A215197 -id:A215197 - cp's OEIS frontend

A274357 Numbers n such that n and n+1 both have 8 divisors.

104, 135, 189, 230, 231, 285, 296, 344, 374, 375, 429, 434, 609, 645, 663, 664, 741, 776, 782, 805, 874, 902, 903, 969, 986, 1001, 1015, 1022, 1029, 1065, 1085, 1095, 1105, 1106, 1112, 1130, 1161, 1208, 1221, 1245, 1265, 1269, 1309, 1310, 1334, 1335, 1374, 1406, 1431

Offset: 1

Charles R Greathouse IV, Jun 19 2016

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Intersection of A005237 and A030626.
Numbers n such that n and n+1 both have k divisors: A039832 (k=4), A049103 (k=6), A274357 (k=8), A215197 (k=10), A174456 (k=12), A274358 (k=14), A274359 (k=16), A274360 (k=18), A274361 (k=20), A274366 (k=22), A274362 (k=24), A274363 (k=26), A274364 (k=28), A274365 (k=30).
Cf. A000005.

SequencePosition[DivisorSigma[0,Range[2000]],{8,8}][[All,1]] (* Harvey P. Dale, Sep 07 2021 *)

PARI
```
is(n)=numdiv(n)==8 && numdiv(n+1)==8
```

A075036 Smaller of two smallest consecutive numbers with 2n divisors.

Original entry on oeis.org

2, 14, 44, 104, 2511, 735, 29888, 2295, 6075, 5264, 2200933376, 5984, 689278976, 156735, 180224, 21735, 2035980763136, 223244, 9399153082499072, 458864, 41680575, 701443071, 2503092614937444351, 201824, 2707370000, 29785673727, 46977524, 5475519, 1737797404898095794225152

Offset: 1

Amarnath Murthy, Sep 03 2002

nonn

There cannot be two consecutive numbers with the same odd number of divisors as both cannot be squares.
These numbers have the property that a(n) * (a(n) + 1) has 4*n^2 divisors. - David A. Corneth, Jun 24 2016
Conjecture: if a term k is even, the highest p-adic order of k (the maximum may be attained by several p's) occurs at p=2 and the highest p-adic order of k+1 occurs at p=3. If a term k is odd, the highest p-adic order of k occurs at p=3 and the highest p-adic order of k+1 occurs at p=2. - Chai Wah Wu, Mar 12 2019
a(49) = 378401464109375, a(58) = 79921490583489592950783. - Jon E. Schoenfield, May 07 2022
a(51) = 34210814718574592, a(55) = 2481402804069375, a(57) = 394311388855795712. - Jon E. Schoenfield, Nov 06 2023 - Nov 08 2023

			a(4) = 104 as tau(104) = tau(105) = 8.

Chai Wah Wu, Table of n, a(n) for n = 1..48
Ray Chandler, Known terms and upper limits.

Cf. A000005, A005237, A039832, A049103, A174456, A215197, A215199, A274357, A274358, A274359.

a[n_] := (For[k=1, ! (DivisorSigma[0, k] == 2*n && DivisorSigma[0, k+1] == 2*n), k++]; k); Array[a, 10] (* Giovanni Resta, Jun 24 2016 *)

a(n) = my(k=1); while(numdiv(k)!=2*n || numdiv(k+1)!=2*n, k++); k \\ Felix Fröhlich, Jun 24 2016

a(n) <= A215199(n-1) for n > 1. Conjecture: if p is prime, then a(p) = A215199(p-1). This conjecture is true if the conjecture in A215199 is true. The b-file of A215199 thus shows that a(p) = A215199(p-1) for prime p < 1279. - Chai Wah Wu, Mar 12 2019

a(5)-a(24) from Max Alekseyev, Mar 12 2009
a(25)-a(28) from Giovanni Resta, Jun 24 2016
a(29) from Chai Wah Wu, Mar 12 2019

A215199 Smallest number k such that k and k+1 are both of the form p*q^n where p and q are distinct primes.

Original entry on oeis.org

14, 44, 135, 2511, 8991, 29888, 916352, 12393728, 155161088, 2200933376, 6856828928, 689278976, 481758175232, 3684603215871, 35419114668032, 2035980763136, 174123685117952, 9399153082499072, 19047348965998592, 203368956137832447, 24217192574746623, 2503092614937444351

Offset: 1

Michel Lagneau, Aug 05 2012

nonn

a(15) <= 35419114668032. - Donovan Johnson, Aug 22 2012
If k is a term such that k = p*q^n and k+1 = r*s^n, where p,q,r,s are primes, then clearly q != s. Conjecture: q and s are either 2 or 3 for all terms. - Chai Wah Wu, Mar 10 2019
Since q^n and s^n are coprime, the Chinese Remainder Theorem can be used to find candidate terms to test, i.e., numbers k such that k+1 == 0 (mod s^n) and k+1 == 1 (mod q^n) (see Python code). - Chai Wah Wu, Mar 12 2019
From David A. Corneth, Mar 13 2019: (Start)
Conjecture: Let 1 <= D < 2^n be the denominator of N/D of (3/2)^n. Without loss of generality, if the conjecture above holds that (q, s) = (2, 3) then r = D + k*2^n for some n.
Example: for n = 100, we have the continued fraction of (3/2)^100 to be 406561177535215237, 2, 1, 1, 14, 9, 1, 1, 2, 2, 1, 4, 1, 2, 6, 5, 1, 195, 3, 26, 39, 6, 1, 1, 1, 2, 7, 1, 4, 2, 1, 11, 1, 25, 6, 1, 4, 3, 2, 112, 1, 2, 1, 3, 1, 3, 4, 8, 1, 1, 12, 2, 1, 3, 2, 2 from which we compute D = 519502503658624787456021964081. We find r = 1100840223501761745286594404230449 = D + 868 * 2^100 giving a(100) + 1 = r*3^100. (End)

