A061286 -id:A061286 - cp's OEIS frontend

A358699 a(n) is the largest prime factor of 2^(prime(n) - 1) - 1.

3, 5, 7, 31, 13, 257, 73, 683, 127, 331, 109, 61681, 5419, 2796203, 8191, 3033169, 1321, 599479, 122921, 38737, 22366891, 8831418697, 2931542417, 22253377, 268501, 131071, 28059810762433, 279073, 54410972897, 77158673929, 145295143558111, 2879347902817, 10052678938039

Offset: 2

Hugo Pfoertner, Nov 27 2022

nonn

Amiram Eldar, Table of n, a(n) for n = 2..197 (terms 2..120 from Hugo Pfoertner)

Cf. A005420, A006093, A006530, A061286, A071243, A086019, A098102.

Subsequence of A005420 and of A274906.

forprime (p=3, 140, my(f=factor(2^(p-1)-1)); print1(f[#f[,1],1],", "))

from sympy import primefactors, sieve
def A358699(n): return primefactors(2**(sieve[n]-1)-1)[-1] # Karl-Heinz Hofmann, Nov 28 2022

a(n) = A006530(A098102(n)). - Michel Marcus, Nov 28 2022

a(n) = A005420(A006093(n)). - Amiram Eldar, Dec 01 2022

A061236 Smallest number with prime(n)^3 divisors where prime(n) is n-th prime.

Original entry on oeis.org

24, 900, 810000, 729000000, 590490000000000, 531441000000000000, 430467210000000000000000, 387420489000000000000000000, 313810596090000000000000000000000, 228767924549610000000000000000000000000000, 205891132094649000000000000000000000000000000

Offset: 1

Labos Elemer, Jun 01 2001

nonn

			If p = 2, then d(128) = d(24) = d(30) = 8 and a(1) = 24 < 30 is the smallest.
If p = 5, then 2^124 > (2^24)*(3^4) > 30^4 = 810000 = a(3).

Amiram Eldar, Table of n, a(n) for n = 1..123

Cf. A000005, A005179, A003680, A030078, A037992, A061283, A061286, A061148, A061149, A061234.

For p = 2, 24 is the solution. If a prime p > 2, the suitable powers of 30 are the least solutions: a(n) = Min{x | d(x) = A000005(x) = p(n)^3} = 30^(prime(n)-1). d(2^(ppp-1)) = d(2^(pp-1)*3^(p-1)) = d(30^(p-1)) = p^3 and 2^(ppp-1) > 2^(pp-1)*3^(p-1) > 30^(p-1) holds if p > 2.

a(n) = A005179(A030078(n)) = A005179(prime(n)^3). - Amiram Eldar, Jan 23 2025

a(10)-a(11) from Amiram Eldar, Jan 23 2025

A071243 Record terms in A005179.

Original entry on oeis.org

1, 2, 4, 6, 16, 64, 1024, 4096, 65536, 262144, 4194304, 268435456, 1073741824, 68719476736, 1099511627776, 4398046511104, 70368744177664, 4503599627370496, 288230376151711744, 1152921504606846976

Offset: 1

Amarnath Murthy, May 20 2002

nonn

The only term that is not a power of 2 is the 4th term 6.

Essentially the same as A061286 with terms {1,6} added.

Cf. A005179.

For n>2, a(n+2) = A061286(n) = 2^(A000040(n)-1) = 2^A006093(n).

Edited, corrected and extended by Ray Chandler, Jun 28 2008

A121858 Smallest odd number having prime(n) divisors, where prime(n) is the n-th prime=A000040(n).

Original entry on oeis.org

3, 9, 81, 729, 59049, 531441, 43046721, 387420489, 31381059609, 22876792454961, 205891132094649, 150094635296999121, 12157665459056928801, 109418989131512359209, 8862938119652501095929, 6461081889226673298932241

Offset: 1

Lekraj Beedassy, Aug 30 2006

nonn

a(n) is also the smallest number k with the property that the symmetric representation of sigma(k) has prime(n) subparts. - Omar E. Pol, Oct 08 2022

Cf. A006093, A061286.

Cf. A000040, A038546, A237593, A279387.

