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 28 results. Next

A081375 a(n) is the least number k such that A081373(k) = n.

Original entry on oeis.org

1, 2, 6, 12, 30, 42, 72, 78, 84, 90, 190, 216, 222, 228, 234, 252, 270, 540, 546, 570, 630, 738, 744, 770, 792, 858, 900, 924, 930, 990, 1050, 1638, 1710, 1890, 1980, 2100, 2310, 2418, 2442, 2508, 2562, 2574, 2604, 2700, 2772, 2790, 2850, 2970, 3150
Offset: 1

Views

Author

Labos Elemer, Mar 24 2003

Keywords

Links

Crossrefs

Cf. A000010, A008479, A081373, A081374, A067004.

Programs

  • Mathematica
    f[x_] := Count[Table[EulerPhi[j]-EulerPhi[x], {j, 1, x}], 0] t=Table[0, {50}]; Do[s=f[n]; If[s<51&&t[[s]]==0, t[[s]]=n], {n, 1, 4000}]; t
  • PARI
    lista(len) = {my(v = vector(len), c = 0, k = 1, i); while(c < len, i = #select(x -> x <= k, invphi(eulerphi(k))); if(i <= len && v[i] == 0, c++; v[i] = k); k++); v;} \\ Amiram Eldar, Nov 08 2024, using Max Alekseyev's invphi.gp

A303777 Ordinal transform of A081373; ordinal transform of {the ordinal transform of A000010}.

Original entry on oeis.org

1, 1, 2, 2, 3, 1, 4, 3, 4, 2, 5, 1, 6, 3, 7, 5, 8, 2, 9, 4, 6, 7, 10, 3, 11, 5, 8, 4, 12, 1, 13, 9, 10, 6, 14, 2, 15, 7, 11, 5, 16, 1, 17, 8, 9, 12, 18, 3, 13, 6, 19, 7, 20, 8, 14, 4, 15, 16, 21, 2, 22, 17, 10, 18, 23, 5, 24, 11, 25, 3, 26, 1, 27, 9, 12, 6, 19, 1, 28, 10, 29, 11, 30, 1, 31, 13, 32, 7, 33, 1, 20, 21, 14, 22, 15, 8
Offset: 1

Views

Author

Antti Karttunen, Apr 30 2018

Keywords

Links

Crossrefs

Cf. A000010, A081373, A081375 (positions of ones), A210719 (positions of records), A286610.

Programs

  • Maple
    b:= proc() 0 end: g:= proc() 0 end:
h:= proc(n) option remember; local t;
      t:= numtheory[phi](n); b(t):= b(t)+1
    end:
a:= proc(n) option remember; local t;
      t:= h(n); g(t):= g(t)+1
    end:
seq(a(n), n=1..120);  # Alois P. Heinz, Apr 30 2018
  • Mathematica
    b[] = 0; g[] = 0;
h[n_] := h[n] = With[{t = EulerPhi[n]}, b[t] = b[t]+1];
a[n_] := a[n] = With[{t = h[n]}, g[t] = g[t]+1];
Array[a, 120] (* Jean-François Alcover, Dec 19 2021, after Alois P. Heinz *)
  • PARI
    up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
v081373 = ordinal_transform(vector(up_to,n,eulerphi(n)));
A081373(n) = v081373[n];
v303777 = ordinal_transform(vector(up_to,n,A081373(n)));
A303777(n) = v303777[n];

A319339 Filter sequence combining A081373(n) with A246277(n).

Original entry on oeis.org

1, 2, 3, 4, 3, 5, 3, 6, 4, 7, 3, 8, 3, 9, 10, 11, 3, 12, 3, 13, 14, 15, 3, 16, 17, 18, 6, 19, 3, 20, 3, 21, 22, 23, 10, 24, 3, 25, 15, 26, 3, 27, 3, 28, 29, 30, 3, 31, 4, 32, 33, 34, 3, 35, 14, 36, 37, 38, 3, 39, 3, 40, 13, 41, 42, 43, 3, 44, 45, 46, 3, 47, 3, 48, 49, 50, 51, 52, 3, 53, 54, 55, 3, 56, 57, 58, 59, 60, 3, 61, 14, 62, 63, 64, 18, 65, 3, 66, 19
Offset: 1

Views

Author

Antti Karttunen, Sep 24 2018

Keywords

Comments

Restricted growth sequence transform of ordered pair [A081373(n), A246277(n)].

Links

Crossrefs

Cf. A081373, A246277, A300247, A305801, A305897, A318500.

