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.

A334901 Infinitary practical numbers: numbers m such that every number 1 <= k <= isigma(m) is a sum of distinct infinitary divisors of m, where isigma is A049417.

Original entry on oeis.org

1, 2, 6, 8, 24, 30, 40, 42, 54, 56, 66, 72, 78, 88, 104, 120, 128, 168, 210, 216, 264, 270, 280, 312, 330, 360, 378, 384, 390, 408, 440, 456, 462, 480, 504, 510, 520, 546, 552, 570, 594, 600, 616, 640, 672, 680, 690, 696, 702, 714, 728, 744, 750, 760, 792, 798
Offset: 1

Views

Author

Amiram Eldar, May 16 2020

Keywords

Comments

Includes the powers of 2 of the form 2^(2^k - 1) for k = 0, 1, ... (A058891). The other terms are a subset of infinitary abundant numbers (A129656) and infinitary pseudoperfect numbers (A306983).

Links

Crossrefs

The infinitary version of A005153.
Cf. A049417, A058891, A129656, A286652, A306983, A334898.

Programs

  • Mathematica
    bin[n_] := 2^(-1 + Position[Reverse @ IntegerDigits[n, 2], ?(# == 1 &)] // Flatten); f[p, e_] := p^bin[e]; icomp[n_] := Flatten[f @@@ FactorInteger[n]]; fun[p_, e_] := Module[{b = IntegerDigits[e, 2]}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]]; isigma[1] = 1; isigma[n_] := Times @@ fun @@@ FactorInteger[n]; infPracQ[n_] := Module[{f, p, e, prod = 1, ok = True}, If[n < 1 || (n > 1 && OddQ[n]), False, If[n == 1, True, r = Sort[icomp[n]]; Do[If[r[[i]] > 1 + isigma[prod], ok = False; Break[]]; prod = prod*r[[i]], {i, Length[r]}]; ok]]]; Select[Range[1000], infPracQ]

A287173 Unitary practical numbers that are nonsquarefree.

Original entry on oeis.org

1050, 1470, 1650, 1950, 3234, 3822, 8250, 9438, 9750, 11550, 13650, 16170, 17850, 19110, 19950, 21450, 24150, 24990, 25410, 27930, 28050, 30450, 31350, 32550, 33150, 33810, 35490, 37050, 37950, 38850, 42042, 42630, 43050, 44850, 45150, 45570, 47190, 47850
Offset: 1

Views

Author

Amiram Eldar, May 24 2017

Keywords

Comments

The squarefree terms of both practical numbers (A005153) and unitary practical numbers (A286652) are the same, A265501.

Links

Crossrefs

Cf. A005153, A034448, A265501, A287173, A286652.

Programs

  • Mathematica
    usigma[n_] := Block[{d = Divisors[n]}, Plus @@ Select[d, GCD[ #, n/# ] == 1 &]]; uPracticalQ[n_] := Module[{f, p, e, prod=1, ok=True}, If[n<1 || (n>1 && OddQ[n]), False, If[n==1, True, f=FactorInteger[n]; {p, e} = Transpose[f]; r = Sort[p^e]; Do[If[r[[i]] > 1+usigma[prod], ok=False; Break[]]; prod=prod*r[[i]], {i, Length[p]}]; ok]]]; aQ[n_]:=!SquareFreeQ[n]&&uPracticalQ[n];Select[Range[100000], aQ]

A334898 Bi-unitary practical numbers: numbers m such that every number 1 <= k <= bsigma(m) is a sum of distinct bi-unitary divisors of m, where bsigma is A188999.

Original entry on oeis.org

1, 2, 6, 8, 24, 30, 32, 40, 42, 48, 54, 56, 66, 72, 78, 88, 96, 104, 120, 128, 160, 168, 192, 210, 216, 224, 240, 264, 270, 280, 288, 312, 320, 330, 336, 352, 360, 378, 384, 390, 408, 416, 432, 440, 448, 456, 462, 480, 486, 504, 510, 512, 520, 528, 544, 546, 552
Offset: 1

Views

Author

Amiram Eldar, May 16 2020

Keywords

Comments

Includes 1 and all the odd powers of 2 (A004171). The other terms are a subset of bi-unitary abundant numbers (A292982) and bi-unitary pseudoperfect numbers (A292985).

Links

Crossrefs

The bi-unitary version of A005153.
Cf. A004171, A188999, A286652, A292982, A292985, A334901.

Programs

  • Mathematica
    biunitaryDivisorQ[div_, n_] := If[Mod[#2, #1] == 0, Last @ Apply[Intersection, Map[Select[Divisors[#], Function[d, CoprimeQ[d, #/d]]] &, {#1, #2/#1}]] == 1, False] & @@ {div, n}; bdivs[n_] := Module[{d = Divisors[n]}, Select[d, biunitaryDivisorQ[#, n] &]]; bPracQ[n_] := Module[{d = bdivs[n], sd, x}, sd = Plus @@ d; Min @ CoefficientList[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, sd}], x] >  0]; Select[Range[1000], bPracQ]

A342400 a(n) is the number of distinct sums of distinct unitary divisors of n.

Original entry on oeis.org

1, 3, 3, 3, 3, 12, 3, 3, 3, 15, 3, 13, 3, 15, 15, 3, 3, 15, 3, 13, 15, 15, 3, 15, 3, 15, 3, 15, 3, 72, 3, 3, 15, 15, 15, 15, 3, 15, 15, 15, 3, 96, 3, 15, 15, 15, 3, 15, 3, 15, 15, 15, 3, 15, 15, 13, 15, 15, 3, 108, 3, 15, 15, 3, 15, 144, 3, 15, 15, 142, 3, 13
Offset: 1

Views

Author

Amiram Eldar, Mar 10 2021

Keywords

Links

Crossrefs

The unitary version of A119347.
Cf. A034448, A077610, A286652.

Programs

  • Mathematica
    a[n_] := Module[{d = Select[Divisors[n], CoprimeQ[#, n/#] &], x, s, m, c}, m = Length[d]; s = Plus @@ d; c = Rest@CoefficientList[Series[Product[1 + x^d[[i]], {i, 1, m}], {x, 0, s}], x]; Count[c, _?(# > 0 &)]]; Array[a, 100]

Formula

a(n) <= A034448(n) with equality if and only if n is a unitary practical number (A286652).
a(p^e) = 3 for a prime p and e >= 1.

A346245 Numbers k for which A003415(k) > k*A003557(k).

Original entry on oeis.org

30, 210, 330, 390, 462, 510, 546, 570, 690, 714, 798, 858, 870, 930, 966, 1110, 1218, 1230, 1290, 1302, 1410, 1554, 1590, 1722, 1770, 1830, 2010, 2130, 2190, 2310, 2370, 2490, 2670, 2730, 2910, 3030, 3090, 3210, 3270, 3390, 3570, 3810, 3930, 3990, 4110, 4170, 4290, 4470, 4530, 4710, 4830, 4890, 5010, 5190, 5370, 5430
Offset: 1

Views

Author

Antti Karttunen, Jul 19 2021

Keywords

Comments

Numbers k such that A342001(k) > k.
Numbers k such that their arithmetic derivative (A003415(k)) is larger than A102631(k), k^2 / (squarefree kernel of k).

Crossrefs

Positions of negative terms in A346244.
Cf. A003415, A003557, A102631, A342001.
Seems to have many common terms with A181629, A265501 and A286652.
Showing 1-5 of 5 results.

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