A008329 -id:A008329 - 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

A119711 Primes p such that p+1, p+2 and p+3 have equal number of divisors.

Original entry on oeis.org

229, 241, 373, 1831, 2053, 2503, 3109, 5861, 6053, 6151, 6871, 8293, 8821, 9161, 9829, 12049, 13591, 13781, 14293, 14887, 16087, 17737, 19141, 19333, 20389, 21493, 23333, 23509, 24151, 25771, 27109, 28807, 29269, 31337, 33413, 33941, 35509

Offset: 1

Zak Seidov, Jul 29 2006

nonn

			229 is OK since 230, 231 and 232 all have 8 divisors: {1,2,5,10,23,46,115,230}, {1,3,7,11,21,33,77,231} and {1,2,4,8,29,58,116,232}.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A008329, A049234.

Select[Prime@Range@5000,DivisorSigma[0,#+1]==DivisorSigma[0,#+2]==DivisorSigma[0,#+3]&]

A119705 Primes p such that the number of divisors of p+1 equals number of divisors of p+2.

Original entry on oeis.org

13, 37, 43, 97, 103, 157, 229, 241, 331, 373, 433, 541, 547, 877, 907, 1021, 1129, 1201, 1373, 1381, 1433, 1489, 1543, 1597, 1613, 1621, 1741, 1831, 1951, 1987, 2017, 2053, 2161, 2377, 2503, 2539, 2557, 2633, 2677, 2713, 2857, 2953, 3061, 3067, 3109, 3169

Offset: 1

Zak Seidov, Jul 29 2006

nonn

Primes p such that A008329(p) = A049234(p).

			13 is a term because 14 and 15 each have 4 divisors: {1, 2, 7, 14} and {1, 3, 5, 15}.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A008329, A049234, A119711, A119728, A119730, A119740.

Select[Range[3200], PrimeQ[#] && DivisorSigma[0, # + 1] == DivisorSigma[0, # + 2] &] (* Amiram Eldar, Jan 26 2020 *)

isok(n) = isprime(n) && (numdiv(n+1) == numdiv(n+2)); \\ Michel Marcus, Oct 10 2013

A119728 Primes p such that p+1, p+2, p+3 and p+4 have equal number of divisors.

Original entry on oeis.org

241, 13781, 19141, 21493, 50581, 61141, 76261, 77431, 94261, 95383, 95413, 98101, 104743, 104869, 134581, 141653, 142453, 152629, 153991, 158341, 160933, 165541, 169111, 199831, 201511, 203431, 206551, 229351, 233941, 235111, 253013, 273367

Offset: 1

Zak Seidov, Jul 29 2006

nonn

			241 is a term since 242, 243, 244 and 245 all have 6 divisors:
{1,2,11,22,121,242},{1,3,9,27,81,243},{1,2,4,61,122,244} and {1,5,7,35,49,245}.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A008329, A049234, A119705, A119711, A119730, A119740.

Select[Prime@Range@50000,DivisorSigma[0,#+1]==DivisorSigma[0,#+2]==DivisorSigma[0,#+3]==DivisorSigma[0,#+4]&]

A119730 Primes p such that p+1, p+2, p+3, p+4 and p+5 have equal number of divisors.

Original entry on oeis.org

13781, 19141, 21493, 50581, 142453, 152629, 253013, 298693, 307253, 346501, 507781, 543061, 845381, 1079093, 1273781, 1354501, 1386901, 1492069, 1546261, 1661333, 1665061, 1841141, 2192933, 2208517, 2436341, 2453141, 2545013

Offset: 1

Zak Seidov, Jul 29 2006

nonn

			13781 is a term since 13782, 13783, 13784, 13785 and 13786 all have 8 divisors:
{1,2,3,6,2297,4594,6891,13782}, {1,7,11,77,179,1253,1969,13783},
{1,2,4,8,1723,3446,6892,13784}, {1,3,5,15,919,2757,4595,13785} and
{1,2,61,113,122,226,6893,13786}.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A008329, A049234, A119705, A119711, A119728, A119740.

Select[Prime@Range[1000000],DivisorSigma[0,#+1]==DivisorSigma[0,#+2]==DivisorSigma[0,#+3]==DivisorSigma[0,#+4]==DivisorSigma[0,#+5]&]
endQ[n_]:= Length[Union[DivisorSigma[0, (n + Range[5])]]]==1; Select[Prime[ Range[ 200000]],endQ] (* Harvey P. Dale, Jan 16 2019 *)

A119740 Primes p such that p+1, p+2, p+3, p+4, p+5 and p+6 have equal number of divisors.

Original entry on oeis.org

298693, 346501, 1841141, 2192933, 2861461, 3106981, 3375781, 3435589, 3437813, 3865429, 4597013, 6191461, 7016213, 7074901, 7637941, 7918373, 9196309, 10216901, 12798901, 13747429, 14100661, 14171653, 14770981, 14779189

Offset: 1

Zak Seidov, Jul 29 2006

nonn

			298693 is a term since 298694, 298695, 298696, 298697, 298698 and 298699 all have 8 divisors:
{1,2,11,22,13577,27154,149347,298694}, {1,3,5,15,19913,59739,99565,298695},
{1,2,4,8,37337,74674,149348,298696}, {1,7,71,497,601,4207,42671,298697},
{1,2,3,6,49783,99566,149349,298698}, {1,19,79,199,1501,3781,15721,298699}.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A008329, A049234, A119705, A119711, A119728, A119730.

Select[Prime@Range[1000000],DivisorSigma[0,#+1]==DivisorSigma[0,#+2]==DivisorSigma[0,#+3]==DivisorSigma[0,#+4]==DivisorSigma[0,#+5]==DivisorSigma[0,#+6]&]

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

A165319 Primes p where the number of divisors of p+1 is a power of 2.

Original entry on oeis.org

2, 5, 7, 13, 23, 29, 37, 41, 53, 61, 73, 101, 103, 109, 113, 127, 137, 151, 157, 167, 173, 181, 193, 229, 257, 263, 269, 277, 281, 311, 313, 317, 353, 373, 383, 389, 397, 401, 409, 421, 433, 439, 457, 461, 487, 509, 541, 569, 593, 601, 613, 617, 631, 641, 653

Offset: 1

Leroy Quet, Sep 14 2009

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Cf. A165318, A165320.

Cf. A008329. [R. J. Mathar, Sep 20 2009]

b:= proc(n) option remember; is(n=2^ilog2(n)) end:
a:= proc(n) option remember; local p; p:= `if`(n=1, 1, a(n-1));
      do p:= nextprime(p);
         if andmap(b, map(i-> i[2]+1, ifactors(p+1)[2])) then break fi
      od; p
    end:
seq(a(n), n=1..55);  # Alois P. Heinz, Sep 05 2019

okQ[p_] := PrimeQ[p] && IntegerQ[Log[2, DivisorSigma[0, p+1]]];
Select[Prime[Range[200]], okQ] (* Jean-François Alcover, May 20 2020 *)
Select[Prime[Range[150]],IntegerQ[Log2[DivisorSigma[0,#+1]]]&] (* Harvey P. Dale, Jul 30 2025 *)

isok(p) = isprime(p) && (nd = numdiv(p+1)) && (nd == 2^valuation(nd, 2)); \\ Michel Marcus, Sep 05 2019

Extended by R. J. Mathar, Sep 20 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

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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A119711 Primes p such that p+1, p+2 and p+3 have equal number of divisors.

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A119705 Primes p such that the number of divisors of p+1 equals number of divisors of p+2.

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A119728 Primes p such that p+1, p+2, p+3 and p+4 have equal number of divisors.

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A119730 Primes p such that p+1, p+2, p+3, p+4 and p+5 have equal number of divisors.

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A119740 Primes p such that p+1, p+2, p+3, p+4, p+5 and p+6 have equal number of divisors.

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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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