Mauro Fiorentini - cp's OEIS frontend

A259039 Larger of a non-unitary amicable pair.

56, 248, 328, 496, 1016, 2032, 6462, 8128, 17412, 20538, 65528, 131056, 524224, 1048568, 2097136, 2096896, 4194296, 8388592, 8388544, 33554368, 33554176, 134217472, 2147467264, 8589918208

Offset: 1

Mauro Fiorentini, Jun 17 2015

The elements 2097136, 8388592, etc. are intentionally out of numerical order so that a(n) and A259038(n) form an amicable pair.

Cf. A034448, A259037, A259038.

a(23)-a(24) added by Amiram Eldar, Sep 27 2018 from the b-file at A259037.

A259038 Smaller of a non-unitary amicable pair.

Original entry on oeis.org

48, 192, 252, 448, 768, 1792, 3240, 7936, 11616, 11808, 49152, 114688, 507904, 786432, 1835008, 2080768, 3145728, 7340032, 8126464, 32505856, 33292288, 133169152, 2147221504, 8588886016

Offset: 1

Mauro Fiorentini, Jun 17 2015

This sequence is strictly increasing (and A259039, which contains the larger number in each pair, is sorted by this sequence).

Cf. A034448, A259037, A259039.

a(23)-a(24) added by Amiram Eldar, Sep 27 2018 from the b-file at A259037.

A259037 Non-unitary amicable numbers.

Original entry on oeis.org

48, 56, 192, 248, 252, 328, 448, 496, 768, 1016, 1792, 2032, 3240, 6462, 7936, 8128, 11616, 11808, 17412, 20538, 49152, 65528, 114688, 131056, 507904, 524224, 786432, 1048568, 1835008, 2080768, 2096896, 2097136, 3145728, 4194296, 7340032, 8126464, 8388544, 8388592, 32505856, 33292288, 33554176, 33554368, 133169152, 134217472

Offset: 1

Mauro Fiorentini, Jun 17 2015

nonn

A pair of integers x and y is called non-unitary amicable if the sum of the non-unitary divisors of either one is equal to the other. Union of A259038 and A259039.
The sequence lists the non-unitary amicable numbers in increasing order. Note that the pairs x, y are not always adjacent to each other in the list. See also A259038 for the x's, A259039 for the y's. The first time a pair is not adjacent is x = 11616, y = 17412 which correspond to a(17) and a(19), respectively.
No other pair below 10^9.
Ligh & Wall showed that if p and q are different Mersenne exponents (A000043) (i.e., 2^p - 1 and 2^q - 1 are Mersenne primes), then 2^(p+1) * (2^q-1) and 2^(q+1) * (2^p-1) is a nonunitary amicable pair. They also found the pairs (252, 328), (3240, 6462), (11616, 17412), (11808, 20538), which are all the known pairs that are not based on Mersenne primes. - Amiram Eldar, Sep 27 2018

			48 and 56 are in the sequence, as sigma(48)-usigma(48) = 56 and sigma(56)-usigma(56) = 48.

Charles R Greathouse IV, Table of n, a(n) for n = 1..48
Steve Ligh and Charles R. Wall, Functions of Nonunitary Divisors, Fibonacci Quarterly, Vol. 25 (1987), pp. 333-338.
Eric Weisstein's World of Mathematics, Unitary Divisor Function
Wikipedia, Unitary divisor

Subsequence of A013929.

Cf. A000043, A034448, A048146, A259038, A259039.

A048146(n)=my(f=factor(n)); sigma(f)-prod(i=1, #f~, f[i, 1]^f[i, 2]+1)
is(n)=my(k=A048146(n)); k>1 && k!=n && A048146(k)==n \\ Charles R Greathouse IV, Jun 17 2015

A256937 Numbers n such that phi(n) = 4*phi(n+1).

Original entry on oeis.org

629, 1469, 85139, 100889, 139859, 154979, 168149, 304079, 396899, 838199, 1107413, 1323449, 1465463, 2088839, 2160899, 2504879, 2684879, 2693249, 2800181, 3404609, 3512249, 3576869, 3885881, 4241819, 4500509, 4620659, 4822649, 5530709, 5805449

Offset: 1

Mauro Fiorentini, Apr 13 2015

nonn

			phi(629) = 576 = 4*phi(630).

Mauro Fiorentini, Table of n, a(n) for n = 1..154 (all terms for n up to 10^9).

Cf. A171262, A256907.

[n: n in [1..10^7] | EulerPhi(n) eq 4*EulerPhi(n+1)]; // Vincenzo Librandi, Apr 14 2015

A:= NULL:
y:= numtheory:-phi(1):
for n from 1 to 10^6 do
x:= numtheory:-phi(n+1);
if y = 4*x then A:= A, n fi;
y:= x;
od:
A;  # Robert Israel, Apr 15 2015

Select[Range@ 1000000, EulerPhi@ # == 4 EulerPhi[# + 1] &] (* Michael De Vlieger, Apr 13 2015 *)
Position[Partition[EulerPhi[Range[6*10^6]],2,1],?(#[[1]]==4#[[2]]&),{1},Heads->False]//Flatten (* _Harvey P. Dale, Sep 18 2016 *)

s=[]; for(n=1, 1000000, if(eulerphi(n)==4*eulerphi(n+1), s=concat(s, n))); s \\ Colin Barker, Apr 13 2015

[n for n in (1..1000000) if euler_phi(n) == 4*euler_phi(n+1)]; # Bruno Berselli, Apr 14 2015

A256907 Numbers n such that phi(n) = 3*phi(n+1).

Original entry on oeis.org

119, 527, 545, 2849, 3689, 4487, 6649, 18619, 26771, 30377, 44659, 47585, 50507, 76997, 83021, 102167, 112463, 128933, 138773, 163877, 174437, 192881, 193115, 198263, 217967, 236441, 243827, 244001, 254539, 268067, 282359, 287825, 298115, 345059, 410123, 464645

Offset: 1

Mauro Fiorentini, Apr 12 2015

nonn

			phi(545) = 3*phi(546) = 432.

Mauro Fiorentini, Table of n, a(n) for n = 1..755 (all terms with n <= 10^9).

Cf. A171262.

[n: n in [1..2*10^6] | EulerPhi(n) eq 3*EulerPhi(n+1)]; // Vincenzo Librandi, Apr 13 2015

Select[Range@ 1000000, EulerPhi@# == 3 EulerPhi[# + 1] &] (* Michael De Vlieger, Apr 12 2015 *)

s=[]; for(n=1, 1000000, if(eulerphi(n)==3*eulerphi(n+1), s=concat(s, n))); s \\ Colin Barker, Apr 12 2015

A248899 Numbers that are palindromic in bases 10 and 19.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 666, 838, 1771, 432234, 864468, 1551551, 1897981, 2211122, 155292551, 330050033, 453848354, 467535764, 650767056, 666909666, 857383758, 863828368, 47069796074, 62558085526, 67269596276, 87161116178, 96060106069, 121791197121, 127673376721, 139103301931, 234595595432, 246025520642

Offset: 1

Mauro Fiorentini, Mar 06 2015

Next term > 10^12.

			838 = 262 in base 19.

Cf. A007632, A007633, A029961, A029962, A029963, A029964, A029965, A029966, A029967, A029968, A029969, A029970, A029731, A097855, A248889.

[n: n in [0..2*10^7] | Intseq(n) eq Reverse(Intseq(n))and Intseq(n, 19) eq Reverse(Intseq(n, 19))]; // Vincenzo Librandi, Mar 08 2015

IsPalindromic := proc(n, Base)   local Conv, i;
   Conv := convert(n, base, Base);
for i from 1 to nops(Conv) / 2 do:
    if Conv [i] <> Conv [nops(Conv) + 1 - i] then
       return false:
    fi:
od:
return true;
end proc;
Base := 19;
A := [];
for i from 1 to 10^6 do:
   S := convert(i, base, 10);
   V := 0;
   if i mod 10 = 0 then
      next;
   fi;
   for j from 1 to nops(S) do:
      V := V * 10 + S [j];
   od:
   for j from 0 to 10 do:
      V1 := V * 10^(nops(S) + j) + i;
      if IsPalindromic(V1, Base) then
         A := [op(A), V1];
      fi;
   od:
   V1 := (V - (V mod 10)) * 10^(nops(S) - 1) + i;
   if IsPalindromic(V1, Base) then
      A := [op(A), V1];
   fi;
od:
sort(A);

palQ[n_, b_] := Block[{d = IntegerDigits[n, b]}, If[d == Reverse@ d, True, False]]; Select[Range[0, 10^6], And[palQ[#, 10], palQ[#, 19]] &] (* Michael De Vlieger, Mar 07 2015 *)
b1=10; b2=19; lst={}; Do[d1=IntegerDigits[n, b1]; d2=IntegerDigits[n, b2]; If[d1==Reverse[d1]&&d2==Reverse[d2], AppendTo[lst, n]], {n, 0, 10^7}]; lst (* Vincenzo Librandi, Mar 08 2015 *)

isok(n) = (n==0) || ((d = digits(n, 10)) && (Vecrev(d) == d) && (d = digits(n, 19)) && (Vecrev(d) == d)); \\ Michel Marcus, Mar 07 2015

A248889 Palindromic in base 10 and 18.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 171, 323, 343, 505, 595, 686, 848, 1661, 2112, 3773, 23332, 46664, 69996, 262262, 583385, 782287, 859958, 981189, 1254521, 1403041, 1832381, 39388393, 54411445, 55499455, 88844888, 118919811, 191010191

Offset: 1

Mauro Fiorentini, Mar 05 2015

a(54) > 10^12.

			848 in decimal is 2B2 in base 18, so 848 is in the sequence.
1661 in decimal is 525 in base 18, so 1661 is in the sequence.
1771 in decimal is 587 in base 18, which is not a palindrome, so 1771 is not in the sequence.

M. Fiorentini and Chai Wah Wu, Table of n, a(n) for n = 1..68 a(n) for n = 1..53 from M. Fiorentini.

Cf. A007632, A007633, A029961, A029962, A029963, A029964, A029965, A029966, A029967, A029968, A029969, A029970, A028731, A097855, A248899

[n: n in [0..2*10^7] | Intseq(n) eq Reverse(Intseq(n)) and Intseq(n,18) eq Reverse(Intseq(n,18))]; // Vincenzo Librandi, Mar 21 2015

IsPalindromic := proc(n, Base)
    local Conv, i;
    Conv := convert(n, base, Base);
    for i from 1 to nops(Conv) / 2 do
        if Conv [i] <> Conv [nops(Conv) + 1 - i] then
            return false;
        fi:
    od:
    true;
end proc:
Base := 18;
A := [];
for i from 1 to 10^6 do:
   S := convert(i, base, 10);
   V := 0;
   if i mod 10 = 0 then
      next;
   fi;
   for j from 1 to nops(S) do:
      V := V * 10 + S [j];
   od:
   for j from 0 to 10 do:
      V1 := V * 10^(nops(S) + j) + i;
      if IsPalindromic(V1, Base) then
         A := [op(A), V1];
      fi;
   od:
   V1 := (V - (V mod 10)) * 10^(nops(S) - 1) + i;
   if IsPalindromic(V1, Base) then
      A := [op(A), V1];
   fi;
od:
sort(A);

palindromicQ[n_, b_:10] := TrueQ[IntegerDigits[n, b] == Reverse[IntegerDigits[n, b]]]; Select[Range[0, 499], palindromicQ[#] && palindromicQ[#, 18] &] (* Alonso del Arte, Mar 21 2015 *)

isok(n) = (n==0) || ((d = digits(n, 10)) && (Vecrev(d) == d) && (d = digits(n, 18)) && (Vecrev(d) == d)); \\ Michel Marcus, Mar 14 2015

def palgen10(l): # generator of palindromes of length <= 2*l
    if l > 0:
        yield 0
        for x in range(1,l+1):
            n = 10**(x-1)
            n2 = n*10
            for y in range(n,n2):
                s = str(y)
                yield int(s+s[-2::-1])
            for y in range(n,n2):
                s = str(y)
                yield int(s+s[::-1])
def palcheck(n, b): # check if n is a palindrome in base b
    s = digits(n, b)
    return s == s[::-1]
A248889_list = [n for n in palgen10(9) if palcheck(n, 18)]
# Chai Wah Wu, Mar 23 2015

A128909 3D version of A005670. The problem is to dissect an n X n X n cube into smaller integer cubes, the gcd of whose sides is 1, using the smallest number of cubes. The gcd condition exclude dissecting a 6 X 6 X 6 cube into eight 3 X 3 X 3 cubes.

Original entry on oeis.org

1, 8, 20, 15, 50, 27, 71, 22, 39, 57, 125, 34

Offset: 1

Mauro Fiorentini, Apr 23 2007

As far as I know, no term, (except trivial cases) has been proved optimal. Repeated dissection, as in the above example, shows that if the side is a composite number mn, a(mn) <= a(m) + a(n) - 1. It is an open problem to find a number mn for which a(mn) < a(m) + a(n) - 1. Dissecting a cube with side n into a cube with side n - 1 and several unit cubes gives a trivial bound: a(n) <= 3n^2 - 3n + 2. Dissecting a cube with side n = 2k + 1 into a cube with side k + 1, 7 with side k and several unit cubes gives another trivial bound: a(n) <= (9n^2 - 12n + 31) / 4.

			a(4)=15 because a 4 X 4 X 4 cube can be dissected into 8 2 X 2 X 2, one of which can be dissected into 8 1 X 1 X 1.

Ainley, Stephen, Mathematical Puzzles, Prentice Hall, New York, 1983. p. 81.

Cf. A005670.

A028307 Form a triangle with n numbers in top row; all other numbers are the sum of their parents. E.g.: 4 1 2 7; 5 3 9; 8 12; 20. The numbers must be positive and distinct and the final number is to be minimized. Sequence gives final number.

Original entry on oeis.org

1, 3, 8, 20, 43, 98, 212, 465, 1000, 2144, 4497, 9504, 19872, 41455, 85356, 178630, 363467, 757085, 1541998, 3183600, 6515066, 13357593, 27432649, 55914902, 114683858, 233517515, 478061719, 972479046, 1986013932

Offset: 1

Mauro Fiorentini

Suggested by Problem 401 of the All-Soviet-Union Mathematical Competitions 1961-1986. Two different links are available for this collection.

			Solutions for n = 1, 2, ... are:
  1;
  1, 2;
  2, 1, 4;
  4, 1, 2, 7;
  7, 2, 1, 4, 6;
  8, 6, 1, 3, 2, 10;
  ...

Mauro Fiorentini, Further comments
Vladimir A. Pertsel, Problems of the All-Soviet-Union Mathematical Competitions 1961-1986

Cf. A028308, A062896, A340389.

From A.H.M. Smeets, Feb 25 2022: (Start)
a(n) > 2*a(n-1). Proof: Let x, y be the numbers in the second last row, then x >= a(n-1), y >= a(n-1) and x != y, so a(n) = x + y > 2*a(n-1).
It seems that a(n) > (4/3)*(2*a(n-1)-a(n-2)). (End)

More terms from the author, Jul 03 2001

cp's OEIS Frontend

User: Mauro Fiorentini

A259039 Larger of a non-unitary amicable pair.

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A259038 Smaller of a non-unitary amicable pair.

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A259037 Non-unitary amicable numbers.

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A256937 Numbers n such that phi(n) = 4*phi(n+1).

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Magma

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Sage

A256907 Numbers n such that phi(n) = 3*phi(n+1).

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Magma

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A248899 Numbers that are palindromic in bases 10 and 19.

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Magma

Maple

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A248889 Palindromic in base 10 and 18.

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Magma

Maple

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A128909 3D version of A005670. The problem is to dissect an n X n X n cube into smaller integer cubes, the gcd of whose sides is 1, using the smallest number of cubes. The gcd condition exclude dissecting a 6 X 6 X 6 cube into eight 3 X 3 X 3 cubes.

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