A037992 -id:A037992 - cp's OEIS frontend

A358263 Numbers with a record number of noninfinitary square divisors.

1, 16, 144, 256, 1296, 2304, 20736, 57600, 331776, 518400, 2822400, 8294400, 12960000, 25401600, 132710400, 207360000, 228614400, 406425600, 635040000, 2057529600, 3073593600, 6502809600, 10160640000, 27662342400, 31116960000, 51438240000, 76839840000, 248961081600

Offset: 1

Amiram Eldar, Nov 06 2022

nonn

Numbers m such that A358261(m) > A358261(k) for all k < m.

The corresponding record values are 0, 1, 2, 3, 5, 6, 11, 12, 13, 22, 24, 26, 37, 44, 46, 47, 48, ... (see the link for more values).

Amiram Eldar, Table of n, a(n) for n = 1..187
Amiram Eldar, Table of n, a(n), A358261(a(n)) for n = 1..187

Cf. A358261, A358262.
Subsequence of A025487.
Similar sequences: A002182, A002110 (unitary), A037992 (infinitary), A293185, A306736, A307845, A309141, A318278, A322484, A335386, A348632, A358253.

f1[p_, e_] := 1 + Floor[e/2]; f2[p_, e_] := 2^DigitCount[If[OddQ[e], e - 1, e], 2, 1]; f[1] = 0; f[n_] := Times @@ f1 @@@ (fct = FactorInteger[n]) - Times @@ f2 @@@ fct; s = {}; fmax = -1; Do[If[(fn = f[n]) > fmax, fmax = fn; AppendTo[s, n]], {n, 1, 6*10^5}]; s

s(n) = {my(f = factor(n));  prod(i=1, #f~, 1+f[i,2]\2) - prod(i=1, #f~, 2^hammingweight(if(f[i,2]%2, f[i,2]-1, f[i,2])))};
lista(nmax) = {my(smax = -1, sn); for(n = 1, nmax, sn = s(n); if(sn > smax, smax = sn; print1(n, ", "))); }

A359412 Numbers with a record number of divisors that are both infinitary and exponential.

Original entry on oeis.org

1, 8, 216, 27000, 9261000, 12326391000, 27081081027000, 110924107886592000, 544970142046826496000, 3737950204299182936064000, 45479640135708158783090688000, 1109202943269786284560798789632000, 33044264882950203203350756741926912000, 1673791149116076642859325881248823873536000

Offset: 1

Amiram Eldar, Dec 30 2022

nonn

Indices of records in A359411.
a(2)-a(7) are the first 6 terms of A115964.
The first 15 terms are cubes. Are there noncubes in this sequence?
The corresponding record values are 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, ... . Apparently, this sequence of records is the powers of 2 (A000079).

Amiram Eldar, Table of n, a(n) for n = 1..15

Cf. A000079, A000578, A115964, A359411.
Subsequence of A025487.
Similar sequences: A037992, A318278.

s[n_] := DivisorSum[n, 1 &, BitAnd[n, #] == # &]; f[p_, e_] := s[e]; d[1] = 1; d[n_] := Times @@ f @@@ FactorInteger[n];
v = Cases[Import["https://oeis.org/A025487/b025487.txt", "Table"], {, }][[;; , 2]];
seq = {}; dm = 0; Do[If[(dk = d[v[[k]]]) > dm, dm = dk; AppendTo[seq, v[[k]]]], {k, 1, Length[v]}]; seq

A361319 Indices of records in the sequence of infinitary harmonic means A361316(k)/A361317(k).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 54, 56, 60, 84, 105, 120, 168, 210, 264, 270, 280, 360, 420, 540, 660, 756, 840, 1080, 1320, 1512, 1848, 1890, 2310, 2520, 3080, 3640, 3780, 4620, 5460, 5940, 7020, 7560, 9240, 10920, 11880, 14040, 16632, 19656

Offset: 1

Amiram Eldar, Mar 09 2023

nonn

			The infinitary harmonic means of the first 6 positive integers are 1 < 4/3 < 3/2 < 8/5 < 5/3 < 2. The next record, A361316(8)/A361317(8) = 32/15, occurs at 8. Therefore, the first 7 terms of this sequence are 1, 2, 3, 4, 5, 6 and 8.

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

Cf. A361316, A361317.
Similar sequences: A179971, A348654.
Other sequences related to records of infinitary divisors: A037992, A327634.

f[p_, e_] := Module[{b = IntegerDigits[e, 2], m}, m = Length[b]; Product[If[b[[j]] > 0, 2/(1 + p^(2^(m - j))), 1], {j, 1, m}]]; ihmean[1] = 1; ihmean[n_] := n*Times @@ f @@@ FactorInteger[n]; seq[kmax_] := Module[{ih, ihmax = 0, s = {}}, Do[ih = ihmean[k]; If[ih > ihmax, ihmax = ih; AppendTo[s, k]], {k, 1, kmax}]; s]; seq[20000]

ihmean(n) = {my(f = factor(n), b); n * prod(i=1, #f~, b = binary(f[i, 2]); prod(k=1, #b, if(b[k], 2/(f[i, 1]^(2^(#b-k))+1), 1))); };
lista(kmax) = {my(ih, ihmax=0); for(k = 1, kmax, ih = ihmean(k); if(ih > ihmax, ihmax = ih; print1(k, ", ")));}

A376471 Lexicographically earliest strictly increasing sequence of numbers whose partial products are all exponentially 2^n-numbers (A138302).