3^(Prime[Range[20]]-1) (* Harvey P. Dale, Mar 19 2013 *)

a(n) = 3^(prime(n)-1) = 3^A006093(n).

a(n) = A038547(A000040(n)). - Omar E. Pol, Oct 08 2022

A135630 2^(prime(n) - 2) - 1.

Original entry on oeis.org

0, 1, 7, 31, 511, 2047, 32767, 131071, 2097151, 134217727, 536870911, 34359738367, 549755813887, 2199023255551, 35184372088831, 2251799813685247, 144115188075855871, 576460752303423487, 36893488147419103231, 590295810358705651711

Offset: 1

Omar E. Pol, Mar 01 2008

			a(4)=31 because the 4th prime number is 7 and 7-2=5 and 2^5=32 and 32-1=31.

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Cf. A000040, A040976, A061286.

[2^(NthPrime(n)-2)-1: n in [1..30]]; // Vincenzo Librandi, Sep 06 2014

Table[2^(Prime[n]-2)-1,{n,20}]  (* Harvey P. Dale, Feb 25 2011 *)

a(n) = 2^(A000040(n)-2) - 1 = 2^(A040976(n)) - 1 = (2^A000040(n)/4) - 1 = (A061286(n)/2) - 1.

More terms from Harvey P. Dale, Feb 25 2011

A137490 Numbers with 27 divisors.

Original entry on oeis.org

900, 1764, 2304, 4356, 4900, 6084, 6400, 10404, 11025, 12100, 12544, 12996, 16900, 19044, 23716, 26244, 27225, 28900, 30276, 30976, 33124, 34596, 36100, 38025, 43264, 49284, 52900, 53361, 56644, 60516, 65025, 66564, 70756, 73984, 74529

Offset: 1

R. J. Mathar, Apr 22 2008

nonn

Maple implementation: see A030513.

Numbers of the form p^26 (subset of A089081), p^2*q^2*r^2 (like 900, 1764, 4356, squares of A007304) or p^2*q^8 (like 2304, 6400, subset of the squares of A030628) where p, q and r are distinct primes. - R. J. Mathar, Mar 01 2010

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000005, A030513-A030516, A030626, A030627, A030634-A030638, A005179, A003680, A096932, A061286, A061283.

Select[Range[150000],DivisorSigma[0,#]==27&] (* Vladimir Joseph Stephan Orlovsky, May 06 2011 *)

isA137490(n) = numdiv(n)==27 \\ Michael B. Porter, Apr 10 2010

A000005(a(n)) = 27.

Sum_{n>=1} 1/a(n) = (P(2)^3 + 2*P(6) - 3*P(2)*P(4))/6 + P(2)*P(8) - P(10) + P(26) = 0.00453941..., where P is the prime zeta function. - Amiram Eldar, Jul 03 2022

A185445 Smallest number having exactly t divisors, where t is the n-th triprime (A014612).

Original entry on oeis.org

24, 60, 180, 240, 900, 960, 720, 2880, 15360, 3600, 6480, 61440, 14400, 46080, 983040, 25920, 32400, 3932160, 184320, 62914560, 233280, 230400, 2949120, 129600, 414720, 11796480, 4026531840, 921600, 16106127360, 810000, 1658880, 188743680, 1166400, 1030792151040, 14745600, 3732480

Offset: 1

Jonathan Vos Post, Feb 03 2011

This is the 3rd row of an infinite array A[k,n] = smallest number having exactly j divisors where j is the n-th natural number with exactly k prime factors (with multiplicity).

The first row is A061286, the second row is A096932.

			a(10) is 3600 because the 10th triprime is 45, and the smallest number with exactly 45 factors is 3600 = 2^4 * 3^2 * 5^2.
a(20) is 62914560 because the 10th triprime is 92, and the smallest number with exactly 92 factors is 62914560 = 2^22 * 3 * 5.

Cf. A005179, A014612, A061286, A096932.

from math import isqrt, prod
from sympy import isprime, primepi, primerange, integer_nthroot, prime, divisors
def A185445(n):
    def mult_factors(n):
        if isprime(n):
            return [(n,)]
        c = []
        for d in divisors(n,generator=True):
            if 1Chai Wah Wu, Aug 17 2024

a(n) = A005179(A014612(n)).

