cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A281019 Partial products of A051953; a(1) = 1.

Original entry on oeis.org

1, 1, 1, 2, 2, 8, 8, 32, 96, 576, 576, 4608, 4608, 36864, 258048, 2064384, 2064384, 24772608, 24772608, 297271296, 2675441664, 32105299968, 32105299968, 513684799488, 2568423997440, 35957935964160, 323621423677440, 5177942778839040, 5177942778839040
Offset: 1

Views

Author

Jaroslav Krizek, Jan 13 2017

Keywords

Links

Crossrefs

Cf. A051953(n) = number of cototatives of n.
Cf. A063985.

Programs

  • Magma
    [1] cat [&*[#[h: h in [2..k] | GCD(h,k) ne 1]: k in [2..n]]: n in [2..100]]
  • Mathematica
    Table[If[n==1, 1, Product[i - EulerPhi[i], {i, 2, n}]], {n, 1, 29}] (* Indranil Ghosh, Mar 09 2017 *)
  • PARI
    for (n=1, 29, print1(if(n==1, 1, prod(i=2, n, i - eulerphi(i))),", ")); \\ Indranil Ghosh, Mar 09 2017

Formula

a(1) = 1; for n>1, a(n) = Product_{i=2..n} A051953(i).

A332972 Solutions k of the equation cototient(k) = cototient(k-1) + cototient(k-2) where cototient(k) is A051953.

Original entry on oeis.org

3, 4, 105, 165, 195, 2205, 2835, 38805, 131145, 407925, 936495, 1025505, 1231425, 1276905, 1788255, 1925565, 2521695, 2792145, 2847585, 3289935, 5003745, 5295885, 5710089, 6315309, 6986889, 13496385, 17168085, 19210065, 20171385, 22348365, 26879685, 27798705
Offset: 1

Views

Author

Amiram Eldar, Mar 04 2020

Keywords

Examples

			3 is a term since cototient(3) = 1 and cototient(1) + cototient(2) = 0 + 1 = 1.
105 is a term since cototient(105) = 57 and cototient(103) + cototient(104) = 1 + 56 = 57.

Links

Crossrefs

Cf. A051953, A065557, A065900, A075565, A076136, A076251, A145469, A291126, A291176, A292033, A294995.

Programs

  • Mathematica
    cotot[n_] := n - EulerPhi[n]; Select[Range[3, 10^6], cotot[#] == cotot[# - 1] + cotot[# - 2] &]

A051999 Minimal value w such that A051953(w) = w - phi(w) is prime and w has n prime divisors.

Original entry on oeis.org

4, 15, 255, 5865, 170085, 5437705, 226473065, 10380578845, 494390700895, 43592479037107
Offset: 1

Views

Author

Labos Elemer, Jan 05 2000

Keywords

Comments

a(11) <= 2995513256722805. - Donovan Johnson, Feb 06 2010

Examples

			a(1)=2^2, a(2)=3*5, a(3)=3*5*17, a(4)=3*5*17*23, a(5)=3*5*17*23*29, a(6)=5437705=5*7*13*17*19*37 with 1,2,3,4,5,6 prime divisors, respectively. The generated primes of form w - phi(w) are as follows: 2, 7, 127, 3049, 91237, 2452721.

Crossrefs

Cf. A050530, A051953.

Extensions

a(7)-a(10) from Donovan Johnson, Feb 06 2010

A053145 When cototient function (A051953) is iterated with initial value A002110(n), a(n) = number of iterations required to reach the stationary value=0.

Original entry on oeis.org

2, 4, 7, 11, 17, 19, 30, 31, 40, 42, 60, 64, 79, 76, 94, 124, 130, 124, 156, 172, 139, 191, 194, 220, 212, 211, 260, 268, 276, 299, 295, 320, 335, 328, 357, 366, 378, 404, 427, 438, 437, 463, 476, 492, 505, 453, 490, 541, 563, 587, 542, 599, 638, 659
Offset: 1

Views

Author

Labos Elemer, Feb 28 2000

Keywords

Examples

			n=6, A002110(6)=30030; the corresponding iteration chain is {30030, 24270, 17806, 9238, 4798, 2400, 1760, 1120, 736, 384, 256, 128, 64, 32, 16, 8, 4, 2, 1, 0}. Its length is 20, so a(6) = 20-1 = 19.

Crossrefs

Cf. A002110, A051953.

Programs

  • Mathematica
    Array[-1 + Length@ NestWhileList[# - EulerPhi@ # &, Product[Prime@ i, {i, #}], # > 0 &] &, 30] (* Michael De Vlieger, Nov 20 2017 *)

Formula

a(n) is smallest number such that Nest[A051953, A002110(n), a(n)]=0.

