A085254 -id:A085254 - cp's OEIS frontend

A085252 Number of ways to write n as sum of two powerful numbers (A001694).

0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 2, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 2, 1, 1, 2, 1, 0, 0, 2, 2, 0, 1, 1, 1, 0, 0, 1, 0, 2, 0, 2, 1, 1, 0, 0, 2, 1, 1, 0, 1, 0, 1, 1, 2, 0, 0, 2, 0, 0, 0, 2, 2, 1, 0, 2, 0, 0, 0, 2, 2, 1, 0, 0, 2, 0, 0, 1, 2, 1, 1, 0, 0, 0, 0, 1, 2, 1, 1, 1, 1, 0

Offset: 1

Reinhard Zumkeller, Jun 23 2003

nonn

			a(81) = 2: 81 = 9 + 72 = A001694(4) + A001694(12) = 32 + 49 = A001694(8) + A001694(10).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Powerful Number.

Cf. A001694, A076871, A085253, A085254, A085255.

With[{m = 120}, pow = Select[Range[m], # == 1 || Min[FactorInteger[#][[;; , 2]]] > 1 &]; BinCounts[Select[Plus @@@ Union[Sort /@ Tuples[pow, {2}]], # <= m &], {1, m, 1}]] (* Amiram Eldar, Jan 30 2023 *)

a(A085253(n)) = 0.
a(A076871(n)) > 0.
a(A085254(n)) = 1.
a(A085255(n)) > 1.

A085255 Numbers having at least two representations as a sum of two powerful numbers (A001694).

Original entry on oeis.org

17, 33, 36, 40, 41, 50, 52, 57, 65, 68, 72, 73, 76, 80, 81, 85, 89, 97, 104, 108, 109, 113, 116, 117, 125, 129, 130, 132, 133, 136, 137, 144, 145, 148, 152, 153, 157, 160, 164, 170, 172, 177, 180, 185, 189, 193, 197, 200, 201, 204, 205, 208, 209, 216, 221

Offset: 1

Reinhard Zumkeller, Jun 23 2003

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Powerful Number.

Cf. A001694, A076871, A085252, A085253, A085254.

With[{m = 222}, pow = Select[Range[m], # == 1 || Min[FactorInteger[#][[;; , 2]]] > 1 &]; Position[BinCounts[Select[Plus @@@ Union[Sort /@ Tuples[pow, {2}]], # <= m &], {1, m, 1}], ?(# > 1 &)] // Flatten] (* _Amiram Eldar, Jan 30 2023 *)

A085252(a(n)) > 1.

A115354 a(n) is the smallest number representable in exactly n ways as a sum of 2 powerful(1) numbers.

Original entry on oeis.org

2, 17, 108, 153, 297, 657, 1764, 2052, 4644, 6156, 10800, 16200, 22932, 29000, 11025, 54225, 92025, 68796, 100548, 99225, 44100, 88200, 264600, 431244, 176400, 441000, 666468, 1151172, 352800, 617400, 396900, 926100, 980100, 793800, 1234800

Offset: 1

Giovanni Resta, Jan 21 2006

nonn

Here we are considering powerful numbers (first definition) A001694. Note that, by definition, 1 is powerful.

			a(2)=17, since 17 = 16+1 = 8+9.

Donovan Johnson, Table of n, a(n) for n = 1..120
Eric Weisstein's World of Mathematics, Powerful Number.

Cf. A001694, A085252, A085253, A085254, A085255, A063274, A115355.

pwfQ[n_] := n == 1 || Min[Transpose[FactorInteger@n][[2]]] > 1; lim=200000; pt = Select[Range[lim], pwfQ]; t = Table[0, {i, lim}]; Do[v = pt[[i]]+ pt[[j]]; If[v<=lim, t[[v]]++ ], {i, Length@pt}, {j, i}]; Table[Position[t, k][[1, 1]], {k, 22}]

a(23)-a(35) from Donovan Johnson, Dec 07 2008

A115355 a(n) is the smallest number representable in exactly n ways as a sum of 3 powerful(1) numbers.

Original entry on oeis.org

3, 17, 33, 41, 66, 77, 89, 117, 133, 145, 153, 189, 161, 225, 301, 257, 324, 333, 341, 297, 432, 425, 369, 517, 613, 441, 521, 585, 513, 809, 689, 792, 657, 1001, 801, 881, 1000, 1017, 873, 945, 900, 1265, 1169, 1425, 1089, 1125, 1197, 1481, 1161, 1584

Offset: 1

Giovanni Resta, Jan 21 2006

nonn

Here we are considering powerful numbers (first definition) A001694. Note that, by definition, 1 is powerful.

			a(2) = 17 since 17 = 4+4+9 = 8+8+1.

Donovan Johnson, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Powerful Number.

Cf. A001694, A085252, A085253, A085254, A085255, A063274, A115354.

pwfQ[n_] := n==1 || Min[Transpose[FactorInteger@n][[2]]] > 1; lim = 5000; pt = Select[Range[lim], pwfQ]; t = Table[0, {i, lim}]; Do[v = pt[[i]]+pt[[j]]+pt[[k]]; If[v <= lim, t[[v]]++ ], {i, Length@pt}, {j, i}, {k, j}]; Table[Position[t, k][[1, 1]], {k, 60}]

cp's OEIS Frontend

A085252 Number of ways to write n as sum of two powerful numbers (A001694).

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A085255 Numbers having at least two representations as a sum of two powerful numbers (A001694).

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A115354 a(n) is the smallest number representable in exactly n ways as a sum of 2 powerful(1) numbers.

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A115355 a(n) is the smallest number representable in exactly n ways as a sum of 3 powerful(1) numbers.

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