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 9, 11, 13, 17, 19, 20, 23, 25, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 77, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 208, 211, 223, 227, 229, 233, 239, 241

Offset: 1

Amiram Eldar, Sep 24 2024

nonn

All the primes are terms.

			1 * 2 = 2^1 and 1 = 2^0.
1 * 2 * 3 = 6 = 2^1 * 3^1 and 1 = 2^0.
1 * 2 * 3 * 5 * 6 = 180 = 2^2 * 3^2 * 5^1, 1 = 2^0 and 2 = 2^1.

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

Disjoint union of A000040 and A376472.
Similar sequences:
  Sequence      | Partial products are in | Exponents are in
  --------------+-------------------------+------------------------
  A050376       | A037992                 | A000225 \ {0} (2^n-1)
  A089237       | A268335                 | A005408 (odd numbers)
  {1} U A246551 | A246551                 | A000290 \ {0} (squares)
  this sequence | A138302                 | A000079 (powers of 2)

expPow2Q[n_] := AllTrue[FactorInteger[n][[;; , 2]], # == 2^IntegerExponent[#, 2] &]; a[1] = 1; a[n_] := a[n] = Module[{prod = Times @@ Array[a, n - 1], k = a[n - 1] + 1}, While[! expPow2Q[prod*k], k++]; k]; Array[a, 100]

ispow2(n) = if(n == 0, 1, n >> valuation(n, 2) == 1);
lista(pindmax) = {my(pmax = prime(pindmax), v = vector(pindmax), f, pind, prd); print1(1, ", "); for(k = 2, pmax, f = factor(k); pind = apply(x -> primepi(x), f[,1]); for(i = 1, #pind, v[pind[i]] += f[i, 2]); if(vecprod(apply(x -> ispow2(x), v)) > 0, print1(k, ", "), for(i = 1, #pind, v[pind[i]] -= f[i, 2])));}

A382293 a(n) is the least number k such that A382290(k) = n.

Original entry on oeis.org

1, 8, 128, 3456, 279936, 34992000, 8957952000, 3072577536000, 1920360960000000, 2556000437760000000, 5615532961758720000000, 13482894641182686720000000, 66241461372130539855360000000, 434610228062548471991016960000000, 2980991554281019969386385328640000000

Offset: 0

Amiram Eldar, Mar 21 2025

nonn

Also, a(n) is the least number k such that A382291(k) = 2^n.

Cumulative products of the sorted sequence that contains 1 and prime powers of the form p^3 and p^(2^k) with k >= 2.

Amiram Eldar, Table of n, a(n) for n = 0..150

Subsequence of A025487.

Cf. A037992, A050376, A382290, A382291.

seq[max_] := Module[{t = {}, k = 3, lim}, While[lim = max^(1/k); lim > 2, t = Join[t, Prime[Range[PrimePi[lim]]]^k]; If[k == 3, k = 4, k *= 2]]; t = Sort[t]; FoldList[Times, 1, t]]; seq[10^4] (* after T. D. Noe at A050376 *)

list(mx) = {my(t = [1], k =3, lim); while(lim = mx^(1/k); lim > 2, t = concat(t, apply(x -> x^k, primes(primepi(lim)))); if(k == 3, k = 4, k *= 2)); t = vecsort(t); vector(#t, n, prod(i = 1, n, t[i]));}

A036539 a(n) is the number of numbers k with 2^(n-1) < k <= 2^n having a number of divisors that is a power of 2.

Original entry on oeis.org

1, 1, 4, 5, 11, 22, 44, 89, 178, 351, 702, 1413, 2817, 5634, 11273, 22542, 45077, 90150, 180322, 360621, 721233, 1442482, 2884968, 5769917, 11539863, 23079674, 46159310, 92318616, 184637146, 369274400, 738548882, 1477097703, 2954195153, 5908390134, 11816780283

Offset: 1

Labos Elemer

nonn

			a(5) = 11: The following 11 numbers have numbers of divisors that are powers of 2: 17, 19, 21, 22, 23, 24, 26, 27, 29, 30 and 31 with 2, 2, 4, 4, 2, 8, 4, 4, 2, 8 and 2 divisors, respectively.