Programs

  • PARI
    up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
v081373 = ordinal_transform(vector(up_to,n,eulerphi(n)));
A081373(n) = v081373[n];
A246277(n) = if(1==n, 0, my(f = factor(n), k = primepi(f[1,1])-1); for (i=1, #f~, f[i,1] = prime(primepi(f[i,1])-k)); factorback(f)/2);
v319339 = rgs_transform(vector(up_to,n,[A081373(n),A246277(n)]));
A319339(n) = v319339[n];

A322023 Lexicographically earliest such sequence a that a(i) = a(j) => A081373(i) = A081373(j) and A303756(i) = A303756(j), for all i, j. Here A081373 and A303756 are the ordinal transforms of Euler phi and Carmichael lambda.

Original entry on oeis.org

1, 2, 1, 2, 1, 3, 1, 4, 2, 5, 1, 6, 1, 3, 7, 4, 1, 8, 1, 9, 10, 2, 1, 11, 1, 5, 2, 11, 1, 12, 1, 13, 14, 5, 7, 15, 1, 3, 4, 16, 1, 17, 1, 18, 9, 2, 1, 19, 2, 20, 7, 11, 1, 8, 14, 21, 10, 2, 1, 22, 1, 2, 23, 4, 24, 25, 1, 9, 7, 17, 1, 26, 1, 20, 18, 12, 14, 27, 1, 28, 1, 20, 1, 29, 30, 3, 7, 12, 1, 31, 32, 4, 18, 2, 3, 33, 1, 8, 6, 34, 1, 35, 1, 36, 37
Offset: 1

Views

Author

Antti Karttunen, Nov 29 2018

Keywords

Comments

Restricted growth sequence transform of the ordered pair [A081373(n), A303756(n)].

Links

Crossrefs

Cf. A000010, A002322, A081373, A303756, A319339, A322024, A322025 (ordinal transform).

Programs

  • PARI
    up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A002322(n) = lcm(znstar(n)[2]); \\ From A002322
v081373 = ordinal_transform(vector(up_to,n,eulerphi(n)));
A081373(n) = v081373[n];
v303756 = ordinal_transform(vector(up_to,n,A002322(n)));
A303756(n) = v303756[n];
v322023 = rgs_transform(vector(up_to, n, [A081373(n), A303756(n)]));
A322023(n) = v322023[n];

A322024 Lexicographically earliest such sequence a that a(i) = a(j) => A014197(i) = A014197(j) and A081373(i) = A081373(j), for all i, j. Here A081373(n) gives the number of k, 1 <= k <= n, with phi(k) = phi(n), while A014197(n) gives the number of integers m with phi(m) = n.

Original entry on oeis.org

1, 2, 3, 4, 3, 5, 3, 6, 7, 8, 3, 9, 3, 10, 3, 11, 3, 12, 3, 13, 7, 14, 3, 15, 3, 10, 7, 16, 3, 17, 3, 18, 7, 10, 3, 19, 3, 10, 7, 20, 3, 21, 3, 22, 10, 14, 3, 23, 7, 24, 3, 16, 3, 16, 7, 25, 7, 14, 3, 26, 3, 7, 10, 27, 3, 17, 3, 10, 3, 28, 3, 29, 3, 24, 10, 30, 7, 31, 3, 15, 3, 16, 3, 32, 3, 10, 3, 33, 3, 34, 7, 2, 10, 7, 10, 35, 3, 24, 24, 21, 3, 28, 3, 2, 10
Offset: 1

Views

Author

Antti Karttunen, Nov 29 2018

Keywords

Comments

Restricted growth sequence transform of the ordered pair [A014197(n), A081373(n)].

Links

Crossrefs

Cf. A014197, A081373, A305896, A319339, A322023.

Programs

  • PARI
    up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A014197(n, m=1) = { n==1 && return(1+(m<2)); my(p, q); sumdiv(n, d, if( d>=m && isprime(d+1), sum( i=0, valuation(q=n\d, p=d+1), A014197(q\p^i, p))))}; \\ From A014197
v081373 = ordinal_transform(vector(up_to,n,eulerphi(n)));
A081373(n) = v081373[n];
v322024 = rgs_transform(vector(up_to, n, [A014197(n), A081373(n)]));
A322024(n) = v322024[n];

A303756 Number of values of k, 1 <= k <= n, with A002322(k) = A002322(n), where A002322 is Carmichael lambda.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Apr 30 2018

Keywords

Comments

Ordinal transform of A002322.

Links

Crossrefs

Cf. A002322.
Cf. also A081373, A303755, A303758.

Programs

  • Mathematica
    a[n_] := With[{c = CarmichaelLambda[n]}, Select[Range[n], c == CarmichaelLambda[#]&] // Length];
Array[a, 1000] (* Jean-François Alcover, Sep 19 2020 *)
  • PARI
    up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
A002322(n) = lcm(znstar(n)[2]); \\ From A002322
v303756 = ordinal_transform(vector(up_to,n,A002322(n)));
A303756(n) = v303756[n];

Formula

Except for a(2) = 2, a(n) = A303758(n).

