A330871 -id:A330871 - cp's OEIS frontend

A334900 Numbers k such that k and k+2 are both bi-unitary practical numbers (A334898).

6, 30, 40, 54, 510, 544, 798, 918, 928, 1120, 1240, 1288, 1408, 1480, 1566, 1672, 1720, 1768, 1792, 1888, 1950, 1974, 2046, 2430, 2440, 2560, 2728, 2814, 2838, 2968, 3198, 3318, 4134, 4158, 4264, 4422, 4480, 4758, 5248, 6102, 6270, 6424, 6942, 7590, 7830, 9280

Offset: 1

Amiram Eldar, May 16 2020

nonn

			6 is a term since 6 and 6 + 2 = 8 are both bi-unitary practical numbers.

Amiram Eldar, Table of n, a(n) for n = 1..112

Cf. A287681, A330871, A334882, A334898, A334903.

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]; seq = {}; q1 = bPracQ[2]; Do[q2 = bPracQ[n]; If[q1 && q2, AppendTo[seq, n - 2]]; q1 = q2, {n, 4, 1000, 2}]; seq

A334903 Numbers k such that k and k+2 are both infinitary practical numbers (A334901).

Original entry on oeis.org

6, 40, 54, 918, 1240, 1288, 1408, 1480, 1672, 1720, 1768, 1974, 2440, 2728, 2838, 2968, 3198, 3318, 4134, 4264, 4422, 4480, 4758, 5248, 6102, 6270, 6424, 7590, 7830, 10624, 11128, 13110, 13182, 14248, 15496, 15928, 16254, 16768, 18088, 19864, 21112, 21318, 21630

Offset: 1

Amiram Eldar, May 16 2020

nonn

			6 is a term since 6 and 6 + 2 = 8 are both infinitary practical numbers.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A287681, A330871, A334882, A334900, A334901.

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]]]; seq = {}; q1 = infPracQ[2]; Do[q2 = infPracQ[n]; If[q1 && q2, AppendTo[seq, n - 2]]; q1 = q2, {n, 4, 10^4, 2}]; seq

cp's OEIS Frontend

A334900 Numbers k such that k and k+2 are both bi-unitary practical numbers (A334898).

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