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.

Showing 1-10 of 92 results. Next

A079695 Values never taken by phi(j)/2 for any j: a(n) = A005277(n)/2.

Original entry on oeis.org

7, 13, 17, 19, 25, 31, 34, 37, 38, 43, 45, 47, 49, 57, 59, 61, 62, 67, 71, 73, 76, 77, 79, 85, 87, 91, 93, 94, 97, 101, 103, 107, 109, 115, 117, 118, 121, 122, 123, 124, 127, 129, 133, 137, 139, 142, 143, 145, 149, 151, 152, 154, 157, 159
Offset: 1

Views

Author

Jon Perry, Jan 31 2003

Keywords

Comments

Because the degree of the minimal polynomial of cos(2*Pi/k) is phi(k)/2, the degree can never be a number in this sequence. - Artur Jasinski, Feb 23 2011

Examples

			A005277(1)=14, therefore a(1)=7.

Links

Crossrefs

Cf. A005277 (nontotients), A002180 (complementary sequence).

Programs

  • Haskell
    import Data.List.Ordered (minus)
a079695 n = a079695_list !! (n-1)
a079695_list = [1..] `minus` a002180_list
-- Reinhard Zumkeller, Nov 22 2015
  • Mathematica
    phiQ[m_] := Select[Range[m + 1, 2 m*Product[(1 - 1/(k*Log[k]))^(-1), {k, 2, DivisorSigma[0, m]}]], EulerPhi[#] == m &, 1] != {}; t = Select[Range[2, 320], phiQ]/2; Select[Range@ Max@ t, !MemberQ[t, #] &] (* Michael De Vlieger, Mar 22 2015, after Jean-François Alcover at A002180 *)
  • PARI
    is(n)=!istotient(2*n) \\ Charles R Greathouse IV, Mar 23 2015

A071615 a(n) is the least m such that 2*n*m is a nontotient value; that is, 2*n*a(n) is in A005277.

Original entry on oeis.org

7, 17, 15, 19, 5, 43, 1, 19, 5, 17, 7, 167, 1, 11, 3, 19, 1, 67, 1, 17, 17, 7, 5, 211, 1, 7, 11, 13, 3, 139, 1, 31, 9, 1, 5, 109, 1, 1, 3, 85, 3, 61, 