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.

Previous Showing 11-16 of 16 results.

A189977 Primes p such that d(p+1) = 2*d(p-1), where d(k) counts the divisors of k.

Original entry on oeis.org

2, 23, 149, 293, 311, 439, 557, 569, 743, 773, 857, 1031, 1151, 1493, 1607, 1663, 1709, 1733, 1879, 1913, 2069, 2141, 2423, 2711, 2719, 2729, 2789, 2969, 3191, 3209, 3559, 3607, 3767, 3821, 3833, 3847, 3929, 3967, 4019, 4073, 4229, 4339, 4451, 4517, 4549
Offset: 1

Views

Author

Ctibor O. Zizka, May 03 2011

Keywords

Links

Crossrefs

Cf. A000005, A008328, A008329.

Programs

  • Mathematica
    Select[Prime[Range[PrimePi[5059]]], 2*DivisorSigma[0, # - 1] == DivisorSigma[0, # + 1] &] (* T. D. Noe, May 03 2011 *)
  • PARI
    isok(p) = isprime(p) && (numdiv(p+1) == 2*numdiv(p-1)); \\ Michel Marcus, Jan 13 2018

A252021 Primes p such that (p - 1)/tau(p - 1) is also prime.

Original entry on oeis.org

13, 19, 41, 61, 89, 137, 157, 229, 233, 277, 349, 373, 569, 709, 733, 809, 853, 857, 877, 997, 1049, 1069, 1097, 1193, 1213, 1237, 1433, 1669, 1789, 1913, 2153, 2293, 2389, 2677, 2749, 2777, 2797, 3209, 3229, 3253, 3373, 3449, 3517, 3593, 3733, 3833, 3929, 4073, 4457, 4549, 4597, 4793, 4813, 4909, 4937, 5197, 5273
Offset: 1

Views

Author

Carlos Eduardo Olivieri, Mar 21 2015

Keywords

Examples

			a(1) = 13, since 12/tau(12) = 2.
a(2) = 19, since 18/tau(18) = 3.
a(4) = 61, since 60/tau(60) = 5.

Links

Crossrefs

Cf. A000005, A008328, A033950.

Programs

  • Magma
    [p:p in PrimesUpTo(5300)| ((p-1) mod NumberOfDivisors(p-1) eq 0) and IsPrime((p-1) div NumberOfDivisors(p-1)) ]; // Marius A. Burtea, Dec 30 2019
  • Mathematica
    Select[Prime[Range[1000]], PrimeQ[(# - 1)/DivisorSigma[0, # - 1]] &]

A145340 a(n) = the maximum of d(p(n)-1) and d(p(n)+1), where d(m) is the number of divisors of m and p(n) is the n-th prime.

Original entry on oeis.org

2, 3, 4, 4, 6, 6, 6, 6, 8, 8, 8, 9, 8, 8, 10, 8, 12, 12, 8, 12, 12, 10, 12, 12, 12, 9, 8, 12, 12, 10, 12, 12, 8, 12, 12, 12, 12, 10, 16, 8, 18, 18, 14, 14, 12, 12, 16, 12, 12, 12, 12, 20, 20, 18, 9, 16, 16, 16, 12, 16, 8, 12, 12, 16, 16, 8, 16, 20, 12, 12, 12, 24, 10, 12, 16, 16, 16
Offset: 1

Views

Author

Leroy Quet, Oct 08 2008

Keywords

Crossrefs

Cf. A145339, A008328, A008329, A067889, A103664, A103665.

Programs

  • Mathematica
    Table[Max[DivisorSigma[0, Prime[n]-1], DivisorSigma[0, Prime[n]+1]], {n, 1, 100}] (* Stefan Steinerberger, Oct 11 2008 *)

Extensions

More terms from Stefan Steinerberger and Ray Chandler, Oct 11 2008

A217442 Numbers n such that d(prime(n) - 1) | n, where d(k) is the number of divisors of k.

Original entry on oeis.org

1, 2, 3, 4, 6, 24, 28, 30, 32, 36, 45, 48, 56, 64, 66, 72, 76, 80, 92, 96, 102, 104, 120, 126, 128, 144, 168, 176, 180, 184, 186, 192, 200, 208, 228, 236, 240, 248, 252, 256, 270, 280, 288, 292, 304, 312, 320, 328, 336, 352, 360, 364, 376, 384, 420, 424, 426
Offset: 1

Views

Author

Raphie Frank, Oct 04 2012

Keywords

Comments

For n in {1,2,3,4,6}, n = d(prime(n)-1). There are no others with this property, as conjectured by Raphie Frank and proved by Charles R Greathouse IV on Physics Forums (Nov, 2010).