Cf. A036537, A037992, A327839.

f[n_] := Boole[n == 2^IntegerExponent[n, 2]]; a[n_] := Sum[f[DivisorSigma[0, k]], {k, 2^(n - 1) + 1, 2^n}]; Array[a, 20] (* Amiram Eldar, Aug 16 2024 *)

a(n)=sum(k=2^(n-1)+1,2^n, my(d=numdiv(k)); (d/(1<Joerg Arndt, Feb 27 2017

a(n) ~ c * 2^(n-1), where c = 0.687827... (A327839). - Amiram Eldar, Aug 16 2024

Name clarified and more terms from Joerg Arndt, Feb 27 2017
a(25)-a(28) from Jon E. Schoenfield, Jul 31 2018
a(29)-a(35) from Jon E. Schoenfield, Aug 04 2018

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

A065743 Smallest number with exactly A025475(n) divisors.

Original entry on oeis.org

1, 6, 24, 36, 120, 1296, 900, 840, 46656, 7560, 44100, 60466176, 810000, 83160, 2176782336, 2822400, 1081080, 2821109907456, 729000000, 101559956668416, 17297280, 131621703842267136, 1944810000, 341510400

Offset: 1

Labos Elemer, Nov 15 2001

nonn

Note that 2^(n-1) has n divisors. - David Wasserman, Sep 09 2002

Amiram Eldar, Table of n, a(n) for n = 1..285

Cf. A005179, A025475, A037992, A003680.

a = Table[ 0, {1024} ]; Do[ b = DivisorSigma[ 0, n]; If[ b < 1025 && a[[b]] == 0, a[[b]] = n], {n, 1, 10^8/2} ]; a[[ Select[ Range[2, 1024], !PrimeQ[ # ] && Mod[ #, # - EulerPhi[ # ]] == 0 & ] ]]

a(n) = A005179(A025475(n)).

More terms from David Wasserman, Sep 09 2002

A349111 Powerful superabundant numbers: numbers m such that psigma(m)/m > psigma(k)/k for all k < m, where psigma(k) is the sum of powerful divisors of k (A183097).

Original entry on oeis.org

1, 4, 8, 16, 32, 64, 128, 144, 216, 432, 864, 1296, 1728, 2592, 5184, 10368, 15552, 31104, 54000, 108000, 162000, 216000, 324000, 648000, 1296000, 1944000, 3240000, 3888000, 6480000, 9720000, 19440000, 38880000, 58320000, 74088000, 111132000, 222264000, 444528000, 666792000

Offset: 1

Amiram Eldar, Nov 08 2021

nonn

The corresponding record values are 1, 5/4, 13/8, 29/16, 61/32, 125/64, ...

The least term k with psigma(k)/k > m, for m = 2, 3, ..., is 144, 54000, 666792000, ...

Subsequence of A349112.
Cf. A005934, A183097.
Similar sequences: A002110 (unitary), A037992 (infinitary), A061742, A292984, A329882, A348273.

f[p_,e_] := (p^(e+1)-1)/(p-1) - p; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; seq = {}; rm = 0; Do[r1 = s[n]/n; If[r1 > rm, rm = r1; AppendTo[seq, n]], {n, 1, 10^6}]; seq

A363330 Numbers with a record number of divisors that are both coreful and infinitary.

Original entry on oeis.org

1, 8, 128, 216, 3456, 27000, 279936, 432000, 9261000, 34992000, 148176000, 8957952000, 12002256000, 197222256000, 3072577536000, 7501410000000, 15975002736000, 433297296432000, 1920360960000000, 4089600700416000, 9984376710000000, 35097081010992000, 2128789617370416000

Offset: 1

Amiram Eldar, May 28 2023

nonn

Indices of records in A363329.

The corresponding record values are 1, 3, 7, 9, 21, 27, 49, 63, 81, 147, 189, 315, ... (see the link for more values).

Amiram Eldar, Table of n, a(n) for n = 1..64
Amiram Eldar, Table of n, a(n), A363329(a(n)) for n = 1..64

Cf. A363329.
Subsequence of A025487.
Similar sequences: A005934, A037992.

f[p_, e_] := 2^DigitCount[e, 2, 1] - 1; d[1] = 1; d[n_] := Times @@ f @@@ FactorInteger[n];
v = Cases[Import["https://oeis.org/A025487/b025487.txt", "Table"], {, }][[;; , 2]];
seq = {}; dm = 0; Do[If[(dk = d[v[[k]]]) > dm, dm = dk; AppendTo[seq, v[[k]]]], {k, 1, Length[v]}]; seq

cp's OEIS Frontend

A358263 Numbers with a record number of noninfinitary square divisors.

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A359412 Numbers with a record number of divisors that are both infinitary and exponential.

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A361319 Indices of records in the sequence of infinitary harmonic means A361316(k)/A361317(k).

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A376471 Lexicographically earliest strictly increasing sequence of numbers whose partial products are all exponentially 2^n-numbers (A138302).

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A382293 a(n) is the least number k such that A382290(k) = n.

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A036539 a(n) is the number of numbers k with 2^(n-1) < k <= 2^n having a number of divisors that is a power of 2.

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

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A065743 Smallest number with exactly A025475(n) divisors.

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