A287681 -id:A287681 - cp's OEIS frontend

A334882 Numbers k such that k and k+2 are both primitive practical numbers (A267124).

28, 304, 306, 340, 460, 462, 858, 868, 1482, 1768, 1974, 2440, 2728, 2838, 2860, 3318, 3738, 4134, 4264, 4288, 4420, 4422, 5236, 5694, 6100, 6102, 7590, 8814, 9040, 9042, 10218, 11128, 11620, 11778, 12558, 12978, 13110, 14320, 14382, 14670, 15568, 16048, 16110

Offset: 1

Amiram Eldar, May 14 2020

nonn

			28 is a term since 28 and 28 + 2 = 30 are both primitive practical numbers.

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

Cf. A267124, A287681, A334883.

f[p_, e_] := (p^(e + 1) - 1)/(p - 1); pracQ[fct_] := (ind = Position[fct[[;; , 1]]/(1 + FoldList[Times, 1, f @@@ Most@fct]), ?(# > 1 &)]) == {}; pracTestQ[fct, k_] := Module[{f = fct}, f[[k, 2]] -= 1; pracQ[f]]; primPracQ[n_] := Module[{fct = FactorInteger[n]}, pracQ[fct] && AllTrue[Range@Length[fct], fct[[#, 2]] == 1 || ! pracTestQ[fct, #] &]]; Select[Range[2, 16200, 2], primPracQ[#] && primPracQ[# + 2] &]

A287682 Triples of practical numbers: numbers n such that n-2, n, n+2 are all practical numbers.

Original entry on oeis.org

4, 6, 18, 30, 198, 306, 462, 702, 1482, 2550, 3330, 4422, 5778, 6102, 6498, 9042, 11178, 11778, 14418, 15498, 17298, 17442, 19458, 20862, 21582, 22878, 23322, 23550, 25230, 26622, 26862, 26910, 27378, 30210, 34542, 36738, 38610, 39006, 39102, 40350, 40662

Offset: 1

Amiram Eldar, May 29 2017

nonn

Melfi proved that this sequence is infinite.

Amiram Eldar, Table of n, a(n) for n = 1..1000
Giuseppe Melfi, A survey on practical numbers, Rend. Sem. Mat. Univ. Pol. Torino, 53, (1995), 347-359.
Giuseppe Melfi, Triplets of practical numbers

Cf. A005153, A209236, A287681, A287683.

practicalQ[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]; Do[If[p[[i]] > 1+DivisorSigma[1, prod], ok=False; Break[]]; prod=prod*p[[i]]^e[[i]], {i, Length[p]}]; ok]]];
a={}; p1=False; p2=False; k=2; While[Length[a]<100, p3=practicalQ[k]; If[p1 && p2 && p3, a=AppendTo[a,k-2]]; p1 = p2; p2 = p3; k+=2];a

A287683 5-tuples of practical numbers: numbers n such that n-6, n-2, n, n+2, n+6 are all practical numbers.

Original entry on oeis.org

18, 30, 198, 306, 462, 1482, 2550, 4422, 17298, 23322, 23550, 40350, 52578, 67938, 88506, 92202, 96222, 123006, 131070, 219102, 226182, 237690, 277506, 312702, 359658, 432822, 526878, 533370, 584166, 659934, 1032858, 1051650, 1140414, 1142658, 1243170, 1255422

Offset: 1

Amiram Eldar, May 29 2017

nonn

Melfi conjectured that this sequence is infinite.

Amiram Eldar and Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 100 terms from Amiram Eldar)
Giuseppe Melfi, A survey on practical numbers, Rend. Sem. Mat. Univ. Pol. Torino, 53, (1995), 347-359.
Giuseppe Melfi, On 5-tuples of twin practical numbers, Bollettino della Unione Matematica Italiana, Serie 8, Vol. 2-B, No. 3 (1999), pp. 723-734.
Giuseppe Melfi, 5-uples of practical numbers

Cf. A005153, A287681, A287682.

prQ[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]; Do[If[p[[i]] > 1+DivisorSigma[1, prod], ok=False; Break[]]; prod=prod*p[[i]]^e[[i]], {i, Length[p]}]; ok]]];
quintupleQ[n_] := prQ[n-6]&&prQ[n-2]&&prQ[n]&&prQ[n+2]&&prQ[n+6];
a={}; k=8; While[Length[a]<100, If[quintupleQ[k], a=AppendTo[a,k]]; k+=2];a

A330871 Numbers k such that k and k+1 are both phi-practical numbers (A260653).

