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 1-8 of 8 results.

A076871 Sum of two powerful numbers (definition (1), A001694).

Original entry on oeis.org

2, 5, 8, 9, 10, 12, 13, 16, 17, 18, 20, 24, 25, 26, 28, 29, 31, 32, 33, 34, 35, 36, 37, 40, 41, 43, 44, 45, 48, 50, 52, 53, 54, 57, 58, 59, 61, 63, 64, 65, 68, 72, 73, 74, 76, 80, 81, 82, 85, 88, 89, 90, 91, 96, 97, 98, 99, 100, 101, 104, 106, 108, 109, 112, 113, 116, 117
Offset: 1

Views

Author

N. J. A. Sloane, Nov 25 2002

Keywords

Comments

Complement of A085253. - Reinhard Zumkeller, Jun 23 2003

References

  • Aleksandar Ivić, The Riemann Zeta-Function, Wiley, NY, 1985, see p. 439.

Links

Crossrefs

Cf. A001694, A076872, A085252, A085253, A261883.
Different from A070049.

Programs

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

Formula

A085252(a(n)) > 0. - Reinhard Zumkeller, Jun 23 2003
Blomer shows that there are x/log^k x sums of two powerful numbers up to x, where k = 0.20629947... is A261883. - Charles R Greathouse IV, Sep 04 2015

Extensions

More terms from Vladeta Jovovic, Nov 25 2002

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

Original entry on oeis.org

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

Views

Author

Reinhard Zumkeller, Jun 23 2003

Keywords

Examples

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

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)

Formula

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

A085254 Numbers having a unique representation as sum of two powerful numbers (A001694).

Original entry on oeis.org

2, 5, 8, 9, 10, 12, 13, 16, 18, 20, 24, 25, 26, 28, 29, 31, 32, 34, 35, 37, 43, 44, 45, 48, 53, 54, 58, 59, 61, 63, 64, 74, 82, 88, 90, 91, 96, 98, 99, 100, 101, 106, 112, 121, 122, 124, 126, 127, 128, 134, 135, 140, 141, 146, 149, 150, 155, 161, 162, 169, 171
Offset: 1

Views

Author

Reinhard Zumkeller, Jun 23 2003

Keywords

Links

Crossrefs

Cf. A001694, A076871, A085252, A085253, A085255.

Programs

  • Mathematica
    With[{m = 180}, 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 *)

Formula

A085252(a(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

Views

Author

Reinhard Zumkeller, Jun 23 2003

Keywords

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)

Formula

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

Views

Author

Giovanni Resta, Jan 21 2006

Keywords

Comments

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

Examples

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

Links

Crossrefs

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

Programs

  • Mathematica
    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}]

Extensions

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

Views

Author

Giovanni Resta, Jan 21 2006

Keywords

Comments

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

Examples

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

Links

Crossrefs

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

Programs

  • Mathematica
    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}]

A331802 Integers having no representation as sum of two nonsquarefree numbers.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 14, 15, 19, 23
Offset: 1

Views

Author

Bernard Schott, Feb 23 2020

Keywords

Comments

This sequence is finite with 14 terms and 23 is the largest term (see Prime Curios link); a proof can be found in comments of A331801.

Examples

			With the two smallest nonsquarefree numbers 4 and 8, it is not possible to get 1, 2, 3, 4, 5, 6, 7, 9, 10 and 11 as sum of two nonsquarefree numbers.

Links

Crossrefs

Cf. A005117 (squarefree), A013929 (nonsquarefree), A331801 (complement).
Cf. A000404 (sum of 2 nonzero squares), A018825 (not the sum of 2 nonzero squares).
Cf. A001694 (squareful), A052485 (not squareful), A076871 (sum of 2 squareful), A085253 (not the sum of 2 squareful).

Programs

  • Mathematica
    max = 25; Complement[Range[max], Union @ Select[Total /@ Tuples[Select[Range[max], !SquareFreeQ[#] &], 2], # <= max &]] (* Amiram Eldar, Feb 24 2020 *)

A331801 Integers that are sum of two nonsquarefree numbers.

Original entry on oeis.org

8, 12, 13, 16, 17, 18, 20, 21, 22, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85
Offset: 1

Views

Author

Bernard Schott, Jan 26 2020

Keywords

Comments

Proposition: All integers > 23 are terms of this sequence (see link Prime Curios!).
Proof by exhaustion:
1) For numbers {4*k} with k>=6, then 4*k = 4*(k-1) + 4 is a term as 4*(k-1) and 4 are nonsquarefree;
2) For numbers {4*k+1} with k>=6, then 4*k+1 = 4*(k-2) + 9 is a term as 4*(k-2) and 9 are nonsquarefree;
3) For numbers {4*k+2} with k>=6, then 4*k+2 = 4*(k-4) + 18 is a term as 4*(k-4) and 18 are nonsquarefree;
4) For numbers {4*k+3}; with k=6, 27 = 9+18 is a term as 9 and 18 are nonsquarefree, and with k>=7, 4*k+3 = 4*(k-6) + 27 is also a term as 4*(k-6) and 27 are nonsquarefree.
Conclusion: every integer > 23 is sum of two nonsquarefree numbers (QED).

Examples

			13 = 4 + 9 and 21 = 9 + 12 are terms of this sequence as 4, 9 and 12 are nonsquarefree numbers.

Links

Crossrefs

Cf. A005117 (squarefree), A013929 (nonsquarefree), A331802 (complement).
Cf. A000404 (sum of 2 nonzero squares), A018825 (not the sum of 2 nonzero squares).
Cf. A001694 (squareful), A052485 (not squareful), A076871 (sum of 2 squareful), A085253 (not the sum of 2 squareful).

Programs

  • Mathematica
    max = 85; Union @ Select[Total /@ Tuples[Select[Range[max], !SquareFreeQ[#] &], 2], # <= max &] (* Amiram Eldar, Feb 04 2020 *)
Join[{8,12,13,16,17,18,20,21,22},Range[24,100]] (* or *) Complement[Range[100],{1,2,3,4,5,6,7,9,10,11,14,15,19,23}] (* Harvey P. Dale, Dec 04 2024 *)
  • PARI
    isok(m) = {for (i=1, m-1, if (!issquarefree(i) && !issquarefree(m-i), return (1));); return(0);} \\ Michel Marcus, Jan 31 2020
Showing 1-8 of 8 results.

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