A179651 -id:A179651 - cp's OEIS frontend

A364706 a(n) is the least number k such that the k-th difference between consecutive practical numbers, A179651(k), equals 2*n, or -1 if no such k exists.

2, 5, 16, 33, 85, 46, 331, 188, 171, 300, 1986, 962, 3321, 968, 2316, 6514, 9974, 3219, 12162, 3831, 4588, 20585, 30099, 22005, 30465, 33485, 28874, 35901, 136396, 48483, 120127, 34145, 140589, 233364, 126080, 185421, 607164, 279989, 359002, 327768, 609867, 354143

Offset: 1

Amiram Eldar, Aug 04 2023

nonn

All the practical number except for 1 are even. Therefore all the differences between consecutive practical numbers are either 1 or even.

			a(1) = 2 since A179651(2) = 2 = 2*1.
a(2) = 5 since A179651(5) = 4 = 2*2.
a(3) = 16 since A179651(16) = 6 = 2*3.

Amiram Eldar, Table of n, a(n) for n = 1..150

Cf. A005153, A179651, A330870, A364707.

f[p_, e_] := (p^(e + 1) - 1)/(p - 1); pracQ[n_] := (ind = Position[(fct = FactorInteger[n])[[;; , 1]]/(1 + FoldList[Times, 1, f @@@ Most @ fct]), _?(# > 1 &)]) == {};
seq[len_, nmax_] := Module[{s = Table[0, {len}], n = 2, prev = 2, k = 2, c = 0, i}, While[c < len && n <= nmax, n+=2; If[pracQ[n], i = (n - prev)/2; If[i <= len && s[[i]] == 0, c++; s[[i]] = k]; prev= n; k++]]; s]; seq[20, 10^6]

A179651(a(n)) = 2*n.

A005153(a(n)) = A364707(n).

A364707 a(n) is the least practical number A005153(k) such that A005153(k+1) - A005153(k) = 2*n, or -1 if no such number exists.

Original entry on oeis.org

2, 8, 42, 112, 368, 180, 1806, 936, 840, 1600, 14168, 6216, 25120, 6272, 16770, 52668, 83720, 24240, 103840, 29440, 35910, 184140, 278334, 197912, 282150, 313040, 266112, 337840, 1438722, 468540, 1254016, 319808, 1486584, 2566432, 1321376, 2003688, 7163646, 3121328

Offset: 1

Amiram Eldar, Aug 04 2023

nonn

			a(1) = 2 since A005153(2) = 2 and A005153(3) = 4 = 2 + 2*1.
a(2) = 8 since A005153(5) = 8 and A005153(6) = 12 = 8 + 2*2.
a(3) = 42 since A005153(16) = 42 and A005153(17) = 48 = 42 + 2*3.

Amiram Eldar, Table of n, a(n) for n = 1..150

Cf. A005153, A179651, A330870, A364706.

f[p_, e_] := (p^(e + 1) - 1)/(p - 1); pracQ[n_] := (ind = Position[(fct = FactorInteger[n])[[;; , 1]]/(1 + FoldList[Times, 1, f @@@ Most @ fct]), _?(# > 1 &)]) == {};
seq[len_, nmax_] := Module[{s = Table[0, {len}], n = 2, prev = 2, c = 0, i}, While[c < len && n <= nmax, n+=2; If[pracQ[n], i = (n - prev)/2; If[i <= len && s[[i]] == 0, c++; s[[i]] = prev]; prev= n]]; s]; seq[20, 10^6]

a(n) = A005153(A364706(n)).

cp's OEIS Frontend

A364706 a(n) is the least number k such that the k-th difference between consecutive practical numbers, A179651(k), equals 2*n, or -1 if no such k exists.

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