A066814 -id:A066814 - cp's OEIS frontend

A103199 Primes p such that p-1 has more divisors than any smaller prime-1.

2, 3, 5, 7, 13, 31, 37, 61, 181, 241, 421, 1009, 1321, 1801, 2161, 2521, 6301, 7561, 12601, 15121, 20161, 30241, 45361, 55441, 100801, 110881, 196561, 332641, 498961, 786241, 982801, 1108801, 1580041, 1940401, 1995841, 2402401, 3880801, 4324321, 11476081, 11531521

Offset: 1

Don Reble, Mar 19 2005

nonn

There are infinitely many primes p such that d(p-1) > exp(c*log(p)/log(log(p))), where d(k) is the number of divisors of k, and c > 0 is a constant (Prachar, 1955). Therefore, this sequence is infinite. - Amiram Eldar, Apr 16 2024

David A. Corneth, Table of n, a(n) for n = 1..130 (first 75 terms from Amiram Eldar)
Karl Prachar, Über die Anzahl der Teiler einer natürlichen Zahl, welche die Form p-1 haben, Monatshefte für Mathematik, Vol. 59 (1955), pp. 91-97.

Cf. A000005, A008328, A066814, A067573, A072826.

seq[pmax_] := Module[{d, dm = 0, s = {}, p = 1}, While[p < pmax, p = NextPrime[p]; d = DivisorSigma[0, p-1]; If[d > dm, dm = d; AppendTo[s, p]]]; s]; seq[10^6] (* Amiram Eldar, Apr 16 2024 *)

lista(pmax) = {my(dm = 0, d); forprime(p = 1, pmax, d = numdiv(p-1); if(d > dm, dm = d; print1(p, ", ")));} \\ Amiram Eldar, Apr 16 2024

a(38)-a(40) added and name clarified by Amiram Eldar, Apr 16 2024

A072275 Least m such that one less than m-th prime has n divisors (or 0 if no such prime exists).

Original entry on oeis.org

2, 3, 4, 7, 6, 0, 11, 12, 30, 0, 18, 0, 44, 79, 47, 0, 42, 0, 53, 106, 1581, 0, 82, 211, 1470, 595, 319, 0, 169, 0, 216, 2819, 177975, 38836904, 279, 0, 62947, 41203, 326, 0, 647, 0, 1795, 907, 8719625, 0, 369, 0, 841, 12545382, 6180, 0, 820, 190066763704011

Offset: 2

Lekraj Beedassy, Jul 09 2002

nonn

A000040(a(n)) = A066814(n).

Do[ If[ !PrimeQ[n], k = 1; While[ Length[ Divisors[ Prime[k] - 1]] != n, k++ ]; Print[k], Print[0]], {n, 6, 34}]

More terms from Robert G. Wilson v, Jul 15 2002

a(35) inserted and a(46)-a(55) added by Amiram Eldar, Feb 06 2019

A167675 Least prime p such that p-2 has n divisors, or 0 if no such prime exists.

Original entry on oeis.org

3, 5, 11, 17, 83, 47, 0, 107, 227, 569, 59051, 317, 0, 9479, 2027, 947, 0, 2207, 0, 2837, 88211, 295247, 0, 3467, 50627, 9034499, 11027, 47387, 0, 14177, 0, 15017, 1476227, 215233607, 455627, 17327, 150094635296999123, 15884240051, 89813531, 36857, 0

Offset: 1

T. D. Noe, Nov 09 2009

nonn

This sequence is the idea of Alonso Del Arte. For n>2, a(n) is conjectured to be the smallest number that is orderly (see A167408) for n-1 values of k. For example, 11 is orderly for k=3 and 9. See A056899 for other primes p that are orderly for two k. It is a conjecture because it is not known whether there are composite numbers that are orderly for more than one value of k.

The terms a(n) for prime n are 0 except when 3^(n-1)+2 is prime. Using A051783, we find the exceptional primes to be n=2, 3, 5, 11, 37, 127, 6959.... For these n, a(n) = 3^(n-1)+2. For any n, it is easy to use the factorization of n to find the forms of numbers that have n divisors. For example, for n=38=2*19, we know that the prime must have the form 2+q*r^18 with q and r prime. The smallest such prime is 2+41*3^18.

Cf. A066814 (smallest prime p such that p-1 has n divisors)

nn=25; t=Table[0,{nn}]; Do[p=Prime[n]; k=DivisorSigma[0,p-2]; If[k<=nn && t[[k]]==0, t[[k]]=p], {n,2,10^6}]; t

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

Jaroslav Krizek, Jan 24 2021

nonn

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

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

Cf. A000005 (tau), A080372, A008328.

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

Magma
```
Ax:=func; [Ax(n): n in[1..20]]
```

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 *)

a(n) = my(p=2); while(numdiv(p-1) != 2*n, p=nextprime(p+1)); p; \\ Michel Marcus, Jan 25 2021

tau(a(n) - 1) = 2*n.

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A103199 Primes p such that p-1 has more divisors than any smaller prime-1.

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A072275 Least m such that one less than m-th prime has n divisors (or 0 if no such prime exists).

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A167675 Least prime p such that p-2 has n divisors, or 0 if no such prime exists.

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

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