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 1-5 of 5 results.

A088240 Values associated with sequence A078511.

Original entry on oeis.org

2, 12, 12, 16, 16, 18, 20, 24, 24, 24, 24, 28, 24, 24, 28, 32, 36, 36, 32, 32, 40, 36, 32, 32, 48, 48, 36, 36, 32, 48, 36, 48, 48, 32, 48, 30, 36, 48, 48, 40, 48, 36, 36, 48, 48, 40, 60, 40, 48, 48, 64, 40, 48, 48, 48, 64, 40, 48, 48, 36, 64, 48, 64, 60, 48, 48, 64, 60, 36, 48
Offset: 1

Views

Author

Lekraj Beedassy, Nov 03 2003

Keywords

Links

Crossrefs

Cf. A000005, A001414, A078511.

Programs

  • Mathematica
    s = {}; Do[If[(d = DivisorSigma[0, n]) == Total[Times @@@ FactorInteger[n]], AppendTo[s, d]], {n, 2, 100000}]; s (* Amiram Eldar, Jun 22 2019 *)
  • PARI
    isA078511(n) = {my(f = factor(n)); sum(i=1, #f~, f[i,1]*f[i,2]) == numdiv(n);}
lista(nn) = for (n=1, nn, if (isA078511(n), print1(numdiv(n), ", "))); \\ Michel Marcus, Feb 11 2015

Formula

a(n) = A000005(A078511(n)) = A001414(A078511(n)).

Extensions

More terms from Ray Chandler, Nov 05 2003

A087004 Numbers whose number of divisors equals the sum of their separate prime-power decompositions.

Original entry on oeis.org

2, 60, 120, 180, 504, 720, 11550, 12180, 17940, 19380, 21252, 22230, 26334, 27846, 29172, 32340, 34440, 34580, 43470, 48840, 56430, 59220, 59670, 63240, 66120, 70686, 82824, 85140, 91350, 95700, 95940, 99528, 112840, 113220, 113652, 115368
Offset: 1

Views

Author

Lekraj Beedassy, Oct 13 2003

Keywords

Examples

			504 = 2^3*3^2*7 is in the sequence because d(504) = A000005(504) = (3+1)*(2+1)*(1+1) = 24 = 2^3 + 3^2 + 7. Similarly for 32340 = 2^2*3*5*7^2*11, where d(32340) = 2^2 + 3 + 5 + 7^2 + 11 = 72.

References

  • S. Kahan, "Divisor Advisory", Journal of Recreational Mathematics 30(1) 41-4 1999-2000 Baywood NY.

Links

Crossrefs

Cf. A078511.

Programs

  • Mathematica
    ndppdQ[n_]:=DivisorSigma[0,n]==Total[#[[1]]^#[[2]]&/@FactorInteger[n]]; Select[Range[2,120000],ndppdQ] (* Harvey P. Dale, Nov 22 2013 *)
  • PARI
    isok(n) = my(f = factor(n)); numdiv(n) == sum(i=1, #f~, f[i,1]^f[i,2]); \\ Michel Marcus, Oct 26 2013

Extensions

a(1) = 2 prepended by Michel Marcus, Oct 26 2013

A305026 Numbers k such that sopfr(k) = tau(k)^2.

Original entry on oeis.org

39, 55, 354, 578, 1634, 1644, 6604, 8253, 9825, 12573, 13516, 14749, 15244, 16684, 18669, 18672, 19276, 19564, 21032, 22225, 25305, 28449, 29853, 31688, 33633, 35793, 41261, 41768, 41949, 42813, 48013, 50670, 54048, 59750, 60804, 63609, 63869, 65265, 78832
Offset: 1

Views

Author

Parker Grootenhuis, May 23 2018

Keywords

Comments

For numbers k that satisfy the condition, tau(k) will always be even because tau(k) is odd only if k is a square, but if k is a square then sopfr(k) is even (because every prime appears with an even exponent) and thus it cannot be equal to tau(k)^2 which is odd as tau(k). - Giovanni Resta, May 24 2018

Links

Crossrefs

Cf. A000005, A001414, A078511.

Programs

  • Mathematica
    Rest@ Select[Range[10^5], Total[Times @@@ FactorInteger@ #] == DivisorSigma[0, #]^2 &] (* Michael De Vlieger, May 27 2018 *)
  • PARI
    isok(n) = my(f=factor(n)); sum(k=1,#f~,f[k,1]*f[k,2]) == numdiv(n)^2; \\ Michel Marcus, May 24 2018

Formula

k such that A001414(k) = A000005(k)^2.

A248662 Numbers with the property: tau(n) > sopfr(n), or A000005(n) > A001414(n).

Original entry on oeis.org

1, 120, 144, 180, 216, 240, 252, 288, 300, 336, 360, 420, 432, 480, 504, 540, 576, 600, 630, 648, 660, 672, 720, 756, 792, 810, 840, 864, 900, 960, 1008, 1050, 1080, 1120, 1152, 1176, 1200, 1260, 1296, 1320, 1344, 1350, 1400, 1440, 1500, 1512, 1560
Offset: 1

Views

Author

Richard R. Forberg, Jan 15 2015

Keywords

Comments

The number of divisors exceeds the sum of its prime factors, with repetition.
These are a subset of the abundant numbers = A005101.
The numbers where tau(n) = sopfr(n) are given by A078511.

Links

Crossrefs

Cf. A000005, A001414, A005101, A078511.

Programs

  • Mathematica
    ResultList = {1}; Do[
If[ (DivisorSigma[0, k] > Total[Times @@@ FactorInteger[k]]),
  AppendTo[ResultList, k]], {k, 2, 10000}]; ResultList
  • PARI
    isok(n) = my(f=factor(n)); sum(i=1,#f~,f[i,1]*f[i,2]) < numdiv(n); \\ Michel Marcus, Jun 22 2019

Extensions

a(1) = 1 inserted by Amiram Eldar, Jun 22 2019

A305349 Numbers k such that sopfr(k) = tau(k)^3.

Original entry on oeis.org

183, 295, 583, 799, 943, 7042, 10978, 13581, 18658, 20652, 22402, 22898, 29698, 40162, 43522, 48442, 54778, 59362, 62338, 68098, 74938, 82618, 87418, 89722, 97282, 99298, 102202, 108418, 110842, 113122, 116602, 118498, 122362, 123322, 123778, 128482, 128698
Offset: 1

Views

Author

Parker Grootenhuis, May 30 2018

Keywords

Comments

Numbers k such that A001414(k) = A000005(k)^3.
For numbers k that satisfy the condition, tau(k) will always be even because tau(k) is odd only if k is a square, but if k is a square then sopfr(k) is even (because every prime appears with an even exponent) and thus it cannot be equal to tau(k)^3 which is odd as tau(k).
A squarefree number k = p_1*...*p_j is in the sequence if p_1 + ... + p_j = 8^j. It is likely that 8^j is the sum of j distinct primes for all j >= 2. - Robert Israel, Dec 10 2018

Links

Crossrefs

Cf. A000005, A001414, A078511, A305026.

Programs

  • Maple
    filter:= proc(n) local F;
  F:= ifactors(n)[2];
  add(t[1]*t[2],t=F) = mul(t[2]+1,t=F)^3
end proc:
select(filter, [$1..200000]); # Robert Israel, Dec 10 2018
  • Mathematica
    sopf[n_] := If[n==1,0,Plus@@Times@@@FactorInteger@ n];Select[Range[200000],sopf[#]==DivisorSigma[0,#]^3 &] (* Amiram Eldar, Nov 01 2018 *)
  • PARI
    sopfr(n) = my(f=factor(n)); sum(k=1, matsize(f)[1], f[k, 1]*f[k, 2]);
isok(n) = sopfr(n) == numdiv(n)^3; \\ Michel Marcus, Nov 02 2018
Showing 1-5 of 5 results.

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