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 13 results. Next

A071386 Numbers k such that the cardinality of the set of solutions to phi(x) = k is odd.

Original entry on oeis.org

2, 8, 20, 32, 40, 44, 48, 56, 60, 72, 92, 96, 104, 108, 116, 120, 128, 132, 140, 144, 156, 164, 192, 204, 212, 216, 220, 240, 252, 260, 272, 276, 296, 300, 332, 344, 356, 360, 368, 380, 384, 392, 396, 400, 416, 420, 440, 444, 452, 456, 476, 480, 500, 504, 512
Offset: 1

Views

Author

Labos Elemer, May 23 2002

Keywords

Examples

			k = 40 is a term: InvPhi(40) = {41,55,75,82,88,100,110,132,150} has 9 entries.

Links

Crossrefs

Cf. A000010 (phi), A014197, A058277, A063512, A071387.

Programs

Formula

{ k : Card(InvPhi(k)) mod 2 = 1 }.

A333019 Numbers k such that both k and k + 2 are totient numbers (A002202).

Original entry on oeis.org

2, 4, 6, 8, 10, 16, 18, 20, 22, 28, 30, 40, 42, 44, 46, 52, 54, 56, 58, 64, 70, 78, 80, 82, 100, 102, 104, 106, 108, 110, 126, 128, 130, 136, 138, 148, 160, 162, 164, 166, 176, 178, 190, 196, 198, 208, 210, 220, 222, 224, 226, 238, 250, 260, 262, 268, 270, 280
Offset: 1

Views

Author

Amiram Eldar, Mar 05 2020

Keywords

Examples

			2 is a term since both 2 and 4 are totient numbers.

Links

Crossrefs

Cf. A000010, A002202, A005277, A063512, A083533, A333020, A333021, A333022, A333023, A333024.

Programs

  • PARI
    for(k = 1, 150, if(istotient(2*k) && istotient(2*k+2), print1(2*k,", ")))

A333020 Starts of runs of 3 consecutive even numbers that are all totient numbers (A002202).

Original entry on oeis.org

2, 4, 6, 8, 16, 18, 20, 28, 40, 42, 44, 52, 54, 56, 78, 80, 100, 102, 104, 106, 108, 126, 128, 136, 160, 162, 164, 176, 196, 208, 220, 222, 224, 260, 268, 292, 328, 342, 344, 356, 378, 380, 416, 438, 440, 460, 462, 464, 476, 498, 500, 502, 504, 520, 560, 584
Offset: 1

Views

Author

Amiram Eldar, Mar 05 2020

Keywords

Examples

			2 is a term since 2, 4 and 6 are all totient numbers.

Links

Crossrefs

Cf. A000010, A002202, A005277, A063512, A083533, A333019, A333021, A333022, A333023, A333024.

Programs

  • PARI
    m = 3; v = vector(m); for(k=1, m, v[k] = istotient(2*k)); for(k = m+1, 300, if(Set(v) == [1], print1(2*(k-m),", ")); v = concat(v[2..m], istotient(2*k)))

A333021 Starts of runs of 4 consecutive even numbers that are all totient numbers (A002202).

Original entry on oeis.org

2, 4, 6, 16, 18, 40, 42, 52, 54, 78, 100, 102, 104, 106, 126, 160, 162, 220, 222, 342, 378, 438, 460, 462, 498, 500, 502, 856, 858, 880, 882, 1086, 1276, 1278, 1300, 1422, 1480, 1482, 1566, 1660, 1662, 1804, 1806, 1996, 2058, 2200, 2202, 2236, 2238, 3016, 3018
Offset: 1

Views

Author

Amiram Eldar, Mar 05 2020

Keywords

Examples

			2 is a term since 2, 4, 6 and 8 are all totient numbers.

Links

Crossrefs

Cf. A000010, A002202, A005277, A063512, A083533, A333019, A333020, A333022, A333023, A333024.

Programs

  • PARI
    m = 4; v = vector(m); for(k=1, m, v[k] = istotient(2*k)); for(k = m+1, 1500, if(Set(v) == [1], print1(2*(k-m),", ")); v = concat(v[2..m], istotient(2*k)))

A333022 Starts of runs of 5 consecutive even numbers that are all totient numbers (A002202).

Original entry on oeis.org

2, 4, 16, 40, 52, 100, 102, 104, 160, 220, 460, 498, 500, 856, 880, 1276, 1480, 1660, 1804, 2200, 2236, 3016, 3160, 3460, 4516, 4780, 5500, 5920, 6040, 6196, 6820, 7240, 7636, 7696, 7720, 8536, 8620, 9196, 9460, 9880, 10456, 12916, 13756, 13960, 14416, 15640
Offset: 1

Views

Author

Amiram Eldar, Mar 05 2020

Keywords

Examples

			2 is a term since 2, 4, 6, 8 and 10 are all totient numbers.

Links

Crossrefs

Cf. A000010, A002202, A005277, A063512, A083533, A333019, A333020, A333021, A333023, A333024.

Programs

  • PARI
    m = 5; v = vector(m); for(k=1, m, v[k] = istotient(2*k)); for(k = m+1, 7500, if(Set(v) == [1], print1(2*(k-m),", ")); v = concat(v[2..m], istotient(2*k)))

A333023 Starts of runs of 6 consecutive even numbers that are all totient numbers (A002202).

Original entry on oeis.org

2, 100, 102, 498, 267670, 26734060, 26734062, 31253680, 65974998, 70938496, 118428800, 1232747200, 2764919296, 3149734998, 3149735000, 3413655896, 3415058276, 3755544796, 4446555802, 5727840798, 6156991616, 10080661998, 10464983096, 11054945296, 11953158220
Offset: 1

Views

Author

Amiram Eldar, Mar 05 2020

Keywords

Examples

			2 is a term since 2, 4, 6, 8, 10 and 12 are all totient numbers.

Links

Crossrefs

Cf. A000010, A002202, A005277, A063512, A083533, A333019, A333020, A333021, A333022, A333024.

Programs

  • PARI
    m = 6; v = vector(m); for(k=1, m, v[k] = istotient(2*k)); for(k = m+1, 1.5e5, if(Set(v) == [1], print1(2*(k-m),", ")); v = concat(v[2..m], istotient(2*k)))

A333024 Starts of runs of 7 consecutive even numbers that are all totient numbers (A002202).

Original entry on oeis.org

100, 26734060, 3149734998, 12960114796, 15685683796, 24077884060, 36987943996, 38809984996, 62521251798, 76314338740, 319408651400
Offset: 1

Views

Author

Amiram Eldar, Mar 05 2020

Keywords

Examples

			100 is a term since the 7 even numbers 100, 102, ... 112 are all totient numbers.

Links

Crossrefs

Cf. A000010, A002202, A005277, A063512, A083533, A333019, A333020, A333021, A333022, A333023.

Programs

  • PARI
    m = 7; v = vector(m); for(k=1, m, v[k] = istotient(2*k)); for(k = m+1, 1.5e7, if(Set(v) == [1], print1(2*(k-m),", ")); v = concat(v[2..m], istotient(2*k)))

Extensions

a(9)-a(11) from Giovanni Resta, Mar 07 2020

A071387 Smallest number k for which the set of solutions to phi(x) = k has 2n-1 entries.

Original entry on oeis.org

0, 2, 8, 32, 40, 48, 396, 704, 72, 216, 144, 1056, 360, 384, 1320, 240, 9000, 7128, 480, 1296, 15936, 3072, 864, 7344, 720, 4992, 2016, 6000, 4752, 3024, 9984, 1920, 7560, 22848, 29160, 3360, 13248, 27720, 9072, 9360, 4032, 4800, 16896, 3840, 9504, 23520, 5040
Offset: 1

Views

Author

Labos Elemer, May 23 2002

Keywords

Examples

			For n = 7: 2n-1 = 13, a(7) = Min(InvPhi(13)) = Min({396,400,420,552,560,660}) = 396.

Links

Crossrefs

Cf. A000010 (phi), A014197, A058277, A063512, A071386, A071388, A071389.

Programs

  • PARI
    a(n) = {if (n==1, return (0)); my(k=1); while(#invphi(k) != 2*n-1, k++); k;} \\ Michel Marcus, May 13 2020

Formula

a(n) = Min({x; Card(InvPhi(x)) = 2n-1}); a(1)=0 because of Carmichael conjecture.

Extensions

a(12)-a(47) from Donovan Johnson, Jul 27 2011

A071388 Numbers k such that the cardinality of the set of solutions to phi(x) = k is a prime.

Original entry on oeis.org

1, 2, 8, 10, 20, 22, 28, 30, 32, 44, 46, 48, 52, 54, 56, 58, 66, 70, 72, 78, 82, 92, 96, 102, 104, 106, 110, 116, 120, 126, 130, 132, 136, 138, 140, 148, 150, 156, 164, 166, 172, 178, 190, 196, 198, 204, 210, 212, 216, 220, 222, 226, 228, 238, 240, 250, 260, 262
Offset: 1

Views

Author

Labos Elemer, May 23 2002

Keywords

Comments

All terms except 1 are even. - Robert Israel, Mar 29 2020

Examples

			InvPhi(48) = {65,104,105,112,130,140,144,156,168,180,210} has 11 terms, so 48 is a term.

Links

Crossrefs

Cf. A000010, A007366, A007367, A014197, A058277, A060668, A060670, A060674, A063512, A071386-A071389.

Programs

  • Maple
    filter:= n -> isprime(nops(numtheory:-invphi(n))):
select(filter, [$1..400]); # Robert Israel, Mar 29 2020
  • PARI
    is(k) = isprime(invphiNum(k)); \\ Amiram Eldar, Nov 15 2024, using Max Alekseyev's invphi.gp

A071389 Least number m such that cardinality of InvPhi(m) = prime(n).

Original entry on oeis.org

1, 2, 8, 32, 48, 396, 72, 216, 1056, 1320, 240, 480, 15936, 3072, 7344, 2016, 3024, 9984, 22848, 3360, 13248, 9360, 4800, 9504, 9216, 23328, 7680, 53280, 12480, 29376, 91200, 159744, 22464, 228960, 29952, 179200, 47040, 68544, 15840, 20736, 61440
Offset: 1

Views

Author

Labos Elemer, May 23 2002

Keywords

Examples

			For n = 11: prime(11) = 31, Card(InvPhi(x)) = 31 for {240, 672, ...}; the smallest is 240 = a(11).

Links

Crossrefs

Cf. A000010, A014197, A058277, A063512, A071386, A071387, A071388.

Programs

  • PARI
    lista(len) = {my(p = prime(len), v = vector(p, i, -!isprime(i)), c = 0, k = 1, i); while(c < len, i = invphiNum(k); if(i > 0 && i <= p && v[i] == 0, c++; v[i] = k); k++); select(x -> x > 0, v);} \\ Amiram Eldar, Nov 11 2024, using Max Alekseyev's invphi.gp

Formula

a(n) = Min{x; Card(InvPhi(x)) = prime(n), n-th prime}

Extensions

4 more terms from Emeric Deutsch, Jul 25 2005
More terms from Max Alekseyev, Apr 24 2010
Showing 1-10 of 13 results. Next

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

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