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 1-6 of 6 results.

A363170 The number of powerful abundant numbers (A363169) not exceeding 10^n.

Original entry on oeis.org

0, 3, 23, 82, 297, 1000, 3268, 10534, 33799, 107901, 343155, 1090189, 3460380, 10970774, 34749182, 109991778, 348006756, 1101058505, 3483105232, 11017518803
Offset: 1

Views

Author

Amiram Eldar, May 19 2023

Keywords

Comments

The ratios a(n)/A118896(n) seem to converge to a positive value as n grows: for n = 14..20 they are 0.506417..., 0.506728..., 0.506863..., 0.506890..., 0.506987..., 0.507059..., 0.507120... .
Conjecture: the asymptotic relative density of the abundant numbers within the powerful numbers exists and equals 0.507... .

Examples

			a(2) = 3 since there are 3 powerful abundant numbers not exceeding 10^2: 36, 72 and 100.

Crossrefs

Cf. A001694, A005101, A118896, A302992, A363169.

Programs

  • Mathematica
    seq[nmax_] := Module[{c = 0, p = 10, k = 1, kmax = 10^nmax, s = {}}, While[k <= kmax, If[DivisorSigma[-1, k] > 2 && Min[FactorInteger[k][[;;, 2]]] > 1, c++]; If[k == p, AppendTo[s, c]; p *= 10]; k++]; s]; seq[5]
  • PARI
    is(n) = { my(f = factor(n)); n > 1 && vecmin(f[, 2]) > 1 && sigma(f, -1) > 2; } \\ A363169
lista(nmax) = {my(c = 0, p = 10, k = 1, kmax = 10^nmax); while(k <= kmax, if(is(k), c++); if(k == p, print1(c, ", "); p *= 10); k++); }

A063734 Square abundant numbers.

Original entry on oeis.org

36, 100, 144, 196, 324, 400, 576, 784, 900, 1296, 1600, 1764, 1936, 2304, 2500, 2704, 2916, 3136, 3600, 4356, 4624, 4900, 5184, 5776, 6084, 6400, 7056, 7744, 8100, 8464, 9216, 9604, 10000, 10404, 10816, 11025, 11664, 12100, 12544, 12996, 13456, 14400
Offset: 1

Views

Author

Jason Earls, Aug 13 2001

Keywords

Links

Crossrefs

Intersection of A000290 and A005101.
Subsequence of A363169.
Cf. A381738.

Programs

  • Mathematica
    Select[Range[150]^2,DivisorSigma[1,#]>2#&] (* Harvey P. Dale, Aug 14 2014 *)
  • PARI
    j=[]; for(n=1,25000, if(sigma(n)>(2*n),a=n; if(issquare(a),j=concat(j,a)))); j
  • PARI
    { n=0; for (m=1, 10^9, if (issquare(m) && sigma(m)>(2*m), write("b063734.txt", n++, " ", m); if (n==1000, break)) ) } \\ Harry J. Smith, Aug 29 2009
  • PARI
    { n=0; for (m=1, 10^9, s=m^2; if (sigma(s)>(2*s), write("b063734.txt", n++, " ", s); if (n==1000, break)) ) } \\ Harry J. Smith, Aug 29 2009

Formula

a(n) = A381738(n)^2. - Amiram Eldar, Mar 07 2025

A363176 Primitive abundant numbers (A091191) that are powerful numbers (A001694).

Original entry on oeis.org

196, 15376, 342225, 570375, 1032256, 3172468, 4636684, 63126063, 99198099, 117234117, 171991125, 280495504, 319600125, 327921075, 404529741, 581549787, 635689593, 762155163, 1029447225, 1148667664, 1356949503, 1435045924, 1501500375, 1558495125, 1596961444, 1757705625
Offset: 1

Views

Author

Amiram Eldar, May 19 2023

Keywords

Comments

The least cubefull (A036966) term is a(158) = 26376098024367 = 3^6 * 7^4 * 13^3 * 19^3.
A363175 is a subsequence. Terms that are not in A363175: 196, 15376, 1032256, 274810802176, 1125882727038976, 72057319160283136, ... .

Links

Crossrefs

Intersection of A001694 and A091191.
A363175 is a subsequence.
Subsequence of A363169.
Cf. A036966.

Programs

  • Mathematica
    f1[p_, e_] := (p^(e + 1) - 1)/(p^(e + 1) - p^e); f2[p_, e_] := (p^(e + 1) - p)/(p^(e + 1) - 1);
primAbQ[n_] := (r = Times @@ f1 @@@ (f = FactorInteger[n])) > 2 && r * Max @@ f2 @@@ f <= 2;
seq[max_] := Module[{pow = Union[Flatten[Table[i^2*j^3, {j, 1, max^(1/3)}, {i, 1, Sqrt[max/j^3]}]]]}, Select[Rest[pow], primAbQ]]; seq[10^10]
  • PARI
    isPrimAb(n) = {my(f = factor(n), r, p, e); r = sigma(f, -1); r > 2 && vecmax(vector(#f~, i, p = f[i, 1]; e = f[i, 2]; (p^(e + 1) - p)/(p^(e + 1) - 1))) * r <= 2; }
lista(lim) = {my(pow = List(), t); for(j=1, sqrtnint(lim\1, 3), for(i=1, sqrtint(lim\j^3), listput(pow, i^2*j^3))); select(x->isPrimAb(x), Set(pow)); }

A363171 Numbers k such that A064549(k) is an abundant number (A005101).

Original entry on oeis.org

6, 10, 12, 14, 18, 20, 24, 28, 30, 36, 40, 42, 44, 48, 50, 52, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96, 98, 100, 102, 104, 105, 108, 110, 112, 114, 120, 126, 130, 132, 136, 138, 140, 144, 150, 152, 154, 156, 160, 162, 168, 170, 174, 176, 180, 182, 184, 186
Offset: 1

Views

Author

Amiram Eldar, May 19 2023

Keywords

Comments

First differs from A334166 at n = 21.
The least odd term is a(33) = 105, and the least term that is coprime to 6 is a(11850456) = 37182145.
The ordered values of A003557(A363169(n)): m is a powerful abundant number (A363169) if and only if A003557(m) is in this sequence.
If k is a term then any positive multiple of k is also a term. The primitive terms are in A363172.
The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 2, 30, 322, 3201, 31863, 318336, 3188014, 31855257, 318427893, 3184885813, 31853300276, ... . Apparently, the asymptotic density of this sequence exists and equals 0.3185... .

Links

Crossrefs

Cf. A003557, A064549, A334166.
Subsequences: A005101, A363169, A363172.

Programs

  • Mathematica
    q[n_] := DivisorSigma[-1, n * Times @@ FactorInteger[n][[;; , 1]]] > 2; Select[Range[200], q]
  • PARI
    A064549(n) = { my(f=factor(n)); prod(i=1, #f~, f[i, 1]^(f[i, 2]+1)); };
is(n) = sigma(A064549(n), -1) > 2;

A363175 Primitive abundant numbers (A071395) that are powerful numbers (A001694).

Original entry on oeis.org

342225, 570375, 3172468, 4636684, 63126063, 99198099, 117234117, 171991125, 280495504, 319600125, 327921075, 404529741, 581549787, 635689593, 762155163, 1029447225, 1148667664, 1356949503, 1435045924, 1501500375, 1558495125, 1596961444, 1757705625, 1778362047
Offset: 1

Views

Author

Amiram Eldar, May 19 2023

Keywords

Comments

The least cubefull (A036966) term is a(154) = A363177(1) = 26376098024367 = 3^6 * 7^4 * 13^3 * 19^3.

Links

Crossrefs

Intersection of A001694 and A071395.
Subsequence of A363169 and A363176.
Subsequences: A306796, A306797, A363177.
Cf. A036966.

Programs

  • Mathematica
    f1[p_, e_] := (p^(e + 1) - 1)/(p^(e + 1) - p^e); f2[p_, e_] := (p^(e + 1) - p)/(p^(e + 1) - 1);
primAbQ[n_] := (r = Times @@ f1 @@@ (f = FactorInteger[n])) > 2 && r * Max @@ f2 @@@ f < 2;
seq[max_] := Module[{pow = Union[Flatten[Table[i^2*j^3, {j, 1, max^(1/3)}, {i, 1, Sqrt[max/j^3]}]]]}, Select[Rest[pow], primAbQ]]; seq[10^10]
  • PARI
    isPrimAb(n) = {my(f = factor(n), r, p, e); r = sigma(f, -1); r > 2 && vecmax(vector(#f~, i, p = f[i, 1]; e = f[i, 2]; (p^(e + 1) - p)/(p^(e + 1) - 1))) * r < 2; }
lista(lim) = {my(pow = List(), t); for(j=1, sqrtnint(lim\1, 3), for(i=1, sqrtint(lim\j^3), listput(pow, i^2*j^3))); select(x->isPrimAb(x), Set(pow)); }

A363177 Primitive abundant numbers (A071395) that are cubefull numbers (A036966).

Original entry on oeis.org

26376098024367, 33912126031329, 1910383099764867, 2792098376579421, 5229860083034911875, 6886512413632368153, 8815747507513708671, 28966027524687899919, 42200802302982406288, 89594138836162749375, 224439112362213402759, 288564573037131517833, 512767531125033485625
Offset: 1

Views

Author

Amiram Eldar, May 19 2023

Keywords

Comments

It seems that this sequence is also the intersection of A036966 and A091191 (checked up to 10^27).
Are there terms that are 4-full numbers (A036967)? There are none below 10^27.

Links

Crossrefs

Intersection of A036966 and A071395.
Subsequence of A363169 and A363175.
A306797 is a subsequence.
Cf. A036967, A091191.
Showing 1-6 of 6 results.

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