A037992 -id:A037992 - cp's OEIS frontend

A069782 Numbers k such that gcd(d(k^3), d(k)) = 2^w for some w.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107

Offset: 1

Labos Elemer, Apr 08 2002

The first missing integer is 432 (see in A069781).

			Below 100000 only 314 integers are missing, collected in A069781.

Andrew Howroyd, Table of n, a(n) for n = 1..10000
Tanya Khovanova, Non Recursions

Cf. A000005, A069780, A069781, A037992, A061701.

f[x_] := GCD[DivisorSigma[0, x^3], DivisorSigma[0, x]]; Do[s=f[n]; If[IntegerQ[Log[2, s]], Print[{n, s}]], {n, 1, 100000}]

is(n)=my(f=factor(n)[, 2], g=gcd(prod(i=1, #f, 3*f[i]+1), prod(i=1, #f, f[i]+1))); g>>valuation(g, 2)==1 \\ Charles R Greathouse IV, Oct 16 2015

A327634 Infinitary highly abundant numbers: numbers m such that isigma(m) > isigma(k) for all k < m, where isigma(k) is the sum of infinitary divisors of n (A049417).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 10, 12, 14, 18, 21, 22, 24, 30, 40, 42, 54, 66, 72, 78, 88, 96, 102, 114, 120, 168, 210, 216, 264, 312, 330, 360, 378, 384, 408, 456, 480, 510, 546, 552, 600, 672, 690, 696, 744, 840, 1080, 1320, 1512, 1560, 1848, 1920, 2040, 2184, 2280, 2688

Offset: 1

Amiram Eldar, Sep 20 2019

The infinitary version of A002093.

			The first 10 values of isigma(k) for k = 1 to 10 are: 1, 3, 4, 5, 6, 12, 8, 15, 10, 18. Record values are reached for all these values of k except for 7 and 9, therefore the sequence begins with 1, 2, 3, 4, 5, 6, 8, 10, ...

Amiram Eldar, Table of n, a(n) for n = 1..930

Cf. A002093, A037992, A049417, A285614, A292983.

f[p_, e_] := p^(2^(-1 + Position[Reverse @ IntegerDigits[e, 2], ?(# == 1 &)])); isigma[1] = 1; isigma[n] := Times @@ (Flatten @ (f @@@ FactorInteger[n]) + 1); seq = {};sm = 0; Do[s = isigma[n]; If[s > sm, sm = s; AppendTo[seq, n]], {n, 1, 10^4}]; seq

A061234 Smallest number with prime(n)^2 divisors where prime(n) is the n-th prime.

Original entry on oeis.org

6, 36, 1296, 46656, 60466176, 2176782336, 2821109907456, 101559956668416, 131621703842267136, 6140942214464815497216, 221073919720733357899776, 10314424798490535546171949056, 13367494538843734067838845976576

Offset: 1

Labos Elemer, Jun 01 2001

nonn

			1296 = 2*2*2*2*3*3*3*3 is the smallest number with 25 divisors.

Amiram Eldar, Table of n, a(n) for n = 1..208

Cf. A000005, A000040, A001248, A003680, A005179, A037992, A061148, A061149, A061283, A061286.

a(n) = Min_{x : d(x) = A000005(x) = p(n)^2} = 6^(p(n)-1) because x = 2^(pp-1) > 2^(p-1)3^(p-1) holds if p > 1.

a(n) = A005179(A001248(n)). - Amiram Eldar, Jun 21 2024

A069780 a(n) = gcd(d(n^3), d(n)).

Original entry on oeis.org

1, 2, 2, 1, 2, 4, 2, 2, 1, 4, 2, 2, 2, 4, 4, 1, 2, 2, 2, 2, 4, 4, 2, 8, 1, 4, 2, 2, 2, 8, 2, 2, 4, 4, 4, 1, 2, 4, 4, 8, 2, 8, 2, 2, 2, 4, 2, 2, 1, 2, 4, 2, 2, 8, 4, 8, 4, 4, 2, 4, 2, 4, 2, 1, 4, 8, 2, 2, 4, 8, 2, 2, 2, 4, 2, 2, 4, 8, 2, 2, 1, 4, 2, 4, 4, 4, 4, 8, 2, 4, 4, 2, 4, 4, 4, 4, 2, 2, 2, 1, 2, 8, 2, 8, 8

Offset: 1

Labos Elemer, Apr 08 2002

nonn

Terms are usually powers of 2. Smallest number m such that A069780(m)=2^n is A037992(n). The first n such that a(n) is not a power of 2 equals 432: a(432) = gcd(d(80621568), d(432)) = gcd(130,20) = 10.

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

Cf. A000005, A037992, A069781, A069782.

Table[GCD[DivisorSigma[0, n^3], DivisorSigma[0, n]], {n, 1, 500}]

a(n)=my(f=factor(n)[, 2]); gcd(prod(i=1, #f, 3*f[i]+1), prod(i=1, #f, f[i]+1)) \\ Charles R Greathouse IV, Oct 16 2015

a(n) = gcd(A000005(n^3), A000005(n)).

A069781 Numbers k such that gcd(d(k^3), d(k)) is not a power of 2.

Original entry on oeis.org

432, 576, 648, 1600, 2000, 2160, 2880, 2916, 3024, 3136, 3240, 4032, 4536, 4752, 4800, 5000, 5488, 5616, 6000, 6336, 7128, 7344, 7488, 7744, 8208, 8424, 9408, 9792, 9936, 10125, 10800, 10816, 10944, 11016, 11200, 12312, 12528, 13248, 13392

Offset: 1

Labos Elemer, Apr 08 2002

nonn

The complement of this sequence in the positive integers A000027 is A069782. - M. F. Hasler, Jan 18 2015

The numbers of the form 4*3^(7*m - 1), m >= 1, are terms. - Marius A. Burtea, Oct 18 2019

			For n<100000, gcd[d(n^3),d[n]] = {5,7,10,14,20,28,40,80} which is obtained for n={20736,576,432,2880,54000,20160,2160,15120} respectively.

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

Cf. A000005, A061701, A037992, A069780, A069782.

f:=func; [k:k in [1..14000]| not IsIntegral(Log(2,f(k)))]; // Marius A. Burtea, Oct 18 2019

f[x_] := GCD[DivisorSigma[0, x^3], DivisorSigma[0, x]] Do[s=f[n]; If[ !IntegerQ[Log[2, s]], Print[n]], {n, 1, 100000}]
Select[Range[14000],!IntegerQ[Log[2,GCD[DivisorSigma[0,#^3], DivisorSigma[ 0,#]]]]&] (* Harvey P. Dale, Mar 20 2018 *)

is(n)=my(f=factor(n)[,2], g=gcd(prod(i=1,#f,3*f[i]+1), prod(i=1,#f,f[i]+1))); g!=1<Charles R Greathouse IV, Oct 16 2015

log_2(gcd(A000005(n^3), A000005(n))) is nonintegral.

A335386 Tri-unitary highly composite numbers: where the number of tri-unitary divisors (A335385) increases to a record.

Original entry on oeis.org

1, 2, 6, 24, 120, 840, 7560, 83160, 1081080, 18378360, 349188840, 8031343320, 200783583000, 5822723907000, 180504441117000, 6678664321329000, 273825237174489000, 11774485198503027000, 553400804329642269000, 27116639412152471181000, 1437181888844080972593000

Offset: 1

Amiram Eldar, Jun 04 2020

nonn

Amiram Eldar, Table of n, a(n) for n = 1..32

Analogous sequences: A002182 (highly composite), A002110 (unitary), A037992 (infinitary), A293185 (bi-unitary), A318278 (exponential), A306736 (exponential infinitary), A307845 (exponential unitary), A309141 (nonunitary), A322484 (semi-unitary).

Cf. A335385.

f[p_, e_] := If[e == 3 || e == 6, 4, 2]; d[1] = 1; d[n_] := Times @@ (f @@@ FactorInteger[n]); dm = 0; s = {}; Do[If[(d1 = d[n]) > dm, dm = d1; AppendTo[s, n]], {n, 1, 1100000}]; s

A335385(a(n)) = 2^(n-1).

A374786 Numerator of the mean infinitary abundancy index of the infinitary divisors of n.

Original entry on oeis.org

1, 5, 7, 9, 11, 35, 15, 45, 19, 11, 23, 21, 27, 75, 77, 33, 35, 95, 39, 99, 5, 115, 47, 105, 51, 135, 133, 135, 59, 77, 63, 165, 161, 175, 33, 19, 75, 195, 63, 99, 83, 25, 87, 207, 209, 235, 95, 77, 99, 51, 245, 243, 107, 665, 23, 675, 91, 295, 119, 231, 123, 315

Offset: 1

Amiram Eldar, Jul 20 2024

The infinitary abundancy index of a number k is A049417(k)/k.
The record values of a(n)/A374787(n) are attained at the terms of A037992.
The least number k such that a(k)/A374787(k) is larger than 2, 3, 4, ..., is A037992(6) = 7560, A037992(33) = 1370819010042780920891599455129161859473627856000, ... .