A303757 a(1) = 1 and for n > 1, a(n) = number of values of k, 2 <= k <= n, with A000010(k) = A000010(n), where A000010 is Euler totient function phi.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 2, 2, 3, 1, 4, 1, 3, 1, 2, 1, 4, 1, 3, 2, 2, 1, 4, 1, 3, 2, 4, 1, 5, 1, 2, 2, 3, 1, 5, 1, 3, 2, 4, 1, 6, 1, 3, 3, 2, 1, 5, 2, 4, 1, 4, 1, 4, 2, 5, 2, 2, 1, 6, 1, 2, 3, 2, 1, 5, 1, 3, 1, 6, 1, 7, 1, 4, 3, 5, 2, 8, 1, 4, 1, 4, 1, 9, 1, 3, 1, 5, 1, 10, 2, 2, 3, 2, 3, 5, 1, 4, 4, 6, 1, 6, 1, 2, 3
Offset: 1

Views

Author

Antti Karttunen, Apr 30 2018

Keywords

Comments

Ordinal transform of f, where f(1) = 0 and f(n) = A000010(n) for n > 1.
After a(1)=1 and a(4)=2, the positions of the rest of records is given by A081375(n) = 6, 12, 30, 42, 72, 78, 84, 90, 190, ..., for n >= 3.
Apart from a(2) = 1, the other positions of 1's is given by A210719.

Links

Crossrefs

Cf. A000010, A081373, A081375, A210719, A303754, A303758, A303777.

Programs

  • Mathematica
    With[{s = EulerPhi@ Range@ 105}, MapAt[# + 1 &, Table[Count[s[[2 ;; n]], ?(# == s[[n]] &)], {n, Length@ s}], 1]] (* _Michael De Vlieger, Nov 23 2018 *)
  • PARI
    up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
Aux303757(n) = if(1==n,0,eulerphi(n));
v303757 = ordinal_transform(vector(up_to,n,Aux303757(n)));
A303757(n) = v303757[n];

Formula

Except for a(2) = 1, a(n) = A081373(n).

A331175 Number of values of k, 1 <= k <= n, with A109395(k) = A109395(n), where A109395(n) = n/gcd(n, phi(n)).

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jan 11 2020

Keywords

Comments

Ordinal transform of A109395.

Links

Crossrefs

Cf. A000010, A003277, A009195, A109395.
Cf. also A081373, A330746, A331177.

Programs

  • PARI
    up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
A109395(n) = n/gcd(n, eulerphi(n));
v331175 = ordinal_transform(vector(up_to, n, A109395(n)));
A331175(n) = v331175[n];

Formula

For n >= 1, a(2^n) = n, a(A003277(n)) = 1.

A296214 Numbers k for which there is at least one x < k such that phi(x) = phi(k).

Original entry on oeis.org

2, 4, 6, 8, 9, 10, 12, 14, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32, 33, 34, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 66, 68, 70, 72, 74, 75, 76, 77, 78, 80, 82, 84, 86, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110, 111, 112, 114, 115
Offset: 1

Views

Author

Antti Karttunen, Dec 08 2017

Keywords

Comments

Numbers k for which A081373(k) > 1.
Apart from the initial term 2, this is the complement of union of A000040 (primes) and A069823.

Links

Crossrefs

Cf. A000010, A069822, A069823, A081373.
Cf. A296087 (a subsequence).

Programs

  • PARI
    for(n=1,200,y=0;s=eulerphi(n);for(k=1,(n-1),if(eulerphi(k)==s,y=1;break)); if(y,print1(n,",")));
  • PARI
    is(k) = invphiMin(eulerphi(k)) < k; \\ Amiram Eldar, Nov 15 2024, using Max Alekseyev's invphi.gp

A322871 Ordinal transform of A060681, where A060681(n) = n - A032742(n).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 3, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 2, 3, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 3, 1, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 4, 1, 1, 2, 2, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 3, 2
Offset: 1

Views

Author

Antti Karttunen, Dec 29 2018

Keywords

Links

Crossrefs

Cf. A032742, A060681, A081373, A303756, A322870, A322872, A322873, A322874.

Programs

  • Mathematica
    A060681[n_] := n - n/FactorInteger[n][[1, 1]];
b[_] = 1;
a[n_] := a[n] = With[{t = A060681[n]}, b[t]++];
a /@ Range[1, 105] (* Jean-François Alcover, Dec 19 2021 *)
  • PARI
    up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
A060681(n) = if(1==n,0,(n-(n/vecmin(factor(n)[, 1]))));
v322871 = ordinal_transform(vector(up_to,n,A060681(n)));
A322871(n) = v322871[n];
Showing 1-10 of 28 results. Next

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