Examples

			d(701 - 1)*7 = pi(701) = 126. The 126th prime is 701 and d(701 - 1) = 18; 18 divides 126 (7 times), so 126 is a member of this sequence.

Links

Crossrefs

Cf. A008328.

Programs

  • Mathematica
    Select[Range[352], Mod[#, DivisorSigma[0, Prime[#] - 1]] == 0 &] (* T. D. Noe, Oct 11 2012 *)
  • PARI
    is(n)=n%numdiv(prime(n)-1)==0 \\ Charles R Greathouse IV, Oct 09 2012

Extensions

a(12), a(31), a(39) from Charles R Greathouse IV, Oct 09 2012

A216326 Number of divisors of the degree of the minimal polynomial of 2*cos(Pi/prime(n)), with prime = A000040, n >= 1.

Original entry on oeis.org

1, 1, 2, 2, 2, 4, 4, 3, 2, 4, 4, 6, 6, 4, 2, 4, 2, 8, 4, 4, 9, 4, 2, 6, 10, 6, 4, 2, 8, 8, 6, 4, 6, 4, 4, 6, 8, 5, 2, 4, 2, 12, 4, 12, 6, 6, 8, 4, 2, 8, 6, 4, 16, 4, 8, 2, 4, 8, 8, 12, 4, 4, 6, 4, 12, 4, 8, 16, 2, 8, 10, 2, 4, 8, 8, 2, 4, 12, 12, 12, 4, 16, 4, 16, 4, 4, 12, 12, 8, 8, 2
Offset: 1

Views

Author

Wolfdieter Lang, Sep 27 2012

Keywords

Comments

See a comment on A216325 on the degree delta(n) = A055034(n) of the polynomial C(n,x) of 2*cos(Pi/n) (coefficients in A187360), Here n is prime.
For p prime, delta(p) = (p - 1)/2 if p > 2 and 1 if p = 2. a(n) is the number of divisors of delta(prime(n)), with prime(n) = A000040(n).
a(n) is also the number of distinct Modd p orders, p = prime, in row prime(n) of the table A216320. (For Modd n see a comment on A203571).
See also A008328 for the mod p analog of this sequence.

Examples

			a(6) = 4 because prime(6) = 13, and row n=13 of A216320 is [1  3  2  6  3  6] with 4 distinct numbers (Modd 13 orders).

Crossrefs

Cf. A000005, A055034, A000040.
Cf. A187360, A216320, A216325, A008328 (mod p analog).

Programs

  • PARI
    delta(n) = if (n==1, 1, eulerphi(2*n)/2); \\ A055034
a(n) = numdiv(delta(prime(n))); \\ Michel Marcus, Sep 12 2023

Formula

a(n) = tau(delta(prime(n))), n>=1, with tau = A000005 (number of divisors), delta = A055034 and prime = A000040.

A340870 a(n) is the smallest prime p such that p - 1 has 2*n divisors.

Original entry on oeis.org

3, 7, 13, 31, 113, 61, 193, 211, 181, 241, 13313, 421, 12289, 2113, 1009, 1321, 2424833, 1801, 786433, 2161, 4801, 15361, 155189249, 2521, 6481, 61441, 6301, 8641, 3489660929, 12241, 3221225473, 7561, 64513, 1376257, 58321, 12601, 206158430209, 8650753, 184321
Offset: 1

Views

Author

Jaroslav Krizek, Jan 24 2021

Keywords

Comments

First differs from A080372(n) + 1 for n = 17, where a(17) = 2424833, whereas A080372(17) + 1 = 2162689. - Hugo Pfoertner, Jan 26 2021

Examples

			a(4) = 31 because 31 is the smallest prime p such that p - 1 has 2*4 divisors; tau(30) = 8.

Crossrefs

Cf. A000005 (tau), A080372, A008328.
Cf. A066814 (p-1 has n divisors), A340799 (p+1 has 2*n divisors).

Programs

  • Magma
    Ax:=func; [Ax(n): n in[1..20]]
  • Mathematica
    a={}; For[n=1,n<=40,n++,i=1;While[DivisorSigma[0,Prime[i]-1]!=2n,i++];AppendTo[a,Prime[i]]]; a (* Stefano Spezia, Jan 25 2021 *)
  • PARI
    a(n) = my(p=2); while(numdiv(p-1) != 2*n, p=nextprime(p+1)); p; \\ Michel Marcus, Jan 25 2021

Formula

tau(a(n) - 1) = 2*n.
Previous Showing 11-16 of 16 results.

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

Content is available under The OEIS End-User License Agreement.