cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

A374780 Odd terms in A245214.

Original entry on oeis.org

10854718875, 12131744625, 13408770375, 18091198125, 19538493975, 20219574375, 21070924875, 22347950625, 22915517625, 23880381525, 24902002125, 25327677375, 28307404125, 28733079375, 29462808375, 32564156625, 35118208125, 36395233875, 39800635875, 40226311125
Offset: 1

Views

Author

Amiram Eldar, Jul 19 2024

Keywords

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.

Examples

			10854718875 is a term since it is odd and A374777(10854718875) / A374778(10854718875) = 11975203 / 5955950 = 2.0106... > 2.

Crossrefs

Subsequence of A245214.
Cf. A005231, A374777, A374778.

Programs

  • 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]
  • 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);

A374777 Numerator of the mean abundancy index of the divisors of n.

Original entry on oeis.org

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
Offset: 1

Views

Author

Amiram Eldar, Jul 19 2024

Keywords

Comments

First differs from A318491 at n = 27.
The abundancy index of a number k is sigma(k)/k = A017665(k)/A017666(k).

Examples

			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.

Links

Crossrefs

Cf. A000005, A000203, A013661, A017665, A017666, A038040, A060640, A245214, A318491, A318492, A374778 (denominators), A374779, A374780, A374781.

Programs

  • 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]
  • 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)));}

Formula

Let f(n) = a(n)/A374778(n). Then:
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).
f(n) = A318491(n)/(A318492(n)*A000005(n)).
f(n) = (Sum_{d|n} d*tau(d)) / (n*tau(n)) = A060640(n)/A038040(n).
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).

A245211 a(n) = Sum_{(d

Original entry on oeis.org

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
Offset: 1

Views

Author

Jaroslav Krizek, Jul 23 2014

Keywords

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.

Examples

			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.

Links

Crossrefs

Cf. A000005, A100484, A245212, A245213, A245214.

Programs

  • Magma
    [(&+[d*#([e: e in Divisors(d)]): d in Divisors(n)])-(n*(#[d: d in Divisors(n)])): n in [1..1000]];
  • PARI
    a(n) = sumdiv(n, d, (dJens Kruse Andersen, Aug 13 2014

Formula

a(n) = A060640(n) - A038040(n) = Sum_{d | n} (d * tau(d)) - n*tau(n).
a(n) = A038040(n) - A245212(n).
a(n) = 1 for n = primes.
a(n) = n + 5 for even semiprimes q = 2p > 4 (see A100484) where p = odd prime.

A245212 a(n) = n * tau(n) - Sum_{(d

Original entry on oeis.org

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
Offset: 1

Views

Author

Jaroslav Krizek, Jul 23 2014

Keywords

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
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).

Examples

			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.

Links

Crossrefs

Cf. A000005, A100484, A245211, A245213, A245214.

Programs

  • Magma
    [(2*(n*(#[d: d in Divisors(n)]))-(&+[d*#([e: e in Divisors(d)]): d in Divisors(n)])): n in [1..1000]];
  • PARI
    a(n) = sumdiv(n, d, (-1)^(dJens Kruse Andersen, Aug 13 2014

Formula

a(n) = A038040(n) - A245211(n).
a(n) = 2 * A038040(n) - A060640(n) = 2 * (n * tau(n)) - Sum_{d | n} (d * tau(d)).

A245213 Numbers n such that A245212(n) < n.

Original entry on oeis.org

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
Offset: 1

Views

Author

Jaroslav Krizek, Jul 23 2014

Keywords

Comments

Numbers n such that A245212(n) = (n * tau(n)) - Sum_((d
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

Examples

			Number 72 is in sequence because A245212(72) = 62 < 72.

Links

Crossrefs

Cf. A000005, A100484, A245211, A245212, A245214.

Programs

  • 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]

A374779 Numbers whose divisors have a mean abundancy index that is larger than 3.

Original entry on oeis.org

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
Offset: 1

Views

Author

Amiram Eldar, Jul 19 2024

Keywords

Comments

Numbers k such that A374777(k)/A374778(k) > 3.
The numbers whose mean abundancy index of divisors is larger than 2 are in A245214.
The least odd term in this sequence is 84712751711029943302437712454902728115050897458369518458984375.

Examples

			10886400 is a term since A374777(10886400)/A374778(10886400) = 70644571/23514624 = 3.004... > 3.

Links

Crossrefs

Subsequence of A245214.
Cf. A068403, A374777, A374778.

Programs

  • 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 &]
  • 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;}

A374785 Numbers whose unitary divisors have a mean unitary abundancy index that is larger than 2.

Original entry on oeis.org

223092870, 281291010, 300690390, 6469693230, 6915878970, 8254436190, 8720021310, 9146807670, 9592993410, 10407767370, 10485364890, 10555815270, 11125544430, 11532931410, 11797675890, 11823922110, 12095513430, 12328305990, 12598876290, 12929686770, 13162479330
Offset: 1

Views

Author

Amiram Eldar, Jul 20 2024

Keywords

Comments

Numbers k such that A374783(k)/A374784(k) > 2.
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.
All the terms are nonpowerful numbers (A052485). For powerful numbers (A001694) k, A374783/(k)/A374784(k) < Product_{p prime} (1 + 1/(2*p)) = 1.242534... (A366586).

Examples

			223092870 is a term since A374783(223092870)/A374784(223092870) = 666225/330752 = 2.014... > 2.

Links

Crossrefs

Subsequence of A052485.
Cf. A001221, A001694, A013929, A034683, A070826, A366586, A374783, A374784.
Similar sequences: A245214, A374788.

Programs

  • 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 &]
  • PARI
    is(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + 1/(2*f[i,1]^f[i,2])) > 2;}

Formula

A001221(a(n)) >= 9.

A374788 Numbers whose infinitary divisors have a mean infinitary abundancy index that is larger than 2.

Original entry on oeis.org

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
Offset: 1

Views

Author

Amiram Eldar, Jul 20 2024

Keywords

Comments

Numbers k such that A374786(k)/A374787(k) > 2.
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.

Examples

			7560 is a term since A374786(7560)/A374787(7560) = 1045/512 = 2.041... > 2.

Links

Crossrefs

Cf. A129656, A374786, A374787.
Similar sequences: A245214, A374785.

Programs

  • 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]
  • 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;}
Showing 1-8 of 8 results.

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