Extensions

More terms from Thomas Baruchel, Oct 11 2003
More terms from Sean A. Irvine, Dec 15 2021

A053146 When cototient function (A051953) is iterated with initial value A002110(n), a(n) = number of iterates that are not powers of 2.

Original entry on oeis.org

1, 2, 4, 6, 12, 11, 25, 22, 32, 26, 54, 55, 67, 68, 85, 117, 126, 113, 150, 166, 130, 178, 186, 207, 201, 194, 253, 255, 256, 295, 284, 309, 320, 315, 348, 347, 365, 398, 423, 425, 433, 459, 468, 488
Offset: 1

Views

Author

Labos Elemer, Feb 28 2000

Keywords

Comments

In these iteration chains, the numbers that are not powers of 2 seem to be in the majority.

Examples

			n=6, A002110(6)=30030; the corresponding iteration chain is: {30030, 24270, 17806, 9238, 4798, 2400, 1760, 1120, 736, 384, 256, 128, 64, 32, 16, 8, 4, 2, 1, 0}. It has 20 terms, among which entries from the first to 10th, also the last 0, are not powers of 2, so a(6)=11.

Crossrefs

Cf. A002110, A051953.

Extensions

a(13)-a(44) from Donovan Johnson, May 13 2010

A053147 When cototient function (A051953) is iterated with initial value A002110(n), a(n) is the value of first (largest) power of 2 which appears in the iteration.

Original entry on oeis.org

2, 4, 8, 32, 32, 256, 32, 512, 256, 65536, 64, 512, 4096, 256, 512, 128, 16, 2048, 64, 64, 512, 8192, 256, 8192, 2048, 131072, 128, 8192, 1048576, 16, 2048, 2048, 32768, 8192, 512, 524288, 8192, 64, 16, 8192, 16, 16, 256, 16
Offset: 1

Views

Author

Labos Elemer, Feb 28 2000

Keywords

Comments

In these iteration chains the number of non-2-powers seem to be dominant.
The sequence is not monotonic.

Examples

			For n=10, the iteration chain of 43 terms is {6469693230, 5447823150, 4315810350, ..., 188416, 98304, 65536, 32768, ..., 4, 2, 1, 0} in which the largest power of 2 is 65536 = 2^16.
For n=11 the length is 61, including 54 numbers that are not powers of 2, and 7 powers of 2, of which the largest is 64 = a(11) < a(10) = 65536.

Crossrefs

Cf. A002110, A051953.

Programs

  • Mathematica
    Table[SelectFirst[NestWhileList[# - EulerPhi@ # &, P, # > 0 &], IntegerQ@ Log2@ # &], {P, FoldList[Times, Prime@ Range@ 30]}] (* Michael De Vlieger, Jun 11 2018 *)

Extensions

More terms from Michael De Vlieger, Jun 11 2018
a(41)-a(44) from Jinyuan Wang, Jul 12 2021

A053148 When cototient function (A051953) is iterated with initial value A002110(n), a(n) = exponent of the largest power of 2 which appears in the iteration.

Original entry on oeis.org

1, 2, 3, 5, 5, 8, 5, 9, 8, 16, 6, 9, 12, 8, 9, 7, 4, 11, 6, 6, 9, 13, 8, 13, 11, 17, 7, 13, 20, 4, 11, 11, 15, 13, 9, 19, 13, 6, 4, 13, 4, 4, 8, 4, 24, 20
Offset: 1

Views

Author

Labos Elemer, Feb 28 2000

Keywords

Comments

In these iteration chains, powers of 2 seem to be in the minority.
The sequence is not monotonic.

Examples

			For n=10, the iteration chain of 43 terms is {6469693230, 5447823150, 4315810350, ..., 188416, 98304, 65536, 32768, ..., 4, 2, 1, 0} in which the largest power of 2 is 65536 = 2^16, so a(10)=16;
for n=11 the length is 61, including 54 numbers that are not powers of 2 and 7 powers of 2, of which the largest is 2^6 thus a(11)=6.

Crossrefs

Cf. A002110, A051953.

