Raghavendra Ugare - cp's OEIS frontend

User: Raghavendra Ugare

Raghavendra Ugare has authored 2 sequences.

A239282 a(n) = A045917(n)*prime(n).

0, 3, 5, 7, 22, 13, 34, 38, 46, 58, 93, 111, 123, 86, 141, 106, 236, 244, 134, 213, 292, 237, 332, 445, 388, 303, 515, 321, 436, 678, 381, 655, 822, 278, 745, 906, 785, 815, 1169, 692, 895, 1448, 955, 772, 1773, 796, 1055, 1561, 681, 1374, 1864, 1195, 1446, 2008

Offset: 1

Raghavendra Ugare, Mar 14 2014

nonn

Cf. A045917, A000040

(*Returns the various ways a number (presumed to be even) can be split as a sum of 2 Primes.*)
getGoldBachSplits[n_Integer] := Module[{i, splits = {}, a, b},
  (
   For[i = 1,
    Prime[i] < n,
    i++,
    a = Prime[i] ;
    b = n - Prime[i];
    If[PrimeQ[b],
     If[MemberQ[splits, {b, a}], Null,
      AppendTo[splits, {a, b}]],
     Null];
    ]; (*End For-loop...*)
   splits
   )
  ]
(* Now we generate our series...*)
series[n_] :=
Module[{}, (Table[Prime[i]*Length[getGoldBachSplits[2 i]], {i, 2, n}])]

A191867 Numbers n which are both the sum of two nonzero squares and the concatenation of the decimal representation of two nonzero squares.

Original entry on oeis.org

41, 116, 125, 136, 149, 164, 169, 181, 369, 416, 425, 436, 449, 464, 481, 641, 916, 925, 936, 949, 964, 981, 1009, 1225, 1256, 1289, 1361, 1576, 1616, 1625, 1636, 1649, 1664, 1681, 1961, 2516, 2525, 2536, 2549, 2561, 2564, 2581, 3616, 3625, 3636, 3649, 3664

Offset: 1

Raghavendra Ugare, Jun 18 2011

It would be interesting to investigate such numbers w.r.t. higher powers and larger n.

			The smallest such number is 41, since it is both the sum of two squares (i.e., 4^2, 5^2) and the concatenation of two squares (i.e., 2^2, 1^2).
3649 also belongs to this sequence because it is sum of two squares (i.e., 60^2, 7^2) and the concatenation of two squares (i.e., 6^2, 7^2).

Klaus Brockhaus, Table of n, a(n) for n = 1..1000

Cf. A000404, A048385.

z:=65; T:=Sort([ s: a in [b..z], b in [1..z] | s le z^2 where s is a^2+b^2 ]); SplitToSquares:=function(n); V:=[]; S:=Intseq(n); for j in [1..#S-1] do A:=[ S[k]: k in [1..j] ]; a:=Seqint(A); B:=[ S[k]: k in [j+1..#S] ]; b:=Seqint(B); if a gt 0 and A[#A] gt 0 and IsSquare(a) and IsSquare(b) then Append(~V, []); end if; end for; return V; end function; U:=[ p: j in [1..#T] | P ne [] where P is SplitToSquares(p) where p is T[j] ]; [ U[j]: j in [1..#U] | j eq 1 or U[j-1] ne U[j] ]; // Klaus Brockhaus, Jun 19 2011

(* find numbers that can be split as the SUM of two powers (squares, cubes, etc.) and also as CONCATENATION of the same powers *)
siamesePowers[n_, power_] := Module[
{listOfSumOfPowers, a, b, i, listOfConcatenatedPowers},
listOfSumOfPowers = Outer[Plus, Table[{i^power}, {i, 1, n}], Table[{i^power}, {i, 1, n}]] // Flatten;
concatNumbers[a_, b_] := IntegerDigits[{a, b}] // Flatten // FromDigits;
listOfConcatenatedPowers := Outer[concatNumbers, Table[i^power, {i, 1, n}], Table[i^power, {i, 1, n}]] // Flatten;
(* The intersection of these 2 lists is the set of our special Siamese numbers *)
Intersection[listOfSumOfPowers, listOfConcatenatedPowers]
]
siamesePowers[30, 2] (* Generate the first 30 such numbers for squares *)

is_A191867(n) = for(p=10,n, issquare(n\p) && issquare(n%p) && n%p*10>=p && return(is_A000404(n)); p=p*10-1)  \\ M. F. Hasler, Jun 19 2011

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User: Raghavendra Ugare

A239282 a(n) = A045917(n)*prime(n).

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A191867 Numbers n which are both the sum of two nonzero squares and the concatenation of the decimal representation of two nonzero squares.

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