			For n = 4, 4 has 2 infinitary divisors, 1 and 4. Their infinitary abundancy indices are isigma(1)/1 = 1 and isigma(4)/4 = 5/4, and their mean infinitary abundancy index is (1 + 5/4)/2 = 9/8. Therefore a(4) = numerator(9/8) = 9.

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

Cf. A005117, A037445, A037992, A049417, A050376, A077609, A138302, A327574, A374787 (denominators).

Similar sequences: A374777/A374778, A374783/A374784.

f[p_, e_] := p^(2^(-1 + Position[Reverse@IntegerDigits[e, 2], ?(# == 1 &)])); a[1] = 1; a[n] := Numerator[Times @@ (1 + 1/(2*Flatten@ (f @@@ FactorInteger[n])))]; Array[a, 100]

a(n) = {my(f = factor(n), b); numerator(prod(i = 1, #f~, b = binary(f[i, 2]); prod(k=1, #b, if(b[k], 1 + 1/(2*f[i, 1]^(2^(#b-k))), 1))));}

Let f(n) = a(n)/A374787(n). Then:
f(n) = (Sum_{d infinitary divisor of n} isigma(d)/d) / id(n), where isigma(n) is the sum of infinitary divisors of n (A049417), and id(n) is their number (A037445).
f(n) is multiplicative with f(p^e) = Product{k>=1, e_k=1} (1 + 1/(2*p^(2^(k+1)))), where e = Sum_{k} e_k * 2^k is the binary representation of e, i.e., e_k is bit k of e.
f(n) = (Sum_{d infinitary divisor of n} d*id(d)) / (n*id(n)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} f(k) = Product_{P} (1 + 1/(2*P*(P+1))) = 1.21407233718434377029..., where P are numbers of the form p^(2^k) where p is prime and k >= 0 (A050376). For comparison, the asymptotic mean of the infinitary abundancy index over all the positive integers is 1.461436... = 2 * A327574.
Lim sup_{n->oo} f(n) = oo (i.e., f(n) is unbounded).
f(n) <= A374777(n)/A374778(n) with equality if and only if n is squarefree (A005117).
f(n) >= A374783(n)/A374784(n) with equality if and only if n is in A138302.

A333931 Recursive highly composite numbers: numbers with a record number of recursive divisors (A282446).

Original entry on oeis.org

1, 2, 4, 6, 12, 30, 36, 60, 180, 420, 900, 1260, 4620, 6300, 13860, 44100, 55440, 69300, 180180, 485100, 720720, 900900, 2882880, 3063060, 6306300, 12252240, 15315300, 49008960, 58198140, 107207100, 232792560, 290990700, 931170240, 1163962800, 2036934900, 4655851200

Offset: 1

Amiram Eldar, Apr 10 2020

nonn

The corresponding record values are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 27, 36, 48, 54, ...

Amiram Eldar, Table of n, a(n) for n = 1..200

Subsequence of A025487.
Cf. A282446.
Analogous sequences: A002182 (highly composite), A002110 (unitary), A037992 (infinitary), A293185 (bi-unitary), A309141 (nonunitary), A318278 (exponential).

recDivNum[1] = 1; recDivNum[n_] := recDivNum[n] = Times @@ (1 + recDivNum/@ (Last /@ FactorInteger[n])); rm = 0; s = {}; Do[r = recDivNum[n]; If[r > rm, rm = r; AppendTo[s, n]], {n, 1, 10^4}]; s

A343020 a(n) is the smallest prime p such that tau(p+1) = 2^n.

Original entry on oeis.org

2, 5, 23, 167, 839, 7559, 128519, 1081079, 20540519, 397837439, 8031343319, 188972783999, 3212537327999, 125568306863999, 2888071057871999, 190487121512687999, 4381203794791823999, 215961289494494543999, 13283916764437951631999, 540119185025730854543999, 26465840066260811872655999, 1356699703068812438127791999

Offset: 1

Jaroslav Krizek, Apr 02 2021

nonn

tau(m) = the number of divisors of m (A000005).
Sequences of primes p such that tau(p+1) = 2^n for 2 <= n <= 5:
n = 2: 5, 7, 13, 37, 61, 73, 157, 193, 277, 313, 397, 421, ...
n = 3: 23, 29, 41, 53, 101, 103, 109, 113, 127, 137, 151, ...
n = 4: 167, 263, 269, 311, 383, 389, 439, 461, 509, 569, ...
n = 5: 839, 1319, 1511, 1559, 1847, 1889, 2039, 2309, 2687, ...
Conjecture: a(n) is also the smallest number m such that tau(m+1) = tau(m)^n.

			a(4) = 167 because 167 is the smallest prime p such that tau(p+1) = 16 = 2^4.

