A037992 -id:A037992 - cp's OEIS frontend

A066616 a(1) = 1; a(n) = n*a(n-1) if n does not divide a(n-1), otherwise a(n) = a(n-1).

1, 2, 6, 24, 120, 120, 840, 840, 7560, 7560, 83160, 83160, 1081080, 1081080, 1081080, 17297280, 294053760, 294053760, 5587021440, 5587021440, 5587021440, 5587021440, 128501493120, 128501493120, 3212537328000, 3212537328000

Offset: 1

Amarnath Murthy, Dec 24 2001

nonn

			a(5) = 120; as 6 divides a(5), we have a(6) = a(5) = 120. Though 9 is not coprime to a(8) but still 9 does not divide a(8) so a(9) = 9 * a(8).

Harry J. Smith, Table of n, a(n) for n = 1..200

Cf. A003418, A037992 (duplicates removed).

Replacing A059896 with A059897 in the formula gives A284567.

nxt[{n_,a_}]:={n+1,If[Mod[a,n+1]==0,a,a(n+1)]}; NestList[nxt,{1,1},30][[All,2]] (* Harvey P. Dale, Jul 01 2022 *)

{ for (n=1, 200, if (n==1, a=1, if (a%n, a=n*a)); write("b066616.txt", n, " ", a) ) } \\ Harry J. Smith, Mar 12 2010

a(1) = 1; for n > 1, a(n) = A059896(a(n-1), n). - Peter Munn, Jul 12 2022

More terms from Vladeta Jovovic, Dec 26 2001

A283052 Numbers k such that uphi(k)/phi(k) > uphi(m)/phi(m) for all m < k, where phi(k) is the Euler totient function (A000010) and uphi(k) is the unitary totient function (A047994).

Original entry on oeis.org

1, 4, 8, 16, 32, 36, 72, 144, 216, 288, 432, 864, 1728, 2592, 3600, 5400, 7200, 10800, 21600, 43200, 64800, 108000, 129600, 216000, 259200, 324000, 529200, 1058400, 2116800, 3175200, 5292000, 6350400, 10584000, 12700800, 15876000, 31752000, 63504000, 95256000

Offset: 1

Amiram Eldar, May 19 2017

nonn

This sequence is infinite.
a(1) = 1, a(6) = 36, a(15) = 3600 and a(32) = 6350400 are the smallest numbers n such that uphi(n)/phi(n) = 1, 2, 3 and 4. They are squares of 1, 6, 60, and 2520.
Also, coreful superabundant numbers: numbers k with a record value of the coreful abundancy index, A057723(k)/k > A057723(m)/m for all m < k. The two sequences are equivalent since A057723(k)/k = A047994(k)/A000010(k) for all k. - Amiram Eldar, Dec 28 2020

			uphi(k)/phi(k) = 1, 1, 1, 3/2 for k = 1, 2, 3, 4, thus a(1) = 1 and a(2) = 4 since a(4) > a(m) for m < 4.

Amiram Eldar, Table of n, a(n) for n = 1..46

Cf. A000010, A047994, A057723, A285906.

Similar sequences: A002110, A004394, A037992, A061742, A292984, A329882, A333953.

uphi[n_] := If[n == 1, 1, (Times @@ (Table[#[[1]]^#[[2]] - 1, {1}] & /@
FactorInteger[n]))[[1]]]; a = {}; rmax = 0; For[k = 0, k < 10^9, k++; r = uphi[k]/EulerPhi[k]; If[r > rmax, rmax = r; a = AppendTo[a, k]]]; a

uphi(n) = my(f = factor(n)); prod(i=1, #f~, f[i,1]^f[i,2]-1);
lista(nn) = {my(rmax = 0); for (n=1, nn, if ((newr=uphi(n)/eulerphi(n)) > rmax, print1(n, ", "); rmax = newr););} \\ Michel Marcus, May 20 2017

A347064 Smallest number with at least 2^n divisors.

Original entry on oeis.org

1, 2, 6, 24, 120, 840, 7560, 83160, 1081080, 17297280, 294053760, 5587021440, 128501493120, 3212537328000, 93163582512000, 2888071057872000, 106858629141264000, 4381203794791824000, 184010559381256608000, 7912454053394034144000, 371885340509519604768000

Offset: 0

Jon E. Schoenfield, Aug 15 2021

nonn

Begins to differ from A037992 at n=18; a(18) < A037992(18), but the number of divisors d(a(18)) = 276480 > 262144 = 2^18.

			   n                A037992(n)                      a(n)  d(a(n))      2^n
  --  ------------------------  ------------------------  -------  -------
  18     188391763176048432000     184010559381256608000   276480   262144
  19    8854412869274276304000    7912454053394034144000   552960   524288
  20  433866230594439538896000  371885340509519604768000  1105920  1048576

Amiram Eldar, Table of n, a(n) for n = 0..356

Cf. A000005, A000079, A037992, A061799.

Table[SelectFirst[Table[{n,DivisorSigma[0,n]},{n,0,11*10^5}],#[[2]]==2^k&],{k,0,8}][[;;,1]] (* The program generates the first nine terms of the sequence. *)  (* Harvey P. Dale, Feb 04 2024 *)

a(n) = A061799(2^n). - Michel Marcus, Aug 16 2021

A358253 Numbers with a record number of non-unitary square divisors.

Original entry on oeis.org

1, 8, 32, 128, 288, 864, 1152, 2592, 4608, 10368, 20736, 28800, 41472, 64800, 115200, 259200, 518400, 1036800, 2073600, 4147200, 8294400, 9331200, 12700800, 25401600, 50803200, 101606400, 203212800, 406425600, 457228800, 635040000, 812851200, 914457600, 1270080000

Offset: 1

Amiram Eldar, Nov 05 2022

nonn

Numbers m such that A056626(m) > A056626(k) for all k < m.

The corresponding record values are 0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 16, 20, 22, ... (see the link for more values).

Amiram Eldar, Table of n, a(n) for n = 1..283
Amiram Eldar, Table of n, a(n), A056626(a(n)) for n = 1..283

Cf. A056626, A358252.
Subsequence of A025487.
Similar sequences: A002182 (all divisors), A002110 (unitary), A037992 (infinitary), A046952 (square divisors), A053624 (odd divisors), A293185 (bi-unitary), A309141 (non-unitary), A318278 (exponential).

f1[p_, e_] := 1 + Floor[e/2]; f2[p_, e_] := 2^(1 - Mod[e, 2]); 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, 10^5}]; s

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

A375271 Partial products of A375270.

Original entry on oeis.org

1, 2, 6, 30, 210, 1680, 18480, 240240, 4084080, 77597520, 1784742960, 48188059920, 1397453737680, 43321065868080, 1602879437118960, 65718056921877360, 2825876447640726480, 132816193039114144560, 7039258231073049661680, 415316235633309930039120, 25334290373631905732386320

Offset: 1

Amiram Eldar, Aug 09 2024

nonn

Numbers with a record number of Zeckendorf-infinitary divisors (A318465). Also, indices of records in A318464.

a(n) is the least number k such that A318464(k) = n-1 and A318465(k) = 2^(n-1).

			A375270 begins with 1, 2, 3, 5, ..., so, a(1) = 1, a(2) = 1 * 2 = 2, a(3) = 1 * 2 * 3 = 6, a(4) = 1 * 2 * 3 * 5 = 30.

Amiram Eldar, Table of n, a(n) for n = 1..353

Cf. A037992 (analogous with "Fermi-Dirac primes", A050376), A318464, A318465, A375270.

Subsequence of A025487.

fib[lim_] := Module[{s = {}, f = 1, k = 2}, While[f <= lim, AppendTo[s, f]; k += 2; f = Fibonacci[k]]; s];
seq[max_] := Module[{s = {}, p = 2, e = 1, f = {}}, While[e > 0, e = Floor[Log[p, max]]; If[f == {}, f = fib[e], f = Select[f, # <= e &]]; s = Join[s, p^f]; p = NextPrime[p]]; FoldList[Times, 1, Sort[s]]]; seq[100]

fib(lim) = {my(s = List(), f = 1, k = 2); while(f <= lim, listput(s, f); k += 2; f = fibonacci(k)); Vec(s);}
lista(pmax) = {my(s = [1], p = 2, e = 1, f = [], r = 1); while(e > 0, e = logint(pmax, p); if(#f == 0, f = fib(e), f = select(x -> x <= e, f)); s = concat(s, apply(x -> p^x, f)); p = nextprime(p+1)); s = vecsort(s); for(i = 1, #s, r *= s[i]; print1(r, ", "))}

a(n) = Product_{k=1..n} A375270(k).

A343087 a(n) is the smallest prime p such that tau(p-1) = 2^n.

Original entry on oeis.org

3, 7, 31, 211, 1321, 7561, 120121, 1580041, 24864841, 328648321, 7558911361, 162023621761, 5022732274561, 93163582512001, 4083134943888001, 151075992923856001, 5236072827921936001, 188391763176048432001, 8854412869274276304001, 469283882071536644112001, 29844457947060064452144001, 1917963226026370264485744001

Offset: 1

Jaroslav Krizek, Apr 04 2021 (following a suggestion of Vaclav Kotesovec)

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: 7, 11, 23, 47, 59, 83, 107, 167, 179, ... (A005385(k) for k >= 2).
n = 3: 31, 41, 43, 67, 71, 79, 89, 103, 131, 137, 139, 191, ...
n = 4: 211, 271, 281, 313, 331, 379, 409, 457, 463, 521, 547, ...
n = 5: 1321, 2281, 2311, 2377, 2689, 2731, 2857, 2971, 3001, ...
Conjecture: a(n) is also the smallest number m such that tau(m-1) = tau(m)^n.

			For n = 4; a(4) = 211 because 211 is the smallest prime p such that tau(p - 1) = 2^4; tau(210) = 16.

Bert Dobbelaere, Table of n, a(n) for n = 1..60

Cf. A000005, A000079, A037992, A343020.

Magma
```
Ax:=func; [Ax(n): n in [1..9]]
```

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
    while True:
        if i>=len(primes):
            primes.append(nextprime(primes[-1]))
        if jBert Dobbelaere, Apr 11 2021

a(11) from Vaclav Kotesovec, Apr 05 2021

More terms from Bert Dobbelaere, Apr 11 2021

A348273 Noninfinitary superabundant numbers: numbers m such that nisigma(m)/m > nisigma(k)/k for all k < m, where nisigma(m) is the sum of noninfinitary divisors of m (A348271).

Original entry on oeis.org

1, 4, 12, 16, 36, 48, 144, 720, 3600, 25200, 176400, 226800, 1587600, 1940400, 2494800, 17463600, 32432400, 192099600, 227026800, 2497294800, 3632428800, 32464832400, 39956716800

Offset: 1

Amiram Eldar, Oct 09 2021

The least term k with A348271(k)/k > m for m = 1, 2, 3, .... is 36, 3600, 1587600, ...

Cf. A348271.
Subsequence of A348272.
The noninfinitary version of A004394.
Similar sequences: A002110 (unitary), A037992 (infinitary), A061742 (exponential), A292984, A329882.

f[p_, e_] := Module[{b = IntegerDigits[e, 2], m}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]]; isigma[1] = 1; isigma[n_] := Times @@ f @@@ FactorInteger[n]; s[n_] := DivisorSigma[1,n] - isigma[n]; seq = {}; rm = -1; Do[r1 = s[n]/n; If[r1 > rm, rm = r1; AppendTo[seq, n]],{n, 1, 10^6}]; seq

A348342 Noninfinitary highly composite numbers: where the number of noninfinitary divisors (A348341) increases to a record.

Original entry on oeis.org

1, 4, 12, 16, 36, 48, 144, 240, 576, 720, 1680, 2880, 3600, 5040, 11520, 14400, 15120, 20160, 25200, 45360, 55440, 80640, 100800, 166320, 176400, 226800, 277200, 498960, 720720, 887040, 1108800, 1587600, 1940400, 2494800, 3603600, 6486480, 9979200, 11531520, 14414400

Offset: 1

Amiram Eldar, Oct 13 2021

nonn

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

Amiram Eldar, Table of n, a(n) for n = 1..468
Amiram Eldar, Table of n, a(n), A348341(a(n)) for n = 1..468

Cf. A348341.
Subsequence of A025487.
Similar sequences: A002182, A002110 (unitary), A037992 (infinitary), A293185, A306736, A307845, A309141, A318278, A322484, A335386.

nid[1] = 0; nid[n_] := DivisorSigma[0, n] - Times @@ Flatten[2^DigitCount[#, 2, 1] & /@ FactorInteger[n][[;; , 2]]]; dm = -1; s = {}; Do[If[(d = nid[n]) > dm, dm = d; AppendTo[s, n]], {n, 1, 10^6}]; s

A348632 Nonexponential highly composite numbers: where the number of nonexponential divisors (A160097) increases to a record.

Original entry on oeis.org

1, 6, 12, 24, 30, 60, 120, 210, 240, 360, 420, 720, 840, 1260, 1680, 2520, 3360, 5040, 7560, 9240, 10080, 15120, 18480, 25200, 27720, 36960, 50400, 55440, 83160, 110880, 166320, 221760, 277200, 332640, 480480, 498960, 554400, 665280, 720720, 1081080, 1441440

Offset: 1

Amiram Eldar, Oct 26 2021

nonn

The corresponding record values are 1, 3, 4, 6, 7, 10, 14, 15, 17, 20, 22, 24, ... (see the link for more values).

Amiram Eldar, Table of n, a(n) for n = 1..651
Amiram Eldar, Table of n, a(n), A160097(a(n)) for n = 1..651

Cf. A160097.
Subsequence of A025487.
Similar sequences: A002182, A002110 (unitary), A037992 (infinitary), A293185, A306736, A307845, A309141, A318278, A322484, A335386.

f1[p_, e_] := e + 1; f2[p_, e_] := DivisorSigma[0, e]; ned[1] = 1; ned[n_] := Times @@ f1 @@@ (f = FactorInteger[n]) - Times @@ f2 @@@ f; dm = -1; s = {}; Do[If[(d = ned[n]) > dm, dm = d; AppendTo[s, n]], {n, 1, 10^6}]; s

A353899 Indices of records in A353898.

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, 3063060, 6306300, 12252240, 15315300, 58198140, 107207100, 232792560, 290990700, 1163962800, 2036934900, 5354228880, 6692786100, 22406283900

Offset: 1

Amiram Eldar, May 10 2022

nonn

First differs from A333931 at n=23.

The corresponding record values are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 27, 36, 48, 54, 72, 81, 96, 108, 144, 162, ... (see the link for more values).

Amiram Eldar, Table of n, a(n) for n = 1..145
Amiram Eldar, Table of n, a(n), A353898(a(n)) for n = 1..145

Cf. A333931, A353898.
Subsequence of A025487 and A138302.
Similar sequences: A002182, A002110 (unitary), A037992 (infinitary), A293185, A306736, A307845, A309141, A318278, A322484, A335386.

f[p_, e_] := Floor[Log2[e]] + 2; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; seq = {}; sm = 0; Do[s1 = s[n]; If[s1 > sm, sm = s1; AppendTo[seq, n]], {n, 1, 10^6}]; seq

cp's OEIS Frontend

A066616 a(1) = 1; a(n) = n*a(n-1) if n does not divide a(n-1), otherwise a(n) = a(n-1).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A283052 Numbers k such that uphi(k)/phi(k) > uphi(m)/phi(m) for all m < k, where phi(k) is the Euler totient function (A000010) and uphi(k) is the unitary totient function (A047994).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A347064 Smallest number with at least 2^n divisors.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A358253 Numbers with a record number of non-unitary square divisors.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A375271 Partial products of A375270.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A343087 a(n) is the smallest prime p such that tau(p-1) = 2^n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Python

Extensions

A348273 Noninfinitary superabundant numbers: numbers m such that nisigma(m)/m > nisigma(k)/k for all k < m, where nisigma(m) is the sum of noninfinitary divisors of m (A348271).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

A348342 Noninfinitary highly composite numbers: where the number of noninfinitary divisors (A348341) increases to a record.

Views

Author