A051953 -id:A051953 - cp's OEIS frontend

A319348 Lexicographically earliest infinite sequence such that a(i) = a(j) => A003557(i) = A003557(j) and A051953(i) = A051953(j), for all i, j >= 1.

1, 2, 2, 3, 2, 4, 2, 5, 6, 7, 2, 8, 2, 9, 10, 11, 2, 12, 2, 13, 14, 15, 2, 16, 17, 18, 19, 20, 2, 21, 2, 22, 23, 24, 25, 26, 2, 27, 28, 29, 2, 30, 2, 31, 32, 33, 2, 34, 35, 36, 37, 38, 2, 39, 28, 40, 41, 30, 2, 42, 2, 43, 44, 45, 46, 47, 2, 48, 49, 47, 2, 50, 2, 51, 52, 53, 46, 54, 2, 55, 56, 57, 2, 58, 41, 59, 60, 61, 2, 62, 37, 63, 64, 65, 66, 67, 2, 68, 69, 70, 2

Offset: 1

Antti Karttunen, Sep 29 2018

nonn

Restricted growth sequence transform of ordered pair [A003557(n), A051953(n)].

For all i, j: a(i) = a(j) => A318305(i) = A318305(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A003557, A051953, A319349.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = f[i, 2]-1); factorback(f); };
A051953(n) = (n-eulerphi(n));
v319348 = rgs_transform(vector(up_to,n,[A003557(n),A051953(n)]));
A319348(n) = v319348[n];

For n >= 3, a(n) = A319349(n) - 1.

Name changed by Antti Karttunen, Feb 03 2024

A344178 Difference between the arithmetic derivative of n and the cototient of n: a(n) = A003415(n) - A051953(n).

Original entry on oeis.org

0, 0, 0, 2, 0, 1, 0, 8, 3, 1, 0, 8, 0, 1, 1, 24, 0, 9, 0, 12, 1, 1, 0, 28, 5, 1, 18, 16, 0, 9, 0, 64, 1, 1, 1, 36, 0, 1, 1, 44, 0, 11, 0, 24, 18, 1, 0, 80, 7, 15, 1, 28, 0, 45, 1, 60, 1, 1, 0, 48, 0, 1, 24, 160, 1, 15, 0, 36, 1, 13, 0, 108, 0, 1, 20, 40, 1, 17, 0, 128, 81, 1, 0, 64, 1, 1, 1, 92, 0, 57, 1, 48, 1, 1, 1, 208

Offset: 1

Antti Karttunen, May 23 2021

nonn

Question: Are all terms nonnegative? See also A211991 and A344584.
From Bernard Schott, May 25 2021: (Start)
Answer: Yes, can be proved when n = Product_{i=1..k} p_i^e_i with n' = n * Sum_{i=1..k} (e_i/p_i) and cototient(n) = n * (1 - Product_{i=1..k} (1 - 1/p_i)).
a(n) = 0 iff n is in A008578 (1 together with the primes).
a(n) = 1 iff n is in A006881 (squarefree semiprimes) (End).

Antti Karttunen, Table of n, a(n) for n = 1..10000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A000010, A003415, A051953, A168036, A344584 (inverse Möbius transform).
Cf. also A211991.
Cf. A006881, A008578.

Array[If[# < 2, 0, # Total[#2/#1 & @@@ FactorInteger[#]]] - # + EulerPhi[#] &, 96] (* Michael De Vlieger, May 24 2021 *)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A344178(n) = A003415(n) - (n-eulerphi(n));

a(n) = A003415(n) - A051953(n) = A168036(n) + A000010(n).

A361021 Lexicographically earliest infinite sequence such that a(i) = a(j) => A007814(i) = A007814(j), A001065(i) = A001065(j) and A051953(i) = A051953(j), for all i, j >= 1.

Original entry on oeis.org

1, 2, 3, 4, 3, 5, 3, 6, 7, 8, 3, 9, 3, 10, 11, 12, 3, 13, 3, 14, 15, 16, 3, 17, 18, 19, 20, 21, 3, 22, 3, 23, 24, 25, 26, 27, 3, 28, 29, 30, 3, 31, 3, 32, 33, 34, 3, 35, 36, 37, 38, 39, 3, 40, 29, 41, 42, 43, 3, 44, 3, 45, 46, 47, 48, 49, 3, 50, 51, 52, 3, 53, 3, 54, 55, 56, 48, 57, 3, 58, 59, 60, 3, 61, 42, 62, 63, 64, 3, 65, 38, 66, 67, 68, 69, 70, 3, 71

Offset: 1

Antti Karttunen, Mar 03 2023

nonn

Restricted growth sequence transform of the triplet [A007814(n), A001065(n), A051953(n)].
For all i, j >= 1:
  A305801(i) = A305801(j) => a(i) = a(j),
  a(i) = a(j) => A305895(i) = A305895(j),
  a(i) = a(j) => A319346(i) = A319346(j).

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A001065, A007814, A051953.
Cf. A305801, A305895, A319346.
Cf. also A353560.
Differs from A353520 for the first time at n=254, where a(254) = 187, while A353520(254) = 125.

up_to = 100000;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A007814(n) = valuation(n,2);
A001065(n) = (sigma(n)-n);
A051953(n) = (n-eulerphi(n));
Aux361021(n) = [A007814(n), A001065(n), A051953(n)];
v361021 = rgs_transform(vector(up_to,n,Aux361021(n)));
A361021(n) = v361021[n];

A051961 Smallest number w such that A051953(w) = w - phi(w) is the n-th prime.

Original entry on oeis.org

4, 9, 25, 15, 35, 33, 65, 51, 95, 161, 87, 217, 185, 123, 215, 329, 371, 177, 427, 335, 213, 511, 395, 581, 1501, 485, 303, 515, 321, 545, 255, 635, 917, 411, 1529, 447, 1057, 1099, 455, 1169, 1211, 537, 1991, 573, 965, 591, 435, 2743, 1115, 681, 665

Offset: 1

Labos Elemer, Jan 05 2000

nonn

			The 31st term is 255 since 255 - phi(255) = 127, the 31st prime, and no number less than 255 has this property.

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

Cf. A050530, A051953.

With[{c=Table[n-EulerPhi[n],{n,4000}]},Table[Position[c,p,1,1],{p,Prime[ Range[ 60]]}]]//Flatten (* Harvey P. Dale, Sep 14 2020 *)

a(n) = {my(k = 1); while(k - eulerphi(k) != prime(n), k++); k;} \\ Michel Marcus, Feb 02 2015

A050530(a(n)) = prime(n) and a(n) is the least number with this property.

a(n) = A063507(A000040(n)). - Michel Marcus, Feb 02 2015

A053480 Sum of values when cototient function A051953 is iterated until fixed point is reached.

Original entry on oeis.org

1, 3, 4, 7, 6, 13, 8, 15, 13, 23, 12, 27, 14, 29, 23, 31, 18, 45, 20, 47, 34, 49, 24, 55, 31, 55, 40, 59, 30, 79, 32, 63, 47, 79, 47, 91, 38, 85, 62, 95, 42, 121, 44, 99, 79, 101, 48, 111, 57, 129, 71, 111, 54, 145, 78, 119, 91, 137, 60, 159, 62, 125, 103, 127, 83, 167

Offset: 1

Labos Elemer, Jan 14 2000

nonn

			If n=130 and A051953 is iterated, we obtain {130,82,42,30,22,12,8,4,2,1,0}. The sum of these terms is 130 + 92 + 42 + 30 + 12 + 8 + 4 + 2 + 1 = 333, so a(130)=333.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A051953.

Array[Total@ Most@ NestWhileList[# - EulerPhi@ # &, #, # > 0 &] &, 66] (* Michael De Vlieger, Nov 20 2017 *)

a(n) = Sum[Nest[co, n, j], {j, 1, x[n]}]; co[n]=A051953[n], x[n] is the number of iterations

A063476 Sum_{d |C(n)} d^2, where C(n) is the Cototient function n - phi(n) (A051953).

Original entry on oeis.org

0, 1, 1, 5, 1, 21, 1, 21, 10, 50, 1, 85, 1, 85, 50, 85, 1, 210, 1, 210, 91, 210, 1, 341, 26, 250, 91, 341, 1, 610, 1, 341, 170, 455, 122, 850, 1, 546, 260, 850, 1, 1300, 1, 850, 500, 850, 1, 1365, 50, 1300, 362, 1050, 1, 1911, 260, 1365, 500, 1300, 1, 2562, 1, 1365

Offset: 1

Jason Earls, Jul 27 2001

nonn

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A051953.

C(n)=n-eulerphi(n); j=[]; for(n=1,200,j=concat(j,sumdiv(C(n),d,d^2))); j

{ for (n=1, 1000, if (n>1, a=sumdiv(n-eulerphi(n), d, d^2), a=0); write("b063476.txt", n, " ", a) ) } \\ Harry J. Smith, Aug 22 2009

A063480 C(n+3)=2*C(n), where C(n) is Cototient(n) := n - phi(n) (A051953).

Original entry on oeis.org

39, 55, 111, 183, 219, 459, 471, 579, 831, 867, 939, 1191, 1263, 1371, 1623, 1839, 1983, 2019, 2199, 2271, 2631, 2991, 3279, 3459, 3603, 3639, 3711, 3963, 4143, 4359, 4863, 4947, 4971, 5259, 5619, 5799, 5979, 6051, 6411, 7023, 7107, 7419, 7671, 7779

Offset: 1

Jason Earls, Jul 28 2001

nonn

			C(39) = 15, C(39+3) = 2*15.

Harry J. Smith, Table of n, a(n) for n=1,...,1000

Cf. A059153.

C(n)=n-eulerphi(n); j=[]; for(n=1,20000, if(C(n+3)==2*C(n),j=concat(j,n))); j

{ n=0; c1=c2=c3=1; for (m=1, 10^9, c=c1; c1=c2; c2=c3; c3=m-eulerphi(m); if (c3==2*c, write("b063480.txt", n++, " ", m - 3); if (n==1000, break)) ) } \\ Harry J. Smith, Aug 23 2009

A082506 a(n) = gcd(2^n, n - phi(n)); largest power of 2 dividing cototient(n) = A051953(n).

Original entry on oeis.org

2, 1, 1, 2, 1, 4, 1, 4, 1, 2, 1, 8, 1, 8, 1, 8, 1, 4, 1, 4, 1, 4, 1, 16, 1, 2, 1, 16, 1, 2, 1, 16, 1, 2, 1, 8, 1, 4, 1, 8, 1, 2, 1, 8, 1, 8, 1, 32, 1, 2, 1, 4, 1, 4, 1, 32, 1, 2, 1, 4, 1, 32, 1, 32, 1, 2, 1, 4, 1, 2, 1, 16, 1, 2, 1, 8, 1, 2, 1, 16, 1, 2, 1, 4, 1, 4, 1, 16, 1, 2, 1, 16, 1, 16, 1, 64, 1, 8, 1

Offset: 1

Labos Elemer, Apr 28 2003

nonn

a(n)=1 if and only if n is odd or n = 2. - Robert Israel, May 31 2018

			Different from A069177, analogous sequence with totient, instead of cototient.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000010, A051953, A009195, A083250, A007283, A050339, A053576, A069177.

f:= n -> padic:-ordp(n - numtheory:-phi(n), 2):
map(f, [$1..100]); # Robert Israel, May 31 2018

A098115 a(n) is the length of iteration trajectory when the cototient function (A051953) is applied to the half of the n-th primorial number (A070826(n) = A002110(n)/2).