David A. Corneth, Table of n, a(n) for n = 1..31

Cf. A037992, A080371, A080372, A340799, A343018, A343019.

Magma
```
Ax:=func; [Ax(n): n in [1..7]]
```

Do[p = 1; While[DivisorSigma[0, Prime[p] + 1] != 2^n, p++]; Print[n, " ", Prime[p]], {n, 1, 9}] (* Vaclav Kotesovec, Apr 03 2021 *)

a(n) = my(t=2^n); forprime(p=2, oo, if(numdiv(p+1)==t, return(p))); \\ Jinyuan Wang, Apr 02 2021

from sympy import isprime,nextprime
primes=[2]
def solve(v,k,i,j):
    global record,stack,primes
    if k==0:
        if isprime(v-1):
            record=v
        return True
    sizeok=False
    cnt=True
    while cnt:
        if i>=len(primes):
            primes.append(nextprime(primes[-1]))
        if jBert Dobbelaere, Apr 11 2021

a(11) from Jinyuan Wang, Apr 02 2021

More terms from David A. Corneth, Apr 09 2021

A036538 Number of integers m <= 2^n such that d(m) = 2^k for some k = 0, 1, 2, 3, ...

Original entry on oeis.org

2, 3, 7, 12, 23, 45, 89, 178, 356, 707, 1409, 2822, 5639, 11273, 22546, 45088, 90165, 180315, 360637, 721258, 1442491, 2884973, 5769941, 11539858, 23079721, 46159395, 92318705, 184637321, 369274467, 738548867, 1477097749, 2954195452, 5908390605, 11816780739

Offset: 1

Labos Elemer

nonn

a(n+1)/a(n) is very close to 2; a(n)/2^n is near 0.7.

As n goes to infinity, lim a(n)/2^n = 0.687827... (A327839; see comments in A036537). - Vladimir Shevelev, Feb 28 2017

			Of the numbers 1 .. 2^4 = 16, only 4, 9, 12 and 16 are not in A036537, so a(4) = 16 - 4 = 12.

Cf. A000005, A036537, A036539, A037992, A327839.

IversonBrackets := expr -> subs(true=1, false=0, expr):
A := proc(n) option remember; {seq(2^k, k=0..n)} end:
h := proc(n) option remember; add(evalb(numtheory:-tau(j) in A(n)), j=2^(n-1) + 1..2^n); IversonBrackets(%) end:
a := n -> 1 + add(h(k), k=1..n); seq(a(n), n=1..17); # Peter Luschny, May 14 2018

Table[Count[#, 1] &@ Table[1 - Sign[# - Floor@ #] &@Log[2, #] &@ DivisorSigma[0, x], {x, 1, 2^m}], {m, 1, 20}] (* original program edited by Michael De Vlieger, Mar 01 2017, or *)
1 + Accumulate@ Table[Count[Range[2^(n - 1) + 1, 2^n], k_ /; IntegerQ@ Log2@ DivisorSigma[0, k]], {n, 20}] (* Michael De Vlieger, Feb 28 2017 *)

a(n) = sum(k=1, 2^n, d = numdiv(k); (d<=2) || (ispower(d,,&p) && (p==2))); \\ Michel Marcus, May 14 2018

from sympy import factorint
def A036538(n): return sum(1 for m in range(1,(1<Chai Wah Wu, Jun 22 2023

a(n) = number of 1s in f(tau(k)) mapped across k = 1..2^n, with f(x):= 1-sign(log_2 x - floor( log_2 x )). - Michael De Vlieger, Mar 01 2017

a(20)-a(26) from Michael De Vlieger, Feb 28 2017

a(27)-a(34) from Giovanni Resta, May 14 2018

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A069782 Numbers k such that gcd(d(k^3), d(k)) = 2^w for some w.

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A327634 Infinitary highly abundant numbers: numbers m such that isigma(m) > isigma(k) for all k < m, where isigma(k) is the sum of infinitary divisors of n (A049417).

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A061234 Smallest number with prime(n)^2 divisors where prime(n) is the n-th prime.

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A069780 a(n) = gcd(d(n^3), d(n)).

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A069781 Numbers k such that gcd(d(k^3), d(k)) is not a power of 2.

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A335386 Tri-unitary highly composite numbers: where the number of tri-unitary divisors (A335385) increases to a record.

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A374786 Numerator of the mean infinitary abundancy index of the infinitary divisors of n.

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A333931 Recursive highly composite numbers: numbers with a record number of recursive divisors (A282446).

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