A067004 -id:A067004 - cp's OEIS frontend

A331284 Number of values of k, 1 <= k <= n, with A329605(k) = A329605(n), where A329605 is the number of divisors of primorial inflation of n (A108951).

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 3, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 3, 1, 1, 3, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 3, 2, 1, 2, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 3

Offset: 1

Antti Karttunen, Jan 14 2020

nonn

Ordinal transform of A329605, or equally, of A329606.

Cf. A000005, A108951, A329605, A329606, A331285 (positions of the first occurrences of each n, also positions of records).

Cf. also A067004.

c[n_] := c[n] = If[n == 1, 1, Module[{f = FactorInteger[n], p, e}, If[Length[f] > 1, Times @@ c /@ Power @@@ f, {{p, e}} = f; Times @@ (Prime[Range[PrimePi[p]]]^e)]]];
A329605[n_] := DivisorSigma[0, c[n]];
Module[{b}, b[_] = 0;
a[n_] := With[{t = A329605[n]}, b[t] = b[t] + 1]];
Array[a, 105] (* Jean-François Alcover, Jan 12 2022 *)

up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
A329605(n) = if(1==n,1,my(f=factor(n),e=1,m=1); forstep(i=#f~,1,-1, e += f[i,2]; m *= e^(primepi(f[i,1])-if(1==i,0,primepi(f[i-1,1])))); (m));
v331284 = ordinal_transform(vector(up_to, n, A329605(n)));
A331284(n) = v331284[n];

a(A331285(n)) = n for all n.

A069823 Nonprime numbers k for which there is no x < k such that phi(x) = phi(k).

Original entry on oeis.org

1, 15, 25, 35, 51, 65, 69, 81, 85, 87, 121, 123, 129, 141, 143, 159, 161, 177, 185, 187, 203, 213, 235, 247, 249, 253, 255, 265, 267, 275, 289, 299, 301, 309, 321, 323, 339, 341, 343, 361, 393, 403, 415, 425, 447, 485, 489, 501, 519, 527, 529, 535, 537, 551

Offset: 1

Benoit Cloitre, Apr 28 2002

If p is prime there is no x < p such that phi(x) = phi(p) = p-1 since phi(x) < p-1.
Nonprime numbers k such that A081373(k)=1; i.e., the number of numbers not exceeding k, and with identical phi value to that of k, equals one. - Labos Elemer, Mar 24 2003
For 1 < n, if a(n) is squarefree, then phi(a(n)) is nonsquarefree. The converse is also true: for 1 < n, if phi(a(n)) is squarefree then a(n) is nonsquarefree. - Torlach Rush, Dec 26 2017

			k=25, a nonprime; phi values for k <= 25 are {1,1,2,2,4,2,6,4,6,4,10,4,12,6,8,8,16,6,18,8,12,10,22,8,20}; no phi(k) except phi(25) equals 20, A081373(25)=1, so 25 is a term.

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

Cf. A081373, A067004, A000010.

f[x_] := EulerPhi[x] fc[x_] := Count[Table[f[j]-f[x], {j, 1, x}], 0] t1=Flatten[Position[Table[fc[w], {w, 1, 1000}], 1]] t2=Flatten[Position[PrimeQ[t1], False]] Part[t1, t2]
(* Second program: *)
Union@ Select[Values[PositionIndex@ Array[EulerPhi, 600]][[All, 1]], ! PrimeQ@ # &] (* Michael De Vlieger, Dec 31 2017 *)

for(s=1,600,if((1-isprime(s))*abs(prod(i=1,s-1,eulerphi(i)-eulerphi(s)))>0, print1(s,",")))

A079788 a(n) = count of numbers <= n for which the number of divisors is also <= tau(n).

Original entry on oeis.org

1, 2, 3, 4, 4, 6, 5, 8, 7, 10, 6, 12, 7, 13, 14, 15, 8, 18, 9, 20, 17, 18, 10, 24, 13, 21, 22, 27, 11, 30, 12, 30, 25, 26, 27, 36, 13, 29, 30, 39, 14, 41, 15, 39, 40, 33, 16, 48, 20, 44, 36, 46, 17, 52, 38, 54, 39, 40, 18, 60, 19, 43, 54, 55, 44, 63, 20, 57, 46, 67, 21, 72, 22, 49

Offset: 1

Amarnath Murthy, Feb 03 2003

nonn

			a(7) = 5 as 1, 2, 3, 5 and 7 qualify for the count.

Cf. A138009, A067004.

Do[s = 0; For[i = 1, i <= n, i++, If[DivisorSigma[0, i] <= DivisorSigma[0, n], s++ ]]; Print[s], {n, 1, 50}] (* Ryan Propper, Mar 30 2006 *)

for(n=1,200,m=0;sn=sigma(n,0);for(i = 1,n,if(sigma(i,0)<=sn,m++));print1(m",")) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Apr 28 2007

More terms from Ryan Propper, Mar 30 2006

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Apr 28 2007

cp's OEIS Frontend

A331284 Number of values of k, 1 <= k <= n, with A329605(k) = A329605(n), where A329605 is the number of divisors of primorial inflation of n (A108951).

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A069823 Nonprime numbers k for which there is no x < k such that phi(x) = phi(k).

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A079788 a(n) = count of numbers <= n for which the number of divisors is also <= tau(n).

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