Original entry on oeis.org

1, 2, 3, 15, 255, 735, 2624, 3135, 4095, 4784, 5264, 5984, 7215, 7424, 7904, 9344, 10064, 10335, 10815, 11024, 11984, 12375, 12495, 13695, 16184, 16575, 22575, 22784, 22815, 26144, 26264, 27104, 30015, 30855, 30975, 32384, 33824, 34335, 34544, 38024, 38415, 39104

Offset: 1

Amiram Eldar, Apr 29 2020

nonn

			1 is a term since both 1 and 2 are phi-practical numbers.

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

Cf. A260653, A287681.

phiPracticalQ[n_] := If[n<1, False, If[n==1, True, (lst = Sort @ EulerPhi @ Divisors[n]; ok=True; Do[If[lst[[m]]>Sum[lst[[l]], {l, 1, m-1}]+1, (ok=False; Break[])], {m, 1, Length[lst]}]; ok)]]; Select[Range[40000], phiPracticalQ[#] && phiPracticalQ[#+1] &] (* after Frank M Jackson at A260653 *)

A294225 Practical numbers q with q + 2 and q^2 + 2 both practical.

Original entry on oeis.org

2, 4, 520, 2560, 3100, 4648, 6448, 6784, 7252, 11128, 12400, 15496, 19264, 26128, 26752, 26860, 28768, 31648, 32368, 36160, 37408, 41728, 45400, 48760, 53248, 53584, 54832, 57148, 58828, 63544, 66820, 68440, 68500, 73948, 74176, 80512, 81508, 84208, 93184, 94300, 106780, 112288, 113968, 118528, 131068

Offset: 1

Zhi-Wei Sun, Oct 25 2017

nonn

Conjecture: The sequence has infinitely many terms.
In 1996 G. Melfi proved that there are infinitely many positive integers q with q and q + 2 both practical.
As any practical number greater than 2 is a multiple of 4 or 6, when q > 2, q + 2 and q^2 + 2 are all practical, we must have q^2 + 2 == 0 (mod 6), hence q is not divisible by 3 and thus 4 | q and 6 | (q + 2), therefore q == 4 (mod 12).

			a(1) = 2 since 2, 2 + 2 = 4 and 2^2 + 2 = 6 are all practical.
a(2) = 4 since 4, 4 + 2 = 6 and 4^2 + 2 = 18 are all practical.

Zhi-Wei Sun, Table of n, a(n) for n = 1..2500
G. Melfi, On two conjectures about practical numbers, J. Number Theory 56 (1996) 205-210.
Zhi-Wei Sun, Conjectures on representations involving primes, in: M. B. Nathanson (ed.), Combinatorial and Additive Number Theory II: CANT, New York, NY, USA, 2015 and 2016, Springer Proc. in Math. & Stat., Vol. 220, Springer, New York, 2017.

Cf. A005153, A287681, A294112, A294200.

f[n_]:=f[n]=FactorInteger[n];
Pow[n_, i_]:=Pow[n,i]=Part[Part[f[n], i], 1]^(Part[Part[f[n], i], 2]);
Con[n_]:=Con[n]=Sum[If[Part[Part[f[n], s+1], 1]<=DivisorSigma[1, Product[Pow[n, i], {i, 1, s}]]+1, 0, 1], {s, 1, Length[f[n]]-1}];
pr[n_]:=pr[n]=n>0&&(n<3||Mod[n, 2]+Con[n]==0);
pq[n_]:=pq[n]=pr[n]&&pr[n+2]&&pr[n^2+2];
tab={};Do[If[pq[k],tab=Append[tab,k]],{k,1,132000}];Print[tab]

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

Original entry on oeis.org

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

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A334882 Numbers k such that k and k+2 are both primitive practical numbers (A267124).

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A287682 Triples of practical numbers: numbers n such that n-2, n, n+2 are all practical numbers.

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A287683 5-tuples of practical numbers: numbers n such that n-6, n-2, n, n+2, n+6 are all practical numbers.

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A330871 Numbers k such that k and k+1 are both phi-practical numbers (A260653).

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A294225 Practical numbers q with q + 2 and q^2 + 2 both practical.

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

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Mathematica

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

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