			a(3) = 135 because 135 = 5*3^3 and 136 = 17*2^3;
a(4) = 2511 because 2511 = 31*3^4 and 2512 = 157*2^4.

Chai Wah Wu, Table of n, a(n) for n = 1..1279 (terms 25..32 from David A. Corneth)

Cf. A074172, A215173, A215197, A215198.

psig := proc(n)
    local s,p ;
    s := [] ;
    for p in ifactors(n)[2] do
        s := [op(s),op(2,p)] ;
    end do:
    sort(s) ;
end proc:
A215199 := proc(n)
    local slim,smi,sma,ca,qi,q,p,k ;
    for slim from 0 do
        smi := slim*1000 ;
        sma := (slim+1)*1000 ;
        ca := sma ;
        q := 2 ;
        for qi from 1 do
            p := nextprime(floor(smi/q^n)-1) ;
            while p*q^n < sma do
                if p <> q then
                    k := p*q^n ;
                    if psig(k+1) = [1,n] then
                        ca := min(ca,k) ;
                    end if;
                end if;
                p := nextprime(p) ;
            end do:
            if q^n >= sma then
                break;
            end if;
            q := nextprime(q) ;
        end do:
        if ca < sma then
            return ca ;
        end if;
    end do:
end proc:
for n from 1 do
    print(A215199(n)) ;
end do; # R. J. Mathar, Aug 07 2012

from sympy import isprime, nextprime
from sympy.ntheory.modular import crt
def A215199(n):
    l = len(str(3**n))-1
    l10, result = 10**l, 2*10**l
    while result >= 2*l10:
        l += 1
        l102, result = l10, 20*l10
        l10 *= 10
        q, qn = 2, 2**n
        while qn <= l10:
            s, sn = 2, 2**n
            while sn <= l10:
                if s != q:
                    a, b = crt([qn,sn],[0,1])
                    if a <= l102:
                        a = b*(l102//b) + a
                    while a < l10:
                        p, t = a//qn, (a-1)//sn
                        if p != q and t != s and isprime(p) and isprime(t):
                            result = min(result,a-1)
                        a += b
                s = nextprime(s)
                sn = s**n
            q = nextprime(q)
            qn = q**n
    return result # Chai Wah Wu, Mar 12 2019

a(10)-a(14) from Donovan Johnson, Aug 22 2012
a(15)-a(17) from Chai Wah Wu, Mar 09 2019
a(18)-a(22) from Chai Wah Wu, Mar 10 2019

A215198 Numbers n such that n and n + 1 are both of the form p*q^5 where p and q are distinct primes.

Original entry on oeis.org

8991, 9375, 335583, 364256, 488672, 535328, 677727, 690848, 755487, 768608, 864351, 908576, 924128, 955232, 1097631, 1377567, 1424223, 1608416, 1688607, 1875231, 2121632, 2124063, 2168288, 2277152, 2541536, 2575071, 2621727, 2901663, 3190624, 3241376, 3409375

Offset: 1

Michel Lagneau, Aug 05 2012

nonn

The smaller of adjacent values in A178740. - R. J. Mathar, Aug 08 2012

			8991 is a member as 8991 = 37*3^5 and 8992 = 281*2^5.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A074172, A178740, A215173, A215197.

with(numtheory):for n from 3 to 10^7 do:x:=factorset(n):y:=factorset(n+1):n1:=nops(x):n2:=nops(y):if n1=2 and n2=2 then xx1:=x[1]*x[2]^5 : xx2:=x[2]*x[1]^5:yy1:=y[1]*y[2]^5: yy2:=y[2]*y[1]^5:if (xx1=n or xx2=n) and (yy1=n+1 or yy2=n+1) then printf("%a, ", n):else fi:fi:od:

lst={}; Do[f1=FactorInteger[n]; If[Sort[Transpose[f1][[2]]]=={1, 5}, f2=FactorInteger[n+1]; If[Sort[Transpose[f2][[2]]]=={1, 5}, AppendTo[lst, n]]], {n, 3, 10^7}]; lst
SequencePosition[Table[If[Sort[FactorInteger[n][[;;,2]]]=={1,5},1,0],{n,341*10^4}],{1,1}][[;;,1]] (* Harvey P. Dale, Nov 04 2023 *)

cp's OEIS Frontend

A274357 Numbers n such that n and n+1 both have 8 divisors.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

A075036 Smaller of two smallest consecutive numbers with 2n divisors.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A215199 Smallest number k such that k and k+1 are both of the form p*q^n where p and q are distinct primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Python

Extensions

A215198 Numbers n such that n and n + 1 are both of the form p*q^5 where p and q are distinct primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica