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 1-10 of 12 results. Next

A293184 Numbers k such that bphi(k) = bphi(k+1), where bphi(k) is the bi-unitary analog of Euler's totient function (A116550).

Original entry on oeis.org

1, 14, 20, 57, 187, 188, 916, 1603, 93928, 142891, 432976, 549815, 692259, 773887, 872191, 4297168, 9478088, 127162432, 127991488, 129015616, 132527167
Offset: 1

Views

Author

Amiram Eldar, Oct 01 2017

Keywords

Comments

187 is the first solution to bphi(k) = bphi(k+1) = bphi(k+2).
a(22) > 1.6*10^9, if it exists. - Amiram Eldar, Jul 16 2022

Examples

			14 is in the sequence since bphi(14) = bphi(15) = 9.

Crossrefs

Cf. A001274, A116550, A287055, A294030.

Programs

  • Mathematica
    bphi[1] = 1; bphi[n_] := With[{pp = Power @@@ FactorInteger[n]},   Count[Range[n], m_ /; Intersection[pp, Power @@@ FactorInteger[m]] == {}]]; a={}; b1=0; Do[b2 = bphi[k];If[b1 == b2, a = AppendTo[a, k - 1]]; b1 = b2, {k, 1, 10^3}]; a (* after Jean-François Alcover at A116550 *)
  • PARI
    udivs(n) = {my(d = divisors(n)); select(x->(gcd(x, n/x)==1), d); }
gcud(n, m) = vecmax(setintersect(udivs(n), udivs(m)));
biuphi(n) = if (n==1, 1, sum(k=1, n-1, gcud(n, k) == 1));
isok(n) = biuphi(n) == biuphi(n+1);
lista(nn) = {x = biuphi(1); for (n=2, nn, y = biuphi(n); if (x==y, print1(n-1, ", ")); x = y;);} \\ Michel Marcus, Nov 09 2017

Extensions

a(10) from Michel Marcus, Nov 11 2017
a(11) from Michel Marcus, Nov 12 2017
a(12)-a(21) from Amiram Eldar, Jul 16 2022

A005424 Smallest number that requires n iterations of the bi-unitary totient function (A116550) to reach 1.

Original entry on oeis.org

2, 3, 4, 5, 8, 9, 13, 16, 17, 24, 25, 35, 44, 63, 64, 91, 97, 128, 193, 221, 259, 324, 353, 391, 477, 702, 929, 1188, 1269, 1589, 1613, 2017, 2309, 2623, 3397, 4064, 4781, 5468, 6515, 6887, 9213, 12286, 12887, 14009, 16564, 16897, 17803, 30428, 36256
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Let p(n) = number of unitary divisors k of n, k

References

  • M. Lal, H. Wareham and R. Mifflin, Iterates of the bi-unitary totient function, Utilitas Math., 10 (1976), 347-350.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A116550, A225175, A225176.

Programs

  • Maple
    L := [seq(0,i=0..100)] ;
for n from 1 do
    itr := A225320(n) ;
    if itr < nops(L) then
        if op(itr,L) = 0 then
            L := subsop(itr=n,L) ;
            print(L) ;
        end if;
    end if;
end do: # R. J. Mathar, May 02 2013
  • Mathematica
    A116550[1] = 1; A116550[n_] := With[{pp = Power @@@ FactorInteger[n]}, Count[Range[n], m_ /; Intersection[pp, Power @@@ FactorInteger[m]] == {}]]; A225320[n_] := A225320[n] = If[n == 1, 0, 1+A225320[A116550[n]]]; L = Array[0&, 100]; For[n = 1, n <= 40000, n++, itr = A225320[n]; If[itr < Length[L], If[L[[itr]] == 0, L = ReplacePart[L, itr -> n]; Print[Select[L, Positive] // Last]]]]; Select[L, Positive] (* Jean-François Alcover, Jan 13 2014, after R. J. Mathar *)

A306070 Partial sums of A116550: Sum_{k=1..n} bphi(k) where bphi(k) is the bi-unitary analog of Euler's totient function.

Original entry on oeis.org

1, 2, 4, 7, 11, 14, 20, 27, 35, 41, 51, 59, 71, 80, 89, 104, 120, 132, 150, 164, 178, 193, 215, 232, 256, 274, 300, 321, 349, 364, 394, 425, 448, 472, 497, 526, 562, 589, 617, 648, 688, 709, 751, 786, 820, 853, 899, 935, 983, 1019, 1056, 1098, 1150, 1189
Offset: 1

Views

Author

Amiram Eldar, Jun 19 2018

Keywords

Comments

The bi-unitary version of A002088 and A177754.

Links

Crossrefs

Cf. A002088, A177754, A116550, A306071.

Programs

  • Mathematica
    bphi[1] = 1; bphi[n_] := With[{pp = Power @@@ FactorInteger[n]}, Count[Range[n], m_ /; Intersection[pp, Power @@@ FactorInteger[m]] == {}]]; Accumulate[Table[bphi[n], {n, 1, 100}]] (* after Jean-François Alcover at A116550 *)
phi[x_, n_] := DivisorSum[n, MoebiusMu[#]*Floor[x/#] &]; bphi[n_] := DivisorSum[n, (-1)^PrimeNu[#]*phi[n/#, #] &, CoprimeQ[#, n/#] &]; Accumulate[Array[bphi, 100]] (* Amiram Eldar, Jun 30 2025 *)
  • PARI
    udivs(n) = {my(d = divisors(n)); select(x->(gcd(x, n/x)==1), d); }
gcud(n, m) = vecmax(setintersect(udivs(n), udivs(m)));
bphi(n) = if (n==1, 1, sum(k=1, n-1, gcud(n, k) == 1));
a(n) = sum(k=1, n, bphi(k)); \\ Michel Marcus, Jun 20 2018

Formula

a(n) = A*n^2/2 + O(n*log(n)^2), where A = A306071.

A225175 Largest number which requires n iterations of the bi-unitary totient function (A116550) to reach 1.

Original entry on oeis.org

1, 2, 3, 6, 10, 11, 12, 18, 30, 42, 78, 106, 210, 366, 550, 603, 750, 1290, 2562, 4398, 4305, 7470, 9090, 14322, 24558, 35382, 55482, 78020, 141190, 207519, 301642, 429870, 552693, 684846, 1060710, 1391390, 2385246, 3454044
Offset: 0

Views

Author

N. J. A. Sloane, May 01 2013

Keywords

Comments

a(26) >= 55482. a(27) >= 78020. - R. J. Mathar, May 05 2013

References

  • M. Lal, H. Wareham and R. Mifflin, Iterates of the bi-unitary totient function, Utilitas Math., 10 (1976), 347-350.

Crossrefs

Cf. A116550, A005424, A225176, A225174, A047994, A003271, A225172, A225173.

Extensions

a(26)-a(37) from Donovan Johnson, Dec 07 2013

A225176 Number of numbers which require n iterations of the bi-unitary totient function (A116550) to reach 1.

Original entry on oeis.org

1, 1, 1, 2, 3, 2, 2, 4, 7, 6, 13, 12, 16, 24, 31, 51, 66, 87, 126, 139, 187, 260, 331, 412, 551, 693
Offset: 1

Views

Author

N. J. A. Sloane, May 01 2013

Keywords

References

  • M. Lal, H. Wareham and R. Mifflin, Iterates of the bi-unitary totient function, Utilitas Math., 10 (1976), 347-350.

Crossrefs

Cf. A116550, A005424, A225175, A225174, A047994, A003271, A225172, A225173.

A225320 The number of iterations of the bi-unitary totient A116550 needed to reach 1 starting with n.

Original entry on oeis.org

0, 1, 2, 3, 4, 3, 4, 5, 6, 4, 5, 6, 7, 7, 7, 8, 9, 7, 8, 8, 8, 8, 9, 10, 11, 8, 9, 9, 10, 8, 9, 10, 10, 11, 12, 11, 12, 10, 10, 10, 11, 9, 10, 13, 12, 11, 12, 12, 13, 12, 13, 10, 11, 11, 10, 12, 10, 10, 11, 12, 13, 13, 14, 15, 13, 13, 14, 13, 14, 11, 12, 14, 15, 12, 13, 15, 14, 10, 11, 14
Offset: 1

Views

Author

R. J. Mathar, May 05 2013

Keywords

Examples

			a(6) = 3 because the first step is A116550(6) = 3, the second A116550(3) = 2, the third A116550(2) = 1, where 1 is reached.

Links

Crossrefs

Cf. A005424 (positions of records), A116550.

Programs

  • Maple
    A225320 := proc(n)
    option remember;
    if n = 1 then
        0;
    else
        1+procname(A116550(n)) ;
    end if;
end proc:
  • Mathematica
    A116550[1] = 1; A116550[n_] := With[{pp = Power @@@ FactorInteger[n]}, Count[Range[n], m_ /; Intersection[pp, Power @@@ FactorInteger[m]] == {}]]; a[n_] := a[n] = If[n == 1, 0, 1 + a[A116550[n]]]; Table[a[n], {n, 1, 80}] (* Jean-François Alcover, Dec 16 2013 *)

Formula

The smallest x such that A116550^x(n) = 1, where the operation Op^x denotes x nestings of the operator Op.

A294030 Values of bphi(k) = bphi(k+1), where bphi is the bi-unitary analog of Euler's totient function (A116550).

Original entry on oeis.org

1, 9, 14, 42, 161, 161, 798, 1400, 86156, 123656, 419430, 387868, 508797, 772121, 870233, 4162866, 8754569, 126168912, 126991491, 128007618, 131491736
Offset: 1

Views

Author

Amiram Eldar, Oct 22 2017

Keywords

Comments

The bi-unitary totient function of numbers k such that k and k+1 have the same function value (A293184).

Examples

			9 is in the sequence since 9 = bphi(14) = bphi(15).

Crossrefs

The bi-unitary version of A003275.
Cf. A116550, A293184.

