A008328 -id:A008328 - cp's OEIS frontend

A103664 Primes p such that the number of divisors of p-1 is less than the number of divisors of p+1.

2, 3, 5, 11, 17, 23, 29, 47, 53, 59, 71, 79, 83, 89, 107, 131, 139, 149, 167, 173, 179, 191, 197, 223, 227, 233, 239, 251, 263, 269, 293, 311, 317, 347, 359, 367, 383, 389, 419, 431, 439, 443, 449, 461, 467, 479, 499, 503, 509, 557, 563, 569, 587, 593, 599, 607

Offset: 1

Hugo Pfoertner, Feb 19 2005

nonn

Mathematica coding by Wouter Meeussen and Robert G. Wilson v.

			a(1)=2 because d(1)=1 < d(3)=2; a(2)=3 because d(2)=2 < d(4)=3.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A008328 number of divisors of p-1, A008329 number of divisors of p+1, A103665, A103666, A103667.

with(numtheory): p:=proc(n) if isprime(n) and tau(n-1)Emeric Deutsch, Feb 22 2005

Select[Prime[Range[1, 140]], Length[Divisors[ # - 1]] < Length[Divisors[ # + 1]] &]
Select[Prime[Range[200]],DivisorSigma[0,#-1]Harvey P. Dale, May 31 2019 *)

A103665 Primes p such that the number of divisors of p-1 is greater than the number of divisors of p+1.

Original entry on oeis.org

13, 31, 37, 43, 61, 67, 73, 97, 101, 109, 113, 127, 151, 157, 163, 181, 193, 211, 229, 241, 257, 271, 277, 281, 283, 313, 331, 337, 353, 373, 379, 397, 401, 409, 421, 433, 457, 463, 487, 521, 523, 541, 547, 571, 577, 601, 613, 617, 631, 641, 661, 673, 677

Offset: 1

Hugo Pfoertner, Feb 19 2005

nonn

Mathematica coding by Wouter Meeussen and Robert G. Wilson v.

			a(1)=13 because d(12)=6 > d(14)=4.

Hugo Pfoertner, Table of n, a(n) for n = 1..10000

Cf. A008328 number of divisors of p-1, A008329 number of divisors of p+1, A103664, A103666, A103667.

Select[Prime[Range[2, 140]], Length[Divisors[ # - 1]] > Length[Divisors[ # + 1]] &]
Select[Prime[Range[200]],DivisorSigma[0,#-1]>DivisorSigma[0,#+1]&] (* Harvey P. Dale, Aug 21 2022 *)

forprime (k=2,700,if(numdiv(k-1)>numdiv(k+1),print1(k,", ")))
\\ Hugo Pfoertner, Nov 30 2017

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

Original entry on oeis.org

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

A145337 a(n) = d(p(n)+1) - d(p(n)-1), where d(m) = the number of divisors of m, p(n) = the n-th prime.

Original entry on oeis.org

1, 1, 1, 0, 2, -2, 1, 0, 4, 2, -2, -5, 0, -2, 6, 2, 8, -8, -2, 4, -8, 2, 8, 4, -6, -1, 0, 8, -4, -2, -4, 4, 0, 4, 6, -4, -8, -4, 12, 2, 14, -10, 6, -10, 3, 0, -10, 4, 8, -4, 4, 12, -14, 10, -1, 12, 10, -6, -8, -8, -2, 6, 0, 8, -12, 2, -10, -14, 8, 0, -4, 20, 2, -4, -4, 12, 10, -14, -7, -8

Offset: 0

Leroy Quet, Oct 08 2008

sign

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A008328, A008329, A067889, A103664, A103665, A145338.

A000005 := proc(n) numtheory[tau](n) ; end:
A008328 := proc(n) A000005(ithprime(n)-1) ; end:
A008329 := proc(n) A000005(ithprime(n)+1) ; end:
A145337 := proc(n) A008329(n)-A008328(n) ; end: # R. J. Mathar, Oct 10 2008

DivisorSigma[0,#+1]-DivisorSigma[0,#-1]&/@Prime[Range[80]] (* Harvey P. Dale, Nov 01 2011 *)

a(n) = A008329(n) - A008328(n). - R. J. Mathar, Oct 10 2008

More terms from R. J. Mathar and Ray Chandler, Oct 10 2008

A165318 Primes p where the number of divisors of p-1 is a power of 2.

