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 11 results. Next

A295501 a(n) = phi(4^n-1), where phi is Euler's totient function (A000010).

Original entry on oeis.org

2, 8, 36, 128, 600, 1728, 10584, 32768, 139968, 480000, 2640704, 6635520, 44717400, 132765696, 534600000, 2147483648, 11452896600, 26121388032, 183250539864, 473702400000, 2427720325632, 8834232287232, 45914084232320, 109586090557440, 656100000000000
Offset: 1

Views

Author

Seiichi Manyama, Nov 22 2017

Keywords

Links

Crossrefs

Cf. A000010, A053285.
phi(k^n-1): A053287 (k=2), A295500 (k=3), this sequence (k=4), A295502 (k=5), A366623 (k=6), A366635 (k=7), A366654 (k=8), A366663 (k=9), A295503 (k=10), A366685 (k=11), A366711 (k=12).
Cf. A366602, A366603, A366604, A366608.

Programs

  • Mathematica
    EulerPhi[4^Range[30] - 1] (* Paolo Xausa, Jun 17 2024 *)
  • PARI
    {a(n) = eulerphi(4^n-1)}

Formula

a(n) = n*A027695(n).
a(n) = A053287(2*n) = A053285(n) * A053287(n). - Max Alekseyev, Jan 07 2024

A366607 Sum of the divisors of 4^n+1.

Original entry on oeis.org

3, 6, 18, 84, 258, 1302, 4356, 20520, 65538, 351120, 1110276, 5048232, 17041416, 82623888, 284225796, 1494039792, 4301668356, 20788904016, 73234343952, 332019460560, 1103789883396, 5936210280000, 18679788287496, 84884999116320, 282937726148616
Offset: 0

Views

Author

Sean A. Irvine, Oct 14 2023

Keywords

Examples

			a(3)=84 because 4^3+1 has divisors {1, 5, 13, 65}.

Links

Crossrefs

Cf. A000203, A052539, A057940, A274903, A366603, A366605, A366606, A366608, A366609.
Cf. A069061, A366578, A366617, A366629, A366638, A366657, A366666, A366668, A366689, A366715.

Programs

  • Maple
    a:=n->numtheory[sigma](4^n+1):
seq(a(n), n=0..100);
  • Mathematica
    DivisorSigma[1,4^Range[0,30]+1] (* Paolo Xausa, Oct 14 2023 *)
  • Python
    from sympy import divisor_sigma
def A366607(n): return divisor_sigma((1<<(n<<1))+1) # Chai Wah Wu, Oct 14 2023

Formula

a(n) = sigma(4^n+1) = A000203(A052539(n)).
a(n) = A069061(2*n). - Max Alekseyev, Jan 08 2024

A366630 a(n) = phi(6^n+1), where phi is Euler's totient function (A000010).

Original entry on oeis.org

1, 6, 36, 180, 1296, 6000, 41472, 230496, 1580800, 8359200, 58579200, 310968900, 2175102720, 10971642240, 76065091200, 351048600000, 2811459796992, 14508487949472, 88870766837760, 522016066337712, 3564233663616000, 17479898551382400, 128060205344805888
Offset: 0

Views

Author

Sean A. Irvine, Oct 14 2023

Keywords

Links

Crossrefs

Cf. A062394, A000010, A366623, A366618, A366627, A366628, A366629, A366670.
Cf. A053285, A366579, A366608, A366639, A366658, A366667, A366669, A366690, A366716.

Programs

  • Mathematica
    EulerPhi[6^Range[0, 22] + 1] (* Paul F. Marrero Romero, Oct 17 2023 *)
  • PARI
    {a(n) = eulerphi(6^n+1)}

Formula

a(n) = A000010(A062394(n)). - Paul F. Marrero Romero, Oct 17 2023

A366667 a(n) = phi(9^n+1), where phi is Euler's totient function (A000010).

Original entry on oeis.org

1, 4, 40, 288, 3072, 23600, 259200, 1847104, 21523360, 152845056, 1700870400, 12550120000, 130459631616, 997562438080, 11159367815680, 81159501312000, 926510094425920, 6670865700716544, 73205598106368000, 540340585126398016, 5691215305506816000
Offset: 0

Views

Author

Sean A. Irvine, Oct 15 2023

Keywords

Links

Crossrefs

Cf. A062396, A000010, A366663, A366664, A366665, A366666.
Cf. A053285, A366579, A366608, A366618, A366630, A366639, A366658, A366669, A366690, A366716.

Programs

  • Mathematica
    EulerPhi[9^Range[0, 20] + 1] (* Paul F. Marrero Romero, Nov 04 2023 *)
  • PARI
    {a(n) = eulerphi(9^n+1)}

Formula

a(n) = A000010(A062396(n)). - Paul F. Marrero Romero, Nov 04 2023
a(n) = A366579(2*n). - Max Alekseyev, Jan 08 2024

A366605 Number of distinct prime divisors of 4^n + 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 2, 3, 1, 4, 2, 3, 3, 4, 2, 5, 2, 4, 4, 4, 2, 6, 3, 5, 3, 5, 3, 6, 3, 3, 4, 5, 2, 6, 3, 6, 5, 5, 4, 9, 3, 5, 5, 5, 4, 10, 2, 4, 3, 6, 6, 9, 2, 4, 6, 6, 5, 8, 3, 7, 6, 6, 4, 10, 2, 9, 7, 6, 4, 8, 4, 6, 7, 5, 2, 12, 4, 9, 5, 4, 4, 10, 4, 6, 8, 10
Offset: 0

Views

Author

Sean A. Irvine, Oct 14 2023

Keywords

Links

Crossrefs

Cf. A001221, A052539, A057940, A274903, A366604, A366606, A366607, A366608, A366609.
Cf. A046799, A366580, A366615, A366627, A366636, A366655, A366664, A119704, A366686, A366712.

Programs

  • Mathematica
    PrimeNu[4^Range[0,100]+1] (* Paolo Xausa, Oct 14 2023 *)
  • PARI
    for(n = 0, 100, print1(omega(4^n + 1), ", "))
  • Python
    from sympy import primenu
def A366605(n): return primenu((1<<(n<<1))+1) # Chai Wah Wu, Oct 14 2023

Formula

a(n) = omega(4^n+1) = A001221(A052539(n)).
a(n) = A046799(2*n). - Max Alekseyev, Jan 08 2024

A366606 Number of divisors of 4^n+1.

Original entry on oeis.org

2, 2, 2, 4, 2, 6, 4, 8, 2, 16, 4, 8, 8, 16, 4, 48, 4, 16, 16, 16, 4, 64, 8, 32, 8, 64, 8, 64, 8, 8, 16, 32, 4, 64, 12, 96, 32, 32, 16, 768, 8, 32, 32, 32, 16, 1536, 4, 16, 8, 64, 64, 512, 4, 16, 64, 96, 32, 256, 8, 128, 64, 64, 16, 1024, 4, 768, 128, 64, 16
Offset: 0

Views

Author

Sean A. Irvine, Oct 14 2023

Keywords

Examples

			a(3)=4 because 4^3+1 has divisors {1, 5, 13, 65}.

Links

Crossrefs

Cf. A000005, A052539, A057940, A274903, A366602, A366605, A366607, A366608, A366609.
Cf. A046798, A366577, A366616, A366628, A366637, A366656, A366665, A344897, A366688, A366714.

Programs

  • Maple
    a:=n->numtheory[tau](4^n+1):
seq(a(n), n=0..100);
  • Mathematica
    DivisorSigma[0,4^Range[0,100]+1] (* Paolo Xausa, Oct 14 2023 *)
  • PARI
    a(n) = numdiv(4^n+1);
  • Python
    from sympy import divisor_count
def A366606(n): return divisor_count((1<<(n<<1))+1) # Chai Wah Wu, Oct 14 2023

Formula

a(n) = sigma0(4^n+1) = A000005(A052539(n)).
a(n) = A046798(2*n). - Max Alekseyev, Jan 08 2024

A366618 a(n) = phi(5^n+1), where phi is Euler's totient function (A000010).

Original entry on oeis.org

1, 2, 12, 36, 312, 1040, 7200, 25088, 183808, 557928, 4396800, 15333120, 121680000, 406812744, 2817007200, 8558784000, 76264519680, 254230063200, 1710194342400, 6349120596480, 47334145996800, 127169887444992, 1088029470747648, 3889097389599864
Offset: 0

Views

Author

Sean A. Irvine, Oct 14 2023

Keywords

Links

Crossrefs

Cf. A000010, A034474, A057939, A074478, A295502, A366615, A366616, A366617.
Cf. A053285, A366579, A366608, A366630, A366639, A366658, A366667, A366669, A366690, A366716.

Programs

  • Mathematica
    EulerPhi[5^Range[0,30]+1] (* Harvey P. Dale, Jun 07 2025 *)
  • PARI
    {a(n) = eulerphi(5^n+1)}

A366639 a(n) = phi(7^n+1), where phi is Euler's totient function (A000010).

Original entry on oeis.org

1, 4, 20, 168, 1200, 7600, 43200, 407680, 2712832, 19707408, 112560000, 945677920, 6768230400, 47530457728, 271289229120, 2096760960000, 16569393144832, 116315256993600, 597938524646400, 5699431359135360, 38890647857280000, 270061302781670400
Offset: 0

Views

Author

Sean A. Irvine, Oct 15 2023

Keywords

Links

Crossrefs

Cf. A034491, A000010, A057937, A227575, A366635, A366636, A366637, A366638.
Cf. A053285, A366579, A366608, A366618, A366630, A366658, A366667, A366669, A366690, A366716.

Programs

  • Mathematica
    EulerPhi[7^Range[0,21] + 1] (* Paul F. Marrero Romero, Nov 05 2023 *)
  • PARI
    {a(n) = eulerphi(7^n+1)}

Formula

a(n) = A000010(A034491(n)). - Paul F. Marrero Romero, Nov 06 2023

A366658 a(n) = phi(8^n+1), where phi is Euler's totient function (A000010).

Original entry on oeis.org

1, 6, 48, 324, 3840, 19800, 186624, 1365336, 16515072, 84768120, 760320000, 5632621632, 64258375680, 366369658200, 3105655160832, 20140520400000, 280012271910912, 1495522910085120, 12824556668190720, 95907982079387520, 1080582572777472000, 5688765822212629632
Offset: 0

Views

Author

Sean A. Irvine, Oct 15 2023

Keywords

Links

Crossrefs

Cf. A062395, A000010, A057936, A274905, A366654, A366655, A366656, A366657, A366671.
Cf. A053285, A366579, A366608, A366618, A366630, A366639, A366667, A366669, A366690, A366716.

Programs

  • Mathematica
    EulerPhi[8^Range[0, 21] + 1] (* Paul F. Marrero Romero, Oct 17 2023 *)
  • PARI
    {a(n) = eulerphi(8^n+1)}
  • Python
    from sympy import totient
def A366658(n): return totient((1<<3*n)+1) # Chai Wah Wu, Oct 15 2023

Formula

a(n) = A000010(A062395(n)). - Paul F. Marrero Romero, Nov 06 2023
a(n) = A053285(3*n). - Max Alekseyev, Jan 09 2024

A366609 Smallest prime dividing 4^n + 1.

Original entry on oeis.org

2, 5, 17, 5, 257, 5, 17, 5, 65537, 5, 17, 5, 97, 5, 17, 5, 641, 5, 17, 5, 257, 5, 17, 5, 193, 5, 17, 5, 257, 5, 17, 5, 274177, 5, 17, 5, 97, 5, 17, 5, 65537, 5, 17, 5, 257, 5, 17, 5, 641, 5, 17, 5, 257, 5, 17, 5, 449, 5, 17, 5, 97, 5, 17, 5, 59649589127497217
Offset: 0

Views

Author

Sean A. Irvine, Oct 14 2023

Keywords

Links

Crossrefs

Bisection of A002586.
Cf. A002586, A038371, A052539, A074476, A274903, A366605, A366606, A366607, A366608, A366670, A366671, A366719.
Showing 1-10 of 11 results. Next

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

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