Programs

  • Mathematica
    bphi[1] = 1; bphi[n_] := With[{pp = Power @@@ FactorInteger[n]}, Count[Range[n], m_ /; Intersection[pp, Power @@@ FactorInteger[m]] == {}]]; a={}; b1=0; Do[b2 = bphi[k]; If[b1 == b2, a = AppendTo[a, b1]]; b1 = b2, {k, 1, 10^2}]; a (* after Jean-François Alcover at A116550 *)

Formula

a(n) = A116550(A293184(n)).

Extensions

a(10)-a(11) from Michel Marcus, Nov 14 2017
a(12)-a(21) from Amiram Eldar, Jul 16 2022

A298759 Numbers k such that bphi(k) = k/2, where bphi is the bi-unitary analog of Euler's totient function (A116550).

Original entry on oeis.org

2, 6, 30, 42, 1722, 1806, 19977474
Offset: 1

Views

Author

Amiram Eldar, Jan 26 2018

Keywords

Comments

With Euler's totient function, phi(k) = k/2 only for powers of 2 (A000079, except for 1). With the unitary totient function (A047994) the corresponding sequence is A030163.
a(8) > 2*10^9, if it exists. - Amiram Eldar, Jul 16 2022

Examples

			42 is in the sequence since bphi(42) = 21 = 42/2.

Crossrefs

Cf. A047994, A030163, A116550.

Programs

  • Mathematica
    bphi[1] = 1; bphi[n_] :=  With[{pp = Power @@@ FactorInteger[n]},   Count[Range[n], m_ /; Intersection[pp, Power @@@ FactorInteger[m]] == {}]]; aQ[n_] := bphi[n] == n/2; Select[Range[10000], aQ]
  • PARI
    udivs(n) = {my(d = divisors(n)); select(x->(gcd(x, n/x)==1), d); }
gcud(n, m) = vecmax(setintersect(udivs(n), udivs(m)));
bphi(n) = if (n==1, 1, sum(k=1, n-1, gcud(n, k) == 1));
isok(n) = bphi(n) == n/2; \\ Michel Marcus, Jan 26 2018

Extensions

a(7) from Amiram Eldar, Jul 16 2022

A356747 Numbers m that divide A306070(m) = Sum_{k=1..m} bphi(k), where bphi is the bi-unitary totient function (A116550).

Original entry on oeis.org

1, 2, 141, 1035, 2388, 3973, 5157, 14160, 37023, 68861, 99889, 116106, 117939, 627400, 1561944, 1626983, 5901444, 10054091, 12260525, 32619981, 49775099
Offset: 1

Views

Author

Amiram Eldar, Aug 25 2022

Keywords

Comments

The corresponding quotients A306070(m)/m are 1, 1, 57, 418, ... (see the link for more values).
a(22) > 6.5*10^8, if it exists.

Links

Crossrefs

Cf. A116550, A306070.
Similar sequences: A048290, A306950.

Programs

  • Mathematica
    phi[x_, n_] := DivisorSum[n, MoebiusMu[#]*Floor[x/#] &]; bphi[n_] := DivisorSum[n, (-1)^PrimeNu[#]*phi[n/#, #] &, CoprimeQ[#, n/#] &]; seq = {}; s = 0; Do[s = s + bphi[n]; If[Divisible[s, n], AppendTo[seq, n]], {n, 1, 10^6}]; seq

A384247 The number of integers from 1 to n whose largest divisor that is an infinitary divisor of n is 1.

Original entry on oeis.org

1, 1, 2, 3, 4, 2, 6, 4, 8, 4, 10, 6, 12, 6, 8, 15, 16, 8, 18, 12, 12, 10, 22, 8, 24, 12, 18, 18, 28, 8, 30, 16, 20, 16, 24, 24, 36, 18, 24, 16, 40, 12, 42, 30, 32, 22, 46, 30, 48, 24, 32, 36, 52, 18, 40, 24, 36, 28, 58, 24, 60, 30, 48, 48, 48, 20, 66, 48, 44, 24
Offset: 1

Views

Author

Amiram Eldar, May 23 2025

Keywords

Comments

Analogous to A047994, as A064380 is analogous to A116550.

Links

Crossrefs

Row lengths of A384246.
Cf. A000010, A001146, A006519, A047994, A064380, A091732, A116550, A138302, A268335, A384248.

Programs

  • Mathematica
    f[p_, e_] := p^e*(1 - 1/p^(2^(IntegerExponent[e, 2]))); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
  • PARI
    a(n) = {my(f = factor(n)); n * prod(i = 1, #f~, (1 - 1/f[i,1]^(1 << valuation(f[i,2], 2))));}

Formula

Multiplicative with a(p^e) = p^e * (1 - 1/p^A006519(e)).
a(n) >= A091732(n), with equality if and only if n is in A138302.
a(n) <= A047994(n), with equality if and only if n is in A138302.
a(n) >= A000010(n), with equality if and only if n is an exponentially odd number (A268335).
a(n) is odd if and only if n = 1 or 2^(2^k) for k >= 0 (A001146). a(2^(2^k)) = 2^(2^k)-1.
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} f(1/p) = 0.66718130416373472394..., and f(x) = 1 - (1-x)*Sum_{k>=1} x^(2^k)/(1-x^(2^k)).
Showing 1-10 of 12 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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