Programs

  • Mathematica
    a[n_] := Max@ IntegerExponent[ NestWhileList[# - EulerPhi[#] &, Times @@ Prime[Range[n]], # > 1 &], 2]; Array[a, 25] (* Giovanni Resta, May 30 2018 *)
  • PARI
    A051953(n)= { return(n-eulerphi(n)); } A002110(n)= { return(prod(i=1,n,prime(i))); } ispow2(n)= { local(nbin,nbinl,sd); nbin=binary(n); nbinl=matsize(nbin); sd=sum(i=1,nbinl[2],nbin[i]); if(sd==1, return(nbinl[2]-1), return(0); ); } A053148itr(n)= { local(v,vbin,maxp); v=A002110(n); maxp=ispow2(v); while(v>0, v=A051953(v); maxp=max(maxp,ispow2(v)); ); return(maxp); } { for(n=1,70, print1(A053148itr(n),","); ); } \\ R. J. Mathar, May 19 2006

Extensions

More terms from R. J. Mathar, May 19 2006
a(37)-a(46) from Giovanni Resta, May 31 2018

A053476 Smallest number m such that when A051953 is applied n times to m the result is neither a power of 2 nor 0.

Original entry on oeis.org

9, 21, 42, 82, 130, 330, 450, 666, 1050, 1470, 1950, 2922, 4074, 5586, 7770, 11154, 15810, 22638, 30702, 42570, 53130, 68970, 107690, 159390, 206910, 289830, 395190, 610350, 823290, 1185570, 1522290, 2168250, 3011850, 4103490, 5364450
Offset: 1

Views

Author

Labos Elemer, Jan 14 2000

Keywords

Comments

An analog for A000005 is A049117.

Examples

			a(6)=330 and the iteration of A051953 applied to 330 gives sequence {330,250,150,110,70,46,24,16,8,4,2,1,0}. Six iterations result in the 6th term 24 which still is neither a power of 2 nor 0. For smaller numbers than 330 these 6 iterations yield a power of 2 or the fixed number 0.

Crossrefs

Cf. A051953, see also its iterates.

Formula

Applying cototient-function A051953 n+1 times to a(n), a power of 2 or 0 appears; a(n) is the smallest with this property.

Extensions

More terms from Jud McCranie, Jan 14 2000

A064016 a(n) = Sum_{k <= 10^n} cototient(k), where cototient is A051953.

Original entry on oeis.org

0, 23, 2006, 196308, 19607514, 1960399246, 196036947608, 19603648572758, 1960364533634092, 196036449326991586, 19603644912113783634, 1960364490766613788860, 196036449073440195974090, 19603644907302101080472556, 1960364490729905106089642146, 196036449072986990521291164848
Offset: 0

Views

Author

Robert G. Wilson v, Sep 07 2001

Keywords

Comments

It appears that lim_{n->infinity} (1/n^2) * Sum_{j=1..n} a(j) = 0.1960364... = (1/2 - 3/Pi^2).

Crossrefs

Cf. A002088, A051953, A063985, A064018.

Programs

  • Mathematica
    s = 0; k = 1; Do[ While[ k <= 10^n, s = s + k - EulerPhi[ k ]; k++ ]; Print[ s ], {n, 0, 8} ]

Formula

a(n) = 10^n*(10^n+1)/2 - A002088(10^n) = 10^n*(10^n+1)/2 - A064018(n). - Chai Wah Wu, Apr 18 2021
a(n) = A063985(10^n). - Michel Marcus, Apr 18 2021

Extensions

a(9) from Jud McCranie, Jun 25 2005
a(10)-a(11) from Donovan Johnson, Feb 06 2010
a(12) from Donovan Johnson, Feb 07 2012
a(13)-a(15) using A064018 from Chai Wah Wu, Apr 18 2021

A072073 Number of solutions to cototient(x) = A051953(x) = 2^n.

Original entry on oeis.org

1, 2, 3, 3, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10
Offset: 1

Views

Author

Labos Elemer, Jun 13 2002

Keywords

Comments

a(n) increases at A000043(n).
Since A051953(p) = 1 for p prime, and given that there are an infinite number of primes, we disregard a(0) = oo. - Michael De Vlieger, Mar 25 2020

Examples

			InvCototient(2^0) has an infinite number of entries, so 2^0=1 is left out.
n=14: 2^14=16384, InvCototient(16384) = {24576,28672,31744,32512,32764,32768}, so a(14)=6;

Crossrefs

Cf. A051953, A058764, A000043, A063740, A000079.

Programs

  • Mathematica
    Length /@ Most@ Split@ DeleteCases[Select[Array[# - EulerPhi[#] &, 10^6], IntegerQ@ Log2@ # &], 1] (* Michael De Vlieger, Mar 25 2020 *)

Formula

a(n) = A063740(A000079(n)). - Ridouane Oudra, Jun 02 2024
Previous Showing 51-60 of 308 results. Next

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