Original entry on oeis.org

2, 3, 7, 11, 23, 31, 41, 43, 47, 59, 67, 71, 79, 83, 89, 103, 107, 131, 137, 139, 167, 179, 191, 211, 223, 227, 233, 239, 251, 263, 271, 281, 283, 311, 313, 331, 347, 359, 367, 379, 383, 409, 419, 431, 439, 443, 457, 463, 467, 479, 499, 503, 521, 547, 563, 569

Offset: 1

Leroy Quet, Sep 14 2009

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Cf. A000005, A008328, A036537, A165319, A165320.

isA000079 := proc(n) RETURN( n=1 or numtheory[factorset](n) = {2}) ; end: A165318 := proc(n) if n = 1 then 2; else p := nextprime(procname(n-1)) ; while not isA000079(numtheory[tau](p-1)) do p := nextprime(p) ; od; p ; fi; end: seq(A165318(n),n=1..90) ; # R. J. Mathar, Sep 18 2009

Select[Prime[Range[200]],IntegerQ[Log[2,DivisorSigma[0,#-1]]]&] (* Harvey P. Dale, Oct 14 2018 *)

isok(p) = isprime(p) && apply(x -> x >> valuation(x, 2), numdiv(p-1)) == 1; \\ Amiram Eldar, Jun 26 2025

More terms from R. J. Mathar, Sep 18 2009

A165320 Primes p where neither the number of divisors of p+1 nor the number of divisors of p-1 is a power of 2.

Original entry on oeis.org

17, 19, 97, 149, 163, 197, 199, 241, 293, 307, 337, 349, 449, 491, 523, 557, 577, 739, 773, 811, 881, 883, 991, 1013, 1051, 1061, 1151, 1171, 1249, 1277, 1279, 1423, 1451, 1459, 1471, 1493, 1531, 1549, 1601, 1637, 1667, 1693, 1709, 1733, 1747, 1861, 1949

Offset: 1

Leroy Quet, Sep 14 2009

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Cf. A000005, A008328, A008329, A036537, A165318, A165319.

isA000079 := proc(n) RETURN( n=1 or numtheory[factorset](n) = {2}) ; end: isA165320 := proc(n) RETURN ( isprime(n) and not isA000079(numtheory[tau](n-1)) and not isA000079(numtheory[tau](n+1)) ) ; end: for n from 1 to 10000 do if isA165320(n) then printf("%d,",n) ; fi; od: # R. J. Mathar, Sep 18 2009

fQ[n_] := Union[ IntegerQ@# & /@ Log[2, DivisorSigma[0, {n - 1, n + 1}]]] == {False}; Select[ Prime@ Range@ 300, fQ@# &] (* Robert G. Wilson v, Sep 16 2009 *)
Select[Prime[Range[300]],NoneTrue[Log2[DivisorSigma[0,#+{1,-1}]],IntegerQ]&] (* Harvey P. Dale, May 03 2023 *)

is1(k) = apply(x -> x >> valuation(x, 2), numdiv(k)) > 1;
isok(p) = isprime(p) && is1(p-1) && is1(p+1); \\ Amiram Eldar, Jun 26 2025

More terms from Robert G. Wilson v and R. J. Mathar, Sep 16 2009

A174842 Irregular triangle T(i,n) giving the number of elements of Zp having multiplicative order di, the i-th divisor of p-1, where p is the n-th prime.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 4, 4, 1, 1, 2, 2, 2, 4, 1, 1, 2, 4, 8, 1, 1, 2, 2, 6, 6, 1, 1, 10, 10, 1, 1, 2, 6, 6, 12, 1, 1, 2, 4, 2, 4, 8, 8, 1, 1, 2, 2, 2, 6, 4, 6, 12, 1, 1, 2, 4, 4, 4, 8, 16, 1, 1, 2, 2, 6, 6, 12, 12, 1, 1, 22, 22, 1, 1, 2, 12, 12, 24, 1, 1, 28, 28, 1, 1, 2, 2, 4, 2, 4, 4, 8, 8

Offset: 1

T. D. Noe, Mar 30 2010

The divisors of p-1 are assumed to be in increasing order. The first row, for prime 2, has only one term. All other rows begin with two 1s and end with phi(p-1). There are tau(p-1), the number of divisors of p-1, terms in each row. The sum of the terms in each row is p-1. When p is a prime of the form 4k-1, then the last two terms in the row are equal. When p is a prime of the form 4k+1, then the last two terms in the row have a ratio of 2.

			For prime p=17, the 7th prime, the multiplicative order of the numbers 1 to p-1 is 1, 8, 16, 4, 16, 16, 16, 8, 8, 16, 16, 16, 4, 16, 8, 2. There is one 1, one 2, two 4's, four 8's, and eight 16's. Hence row 7 is 1, 1, 2, 4, 8.

