of static forces of sizes tau(d) operating in places d on the cantilever as the nonnegative number axis of length n with support at point 0 by the schema: a(n) = (n * tau(n)) - Sum_{(d<n) | n} (d * tau(d)).
If a(n) = 0 then n must be > 10^7.
Conjecture: a(n) = sigma(n) iff n is a power of 2 ( A000079). Number n = 72 is the smallest number n such that a(n) < n (see A245213). Number n = 144 is the smallest number n such that a(n) < 0 (see A245214).
FORMULA
a(n) = 2 * A038040(n) - A060640(n) = 2 * (n * tau(n)) - Sum_{d | n} (d * tau(d)).
EXAMPLE
For n = 6 with divisors [1, 2, 3, 6] we have: a(6) = 6 * tau(6) - (3 * tau(3) + 2 * tau(2) + 1 * tau(1)) = 6*4 - (3*2+2*2+1*1) = 13.
PROG
(Magma) [(2*(n*(#[d: d in Divisors(n)]))-(&+[d*#([e: e in Divisors(d)]): d in Divisors(n)])): n in [1..1000]];
Numbers n such that A245212(n) < n.
+10
4
72, 96, 120, 144, 180, 192, 216, 240, 288, 336, 360, 384, 432, 480, 504, 528, 540, 576, 600, 624, 648, 672, 720, 756, 768, 792, 840, 864, 900, 936, 960, 972, 1008, 1056, 1080, 1120, 1152, 1176, 1200, 1224, 1248, 1260, 1280, 1296, 1320, 1344, 1368, 1440, 1512
COMMENTS
Numbers n such that A245212(n) = (n * tau(n)) - Sum_((d<n) | n) (d * tau(d)) < n. If d are divisors of n then values of sequence A245212(n) are the bending moments at point 0 of static forces of sizes tau(d) operating in places d on the cantilever as the nonnegative number axis of length n with support at point 0 by the schema: A245212(n) = (n * tau(n)) - Sum_((d<n) | n) (d * tau(d)).
EXAMPLE
Number 72 is in sequence because A245212(72) = 62 < 72.
PROG
(Magma) [n: n in [1..100000] | (2*(n*(#[d: d in Divisors(n)]))-(&+[d*#([e: e in Divisors(d)]): d in Divisors(n)])) lt n]
Numbers whose divisors have a mean abundancy index that is larger than 3.
+10
2
10886400, 13305600, 14515200, 18144000, 19958400, 21772800, 23587200, 23950080, 24192000, 25401600, 26611200, 27216000, 29030400, 29937600, 30481920, 31449600, 31933440, 32659200, 33264000, 33868800, 35380800, 35925120, 36288000, 37739520, 38102400, 39312000, 39916800
COMMENTS
The numbers whose mean abundancy index of divisors is larger than 2 are in A245214. The least odd term in this sequence is 84712751711029943302437712454902728115050897458369518458984375.
EXAMPLE
10886400 is a term since A374777(10886400)/ A374778(10886400) = 70644571/23514624 = 3.004... > 3.
MATHEMATICA
f[p_, e_] := ((e+1)*p^2 - (e+2)*p + p^(-e))/((e+1)*(p-1)^2); s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[4*10^7], s[#] > 3 &]
PROG
(PARI) is(n) = {my(f = factor(n)); prod(i = 1, #f~, p=f[i, 1]; e=f[i, 2]; (-2*p - e*p + p^2 + e*p^2 + p^(-e))/((e + 1)*(p - 1)^2)) > 3; }
Numbers whose unitary divisors have a mean unitary abundancy index that is larger than 2.
+10
2
223092870, 281291010, 300690390, 6469693230, 6915878970, 8254436190, 8720021310, 9146807670, 9592993410, 10407767370, 10485364890, 10555815270, 11125544430, 11532931410, 11797675890, 11823922110, 12095513430, 12328305990, 12598876290, 12929686770, 13162479330
COMMENTS
The least odd term is A070826(43) = 5.154... * 10^74, and the least term that is coprime to 6 is Product_{k=3..219} prime(k) = 1.0459... * 10^571. The least nonsquarefree ( A013929) term is a(613) = 148802944290 = 2 * 3 * 5 * 7 * 11 * 13 * 17 *19 * 23^2 * 29.
EXAMPLE
223092870 is a term since A374783(223092870)/ A374784(223092870) = 666225/330752 = 2.014... > 2.
MATHEMATICA
f[p_, e_] := 1 + 1/(2*p^e); r[1] = 1; r[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[4*10^8], s[#] > 2 &]
PROG
(PARI) is(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + 1/(2*f[i, 1]^f[i, 2])) > 2; }
Numbers whose infinitary divisors have a mean infinitary abundancy index that is larger than 2.
+10
2
7560, 9240, 10920, 83160, 98280, 120120, 120960, 128520, 143640, 147840, 154440, 157080, 173880, 174720, 175560, 185640, 189000, 190080, 201960, 207480, 212520, 216216, 219240, 224640, 225720, 228480, 231000, 234360, 238680, 251160, 255360, 266112, 266760, 267960
COMMENTS
The least odd term is 17737266779965459404793703604641625, and the least term that is coprime to 6 is 5^7 * (7 * 11 * ... * 23)^3 * 29 * 31 * ... * 751 = 3.140513... * 10^329.
EXAMPLE
7560 is a term since A374786(7560)/ A374787(7560) = 1045/512 = 2.041... > 2.
MATHEMATICA
f[p_, e_] := p^(2^(-1 + Position[Reverse@IntegerDigits[e, 2], _?(# == 1 &)])); q[1] = False; q[n_] := Times @@ (1 + 1/(2*Flatten@ (f @@@ FactorInteger[n]))) > 2; Select[Range[300000], q]
PROG
(PARI) is(n) = {my(f = factor(n), b); prod(i = 1, #f~, b = binary(f[i, 2]); prod(k=1, #b, if(b[k], 1 + 1/(2*f[i, 1]^(2^(#b-k))), 1))) > 2; }
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