Original entry on oeis.org

2, 3, 4, 7, 10, 5, 12, 15, 12, 28, 6, 6, 31, 12, 47, 29, 23, 32, 33, 24, 40, 28, 12, 35, 34, 56, 17, 36, 40, 123, 57, 61, 9, 99, 94, 132, 158, 172, 23, 43, 89, 186, 196, 194, 203, 157, 205, 62, 32, 26, 76, 105, 65, 45, 177, 56, 278

Offset: 1

Labos Elemer, Sep 27 2004

Initial values are here odd numbers. Comparing with the case of primorials (A098202), the lengths are here significantly smaller. The cause of this is unknown, albeit informally "understood": lack of powers of 2 in lists because parity is invariant during this iteration. See also lists for A098200 and A098201.

			For n = 7: list = {255255,163095,77815,16663,895,183,63,27,9,3,1,0}, a(7) = 12, while the comparable length for 510510 is A098202(7) = 43.

Cf. A002110, A051953, A053475, A070826, A098200, A098201, A098202.

g[x_] :=x-EulerPhi[x]; f[x_] :=Length[FixedPointList[g, x]]-1; q[x_] :=Apply[Times, Table[Prime[j], {j, 1, x}]]; t=Table[f[q[w]/2], {w, 1, 37}]
a[n_] := Length@ NestWhileList[(# - EulerPhi[#])&, Times @@ Prime[Range[2, n]], # > 0 &]; Array[a, 30] (* Amiram Eldar, Nov 19 2024 *)

a(n) = {my(p = prod(i=2, n, prime(i)), c = 1); while(p > 0, c++; p -= eulerphi(p)); c;} \\ Amiram Eldar, Nov 19 2024

a(n) = A053475(A070826(n)) = A053475(A002110(n)/2).

a(38)-a(57) from Amiram Eldar, Nov 19 2024

A141846 Triangle read by rows: A051731 * A051953^(n-k) * 0^(n-k), 1 <= k <= n.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 0, 1, 0, 2, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 4, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 0, 0, 4, 0, 0, 1, 0, 0, 0, 0, 0, 3, 0, 1, 0, 0, 1, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 0, 4, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 8

Offset: 1

Gary W. Adamson, Jul 11 2008

Row sums = A001065: (0, 1, 1, 3, 1, 6, 1, 7, 4, 8,...).
n-th row = (p-1) zeros followed by 1, iff n is prime.
For T(n,k), k divides n if k not = 0.

			First few rows of the triangle =
0;
0, 1;
0, 0, 1;
0, 1, 0, 2;
0, 0, 0, 0, 1;
0, 1, 1, 0, 0, 4;
0, 0, 0, 0, 0, 0, 1;
0, 1, 0, 2, 0, 0, 0, 4;
0, 0, 1, 0, 0, 0, 0, 0, 3;
0, 1, 0, 0, 1, 0, 0, 0, 0, 6;
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
0, 1, 1, 2, 0, 4, 0, 0, 0, 0, 0, 8;
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 8;
...

Cf. A051953, A001065.

Triangle read by rows, A051731 * A051953^(n-k); where A051953^(n-k) = an infinite lower triangular matrix with A051953 (0, 1, 1, 2, 1, 4, 1, 4, 3, 6, 1, 8,...) in the main diagonal and the rest zeros. A051731 = inverse Mobius transform.

cp's OEIS Frontend

A319348 Lexicographically earliest infinite sequence such that a(i) = a(j) => A003557(i) = A003557(j) and A051953(i) = A051953(j), for all i, j >= 1.

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A344178 Difference between the arithmetic derivative of n and the cototient of n: a(n) = A003415(n) - A051953(n).

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A361021 Lexicographically earliest infinite sequence such that a(i) = a(j) => A007814(i) = A007814(j), A001065(i) = A001065(j) and A051953(i) = A051953(j), for all i, j >= 1.

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A051961 Smallest number w such that A051953(w) = w - phi(w) is the n-th prime.

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A053480 Sum of values when cototient function A051953 is iterated until fixed point is reached.

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A063476 Sum_{d |C(n)} d^2, where C(n) is the Cototient function n - phi(n) (A051953).

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A063480 C(n+3)=2*C(n), where C(n) is Cototient(n) := n - phi(n) (A051953).

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A082506 a(n) = gcd(2^n, n - phi(n)); largest power of 2 dividing cototient(n) = A051953(n).

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