T. D. Noe, Rows n=1..500 of triangle, flattened
Eric W. Weisstein, MathWorld: Modulo Multiplication Group

A008328 (tau(p-1)), A008330 (phi(p-1)), A174843 (divisors of p-1)

Flatten[Table[EulerPhi[Divisors[p-1]], {p, Prime[Range[100]]}]]

T(i,n) = phi(di), where di is the i-th divisor of prime(n)-1.

A079552 Record values in A079551.

Original entry on oeis.org

0, 1, 3, 6, 10, 14, 20, 25, 31, 35, 41, 49, 58, 66, 74, 78, 84, 88, 100, 108, 116, 128, 136, 140, 148, 160, 169, 177, 181, 193, 203, 215, 223, 231, 239, 245, 257, 269, 279, 283, 289, 293, 311, 319, 333, 342, 354, 370, 378, 382, 394, 402, 410, 430, 438, 447, 451, 457, 473

Offset: 0

N. J. A. Sloane, Jan 24 2003

nonn

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A000005, A006093, A079551.

Partial sums of A008328.

RecurrenceTable[{a[0]==0, a[n]==DivisorSigma[0,Prime[n]-1]+a[n-1]}, a, {n, 100}] (* Jon Maiga, Jan 02 2019 *)

a(n) = A000005(A006093(n)) + a(n-1). - Jon Maiga, Jan 02 2019

A145339 a(n) = the minimum 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

1, 2, 3, 4, 4, 4, 5, 6, 4, 6, 6, 4, 8, 6, 4, 6, 4, 4, 6, 8, 4, 8, 4, 8, 6, 8, 8, 4, 8, 8, 8, 8, 8, 8, 6, 8, 4, 6, 4, 6, 4, 8, 8, 4, 9, 12, 6, 8, 4, 8, 8, 8, 6, 8, 8, 4, 6, 10, 4, 8, 6, 6, 12, 8, 4, 6, 6, 6, 4, 12, 8, 4, 8, 8, 12, 4, 6, 4, 8, 8, 8, 4, 8, 8, 8, 8, 14, 4, 12, 10, 4, 4, 8, 12, 8, 4, 6, 12, 6, 4, 6

Offset: 1

Leroy Quet, Oct 08 2008

nonn

Cf. A145340, A008328, A008329, A067889, A103664, A103665.

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

a(n) = my(p = prime(n)); min(numdiv(p-1), numdiv(p+1)); \\ Michel Marcus, Sep 28 2018

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

A174843 Irregular triangle in which row n lists the divisors of prime(n)-1.

Original entry on oeis.org

1, 1, 2, 1, 2, 4, 1, 2, 3, 6, 1, 2, 5, 10, 1, 2, 3, 4, 6, 12, 1, 2, 4, 8, 16, 1, 2, 3, 6, 9, 18, 1, 2, 11, 22, 1, 2, 4, 7, 14, 28, 1, 2, 3, 5, 6, 10, 15, 30, 1, 2, 3, 4, 6, 9, 12, 18, 36, 1, 2, 4, 5, 8, 10, 20, 40, 1, 2, 3, 6, 7, 14, 21, 42, 1, 2, 23, 46, 1, 2, 4, 13, 26, 52, 1, 2, 29, 58, 1, 2, 3, 4

Offset: 1

T. D. Noe, Mar 30 2010

Row n begins with 1, ends with prime(n)-1, and has A008328(n) terms.

			The first 10 rows:
  1
  1, 2
  1, 2, 4
  1, 2, 3, 6
  1, 2, 5, 10
  1, 2, 3, 4, 6, 12
  1, 2, 4, 8, 16
  1, 2, 3, 6, 9, 18
  1, 2, 11, 22
  1, 2, 4, 7, 14, 28

T. D. Noe, Rows n=1..500 of triangle, flattened

Cf. A008328 (row lengths), A008332 (row sums).

Flatten[Table[Divisors[p-1], {p, Prime[Range[100]]}]]

row(n) = divisors(prime(n)-1); \\ Amiram Eldar, May 02 2025

cp's OEIS Frontend

A103664 Primes p such that the number of divisors of p-1 is less than the number of divisors of p+1.

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A103665 Primes p such that the number of divisors of p-1 is greater than the number of divisors of p+1.

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

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A145337 a(n) = d(p(n)+1) - d(p(n)-1), where d(m) = the number of divisors of m, p(n) = the n-th prime.

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A165318 Primes p where the number of divisors of p-1 is a power of 2.

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A165320 Primes p where neither the number of divisors of p+1 nor the number of divisors of p-1 is a power of 2.

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A174842 Irregular triangle T(i,n) giving the number of elements of Zp having multiplicative order di, the i-th divisor of p-1, where p is the n-th prime.

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A079552 Record values in A079551.

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