A332878 a(1)=1; a(n+1) is the smallest number not yet seen in the sequence which is divisible by 1 + Omega(a(n)).

Original entry on oeis.org

1, 2, 4, 3, 6, 9, 12, 8, 16, 5, 10, 15, 18, 20, 24, 25, 21, 27, 28, 32, 30, 36, 35, 33, 39, 42, 40, 45, 44, 48, 54, 50, 52, 56, 55, 51, 57, 60, 65, 63, 64, 7, 14, 66, 68, 72, 78, 76, 80, 84, 70, 88, 75, 92, 96, 49, 69, 81, 85, 87, 90, 95, 93, 99, 100, 105, 104

Offset: 1

David James Sycamore, Feb 28 2020

nonn

Here Omega = A001222, the number of prime factors counted with multiplicity.
Conjecture: This is a permutation of the natural numbers in which the primes appear in their natural order. Prime p > 2 arises as a(k) if and only if a(k-1) = 2^(p-1), in which case a(k+1) = 2*p. The sequence of numbers k such that a(k) is prime starts 2, 4, 10, 42, ... How does it continue?
a(636) = 11, a(2530) = 13, a(39731) = 17. It appears that the prime p occurs roughly at index 2^(p-2)*(1 + O(1/log p)). It is followed by 2p and then a multiple of 3. The graph of the sequence has several "branches" which can be labeled by odd primes: Most numbers occur on the main (p=3) branch which has an initial slope of about 1.61 increasing to 1.65 in the range 1e4 .. 4e4. A smaller fraction of the numbers lie on a second (p=5) and third (p=7) branch with slope of roughly 1.25 resp. 1.11 around n ~ 4e4, and a very small fraction lies on the branches with even lower slope (about 0.15 for the p=11 and 0.035 for the p=13 branch). - M. F. Hasler, Mar 04 2020

			a(1) = 1 => 1 + Omega(a(1)) = 1, so a(2) must be 2.
Then 1 + Omega(a(2)) = 2, so a(3) = 4.
Since 1 + Omega(4) = 3, a(4) = 3; then 1 + Omega(3) = 2 and a(5) = 6, etc.

M. F. Hasler, Table of n, a(n) for n = 1..10000
M. F. Hasler, Scatterplot of a(1..40000).

Cf. A000040, A001222, A061286.

a[1]=1; a[n_] := a[n] = Block[{s = Array[a, n-1], k, o = 1 + PrimeOmega@ a[n-1]}, k = o; While[ MemberQ[s, k], k += o]; k]; Array[a, 56] (* Giovanni Resta, Mar 04 2020 *)

{A332878_vec(N,a=1,u=1)=vector(N,n, a=n=bigomega(a)+1; while(bittest(u,a), a+=n); u+=1<M. F. Hasler, Mar 04 2020

A343327 Start with a(1)=1; thereafter a(n+1) = Sum_{k=1..n such that d(a(k)) = d(a(n))} d(a(k)) where d = A000005.

Original entry on oeis.org

1, 1, 2, 2, 4, 3, 6, 4, 6, 8, 12, 6, 16, 5, 8, 20, 12, 18, 24, 8, 24, 16, 10, 28, 30, 24, 32, 36, 9, 9, 12, 42, 40, 48, 10, 32, 48, 20, 54, 56, 64, 7, 10, 36, 18, 60, 12, 66, 72, 24, 80, 30, 88, 96, 36, 27, 40, 104, 112, 40, 120, 16, 15, 44, 72, 48, 50, 78, 128, 136, 144, 15

Offset: 1

David James Sycamore, Apr 12 2021

nonn

Every nonzero number occurs in the sequence since all terms in A000005 are included. The multiplicity of any term m (> 2) is d(m)-1 (since m > 2 cannot be created by adding m 1s). Prime 2 occurs twice, consecutively (see example), and thereafter no two adjacent terms can be prime because the term prior to a prime q is the first occurrence of 2^(q-1) (composite for q > 2), and since d(q)=2, the following term is a multiple (> 1) of 2. Thus each prime > 2 is flanked by composite even numbers, and occurs once only. Conjecture: The primes, after the first appearance of 2, appear in their natural order.

The primes appear in order at a(n) for n in {3, 4, 6, 14, 42, 507, 1939, 22454, 90322, ...}, with a(3) = a(4) = 2. No further primes appear for n <= 2^20. - Michael De Vlieger, May 01 2021

			Given a(1)=1, a(2) is also 1, since a(1) has no prior term. a(3)=1+1=2 and a(4)=2 because a(3) has 2 divisors and there is no prior term with 2 divisors. a(5)=2+2=4.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Scatterplot of a(n) for n = 1..2^20 showing trajectories of tau(a(n)).
Michael De Vlieger, Annotated scatterplot of a(n) for n = 1..256, noting and illustrating tau(a(n)) via color function.

Cf. A000005, A009087, A005179, A061286.

Block[{a = {1}, c}, Do[(If[! IntegerQ[c[#]], Set[c[#], 1], c[#]++]; AppendTo[a, # c[#]]) &@ DivisorSigma[0, a[[-1]]], 71]; a] (* Michael De Vlieger, Apr 12 2021 *)

lista(nn) = {my(va = vector(nn)); va[1] = 1; for (n=2, nn, va[n] = sum(k=1, n-1, if (numdiv(va[k]) == numdiv(va[n-1]), numdiv(va[k])));); va;} \\ Michel Marcus, Apr 28 2021

A349763 Numbers k such that d(k) = A000005(k), sigma(k) = A000203(k) and phi(k) = A000010(k) are all deficient numbers (A005100).

Original entry on oeis.org

1, 2, 3, 4, 8, 16, 48, 64, 121, 128, 192, 256, 512, 529, 1024, 2116, 2209, 2809, 3072, 3481, 4096, 6889, 8192, 8836, 11449, 12288, 13924, 14641, 16384, 17161, 18769, 22201, 27556, 27889, 29282, 29929, 32041, 32768, 36481, 45796, 51529, 54289, 57121, 63001, 65536

Offset: 1

Amiram Eldar, Nov 29 2021

nonn

Sándor (2005) proved that this sequence is infinite by showing that any number of the form 2^(p-1), where p is a sufficiently large prime, is a term. d(2^(p-1)) = p and phi(2^(p-1)) = 2^(p-2) are deficient for all primes, while sigma(2^(p-1)) = 2^p - 1 is deficient for a sufficiently large prime, a result of a theorem by Bojanić (1954): lim_{p prime -> oo} sigma(2^p - 1)/(2^p - 1) = 1.

			2 is a term since d(2) = 2, sigma(2) = 3 and phi(2) = 1 are all deficient numbers.

R. Bojanić, Asymptotic evaluations of the sum of divisors of certain numbers (in Serbo-Croatian), Bull. Soc. Math.-Phys. R. P. Macédoine, Vol. 5 (1954), pp. 5-15.

Amiram Eldar, Table of n, a(n) for n = 1..10000
József Sándor, Selected Chapters of Geometry, Analysis and Number Theory, 2005, pp. 132-134.

Cf. A000005, A000010, A000203, A005100, A061286.

Subsequence of A349759.

defQ[n_] := DivisorSigma[1, n] < 2*n; q[n_] := And @@ defQ /@ Join[DivisorSigma[{0, 1}, n], {EulerPhi[n]}]; Select[Range[10^5], q]

isdef(k) = sigma(k) < 2*k;
isok(k) = my(f=factor(k)); isdef(numdiv(f)) && isdef(sigma(f)) && isdef(eulerphi(k)); \\ Michel Marcus, Dec 01 2021

cp's OEIS Frontend

A358699 a(n) is the largest prime factor of 2^(prime(n) - 1) - 1.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Python

Formula

A061236 Smallest number with prime(n)^3 divisors where prime(n) is n-th prime.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

Extensions

A071243 Record terms in A005179.

Views

Author

Keywords

Comments

Crossrefs

Formula

Extensions

A121858 Smallest odd number having prime(n) divisors, where prime(n) is the n-th prime=A000040(n).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

A135630 2^(prime(n) - 2) - 1.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Formula

Extensions

A137490 Numbers with 27 divisors.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A185445 Smallest number having exactly t divisors, where t is the n-th triprime (A014612).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Python

Formula

A332878 a(1)=1; a(n+1) is the smallest number not yet seen in the sequence which is divisible by 1 + Omega(a(n)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica