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 1-4 of 4 results.

A362054 Primitive unitary abundant numbers k (A302573) whose unitary abundancy index usigma(k)/k has a record low value.

Original entry on oeis.org

70, 1092, 1428, 1596, 4030, 5830, 50388, 133042, 216300, 269990, 437745, 442365, 4199030, 22982388, 124540390, 361745930, 507298090, 541900788, 624032630, 1113445430, 3002432810, 6771402960, 13455037365, 17242767300, 60428265370
Offset: 1

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Author

Amiram Eldar, Apr 06 2023

Keywords

Comments

The unitary abundancy index of an integer k is usigma(k)/k, where usigma is the sum-of-unitary-divisors function (A034448).
Terms k of A302573 such that usigma(k)/k < usigma(m)/m for all smaller terms m < k of A302573.

Examples

			The unitary abundancy indices of the first terms are 72/35 > 80/39 > 240/119 > 800/399 > 4032/2015 > 5832/2915 > ... > 2.

Crossrefs

The unitary version of A362053.
Cf. A034448, A302573.

Programs

  • Mathematica
    f1[p_, e_] := 1 + 1/p^e; f2[p_, e_] := p^e/(p^e + 1);
(* Returns the unitary abundancy index of n if n is primitive unitary abundant,and 0 otherwise: *)
uabIndex[n_] := If[(r = Times @@ f1 @@@ (f = FactorInteger[n])) > 2 && r * Max @@ f2 @@@ f < 2, r, 0]; uabIndex[1] = 0;
seq[kmax_] := Module[{s = {}, uab, uabm = 3}, Do[If[0 < (uab = uabIndex[k]) < uabm, uabm = uab; AppendTo[s, k]], {k, 1, kmax}]; s]; seq[10^6]
  • PARI
    uabindex(n) = {my(f = factor(n), r); r = prod(i = 1, #f~, 1 + 1/f[i, 1]^f[i, 2]); if(r <= 2, return(0)); if(vecmax(vector(#f~, i, f[i, 1]^f[i, 2]/(f[i, 1]^f[i, 2] + 1))) * r < 2, r, 0);} \\ Returns the unitary abundancy index of n if n is primitive unitary abundant, and 0 otherwise.
lista(kmax) = {my(uab, uabm = 3); for(k = 1, kmax, uab = uabindex(k); if(uab > 0 && uab < uabm, uabm = uab; print1(k, ", ")));}

A302574 Primitive unitary abundant numbers (definition 2): unitary abundant numbers (A034683) having no unitary abundant proper unitary divisor.

Original entry on oeis.org

30, 42, 66, 70, 78, 102, 114, 138, 150, 174, 186, 222, 246, 258, 282, 294, 318, 354, 366, 402, 420, 426, 438, 474, 498, 534, 582, 606, 618, 642, 654, 660, 678, 726, 750, 762, 780, 786, 822, 834, 840, 894, 906, 924, 942, 978, 990, 1002, 1014, 1020, 1038, 1074
Offset: 1

Views

Author

Amiram Eldar, Apr 10 2018

Keywords

Comments

The unitary analog of A091191.

Examples

			70 is primitive unitary abundant since it is unitary abundant (usigma(70) = 144 > 2*70), and all of its unitary divisors are unitary deficient. 210 is unitary abundant since usigma(210) = 576 > 2*210, but is not in this sequence since 70 is one of its unitary divisors, and 70 is unitary abundant.

Links

Crossrefs

Cf. A034448, A034683, A091191, A129487, A302573.

Programs

  • Mathematica
    usigma[n_] := If[n == 1, 1, Times @@ (1 + Power @@@ FactorInteger[n])]; delta[n_] := usigma[n]-2n; udefQ[n_] := Module[{}, v=Most[Module[{d = Divisors[n]}, Select[ d, GCD[ #, n/# ] == 1 &]]]; u = Max[Map[delta,v]]; u<=0 ]; puaQ[n_] := delta[n] > 0 && udefQ[n]; Select[Range[10000],puaQ]

A372298 Primitive infinitary abundant numbers (definition 1): infinitary abundant numbers (A129656) whose all proper infinitary divisors are infinitary deficient numbers.

Original entry on oeis.org

40, 56, 70, 72, 88, 104, 756, 924, 945, 1092, 1188, 1344, 1386, 1428, 1430, 1596, 1638, 1760, 1870, 2002, 2016, 2080, 2090, 2142, 2176, 2210, 2394, 2432, 2470, 2530, 2584, 2720, 2750, 2944, 2990, 3040, 3128, 3190, 3200, 3230, 3250, 3400, 3410, 3496, 3712, 3770
Offset: 1

Views

Author

Amiram Eldar, Apr 25 2024

Keywords

Examples

			40 is a term since it is an infinitary abundant number and all its proper infinitary divisors, {1, 2, 4, 5, 8, 10, 20}, are infinitary deficient numbers.
24 and 30, which are infinitary abundant numbers, are not primitive, because they are divisible by 6 which is an infinitary perfect number.

Links

Crossrefs

Subsequence of A129656 and A372299.
A372300 is a subsequence.
Cf. A007357, A077609, A129657.
Similar sequences: A071395, A298973, A302573, A307112, A307114, A307115.

Programs

  • Mathematica
    f[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 @@ f @@@ FactorInteger[n]; idefQ[n_] := isigma[n] < 2*n; idivs[1] = {1};
idivs[n_] := Sort@ Flatten@ Outer[Times, Sequence @@ (FactorInteger[n] /. {p_, e_Integer} :> p^Select[Range[0, e], BitOr[e, #] == e &])];
q[n_] := Module[{d = idivs[n]}, Total[d] > 2*n && AllTrue[Most[d], idefQ]]; Select[Range[4000], q]
  • PARI
    isidiv(d, f) = {if (d==1, return (1)); for (k=1, #f~, bne = binary(f[k, 2]); bde = binary(valuation(d, f[k, 1])); if (#bde < #bne, bde = concat(vector(#bne-#bde), bde)); for (j=1, #bne, if (! bne[j] && bde[j], return (0)); ); ); return (1); }
idivs(n) = {my(f = factor(n), d = divisors(f), idiv = []); for (k=1, #d, if (isidiv(d[k], f), idiv = concat(idiv, d[k])); ); idiv; } \\ Michel Marcus at A077609
isigma(n) = {my(f = factor(n), b); prod(i=1, #f~, b = binary(f[i, 2]); prod(k=1, #b, if(b[k], 1+f[i, 1]^(2^(#b-k)), 1)))} ;
is(n) = isigma(n) > 2*n && select(x -> x < n && isigma(x) >= 2*x, idivs(n)) == [];

A361935 Numbers k such that k and k+1 are both primitive unitary abundant numbers (definition 2, A302574).

Original entry on oeis.org

2457405145194, 2601523139214, 3320774552094, 3490250769005, 3733421997305, 3934651766045, 3954730124345, 4514767592334, 4553585751714, 4563327473705, 5226433847634
Offset: 1

Views

Author

Amiram Eldar, Mar 31 2023

Keywords

Comments

There are no more terms below 10^13.
There are no numbers k such that k and k+1 are both unitary abundant numbers with definition 1 (A302573) below 10^13.

Crossrefs

Subsequence of A034683, A302574 and A331412.
Cf. A302573.
Similar sequences: A283418, A330872.

Programs

  • Mathematica
    f1[p_, e_] := 1 + 1/p^e; f2[p_, e_] := p^e/(p^e + 1);
puabQ[n_] := (r = Times @@ f1 @@@ (f = FactorInteger[n])) > 2 && r * Max @@ f2 @@@ f <= 2;
Select[Import["https://oeis.org/A331412/b331412.txt", "Table"][[;; , 2]], puabQ[#] && puabQ[# + 1] &]
Showing 1-4 of 4 results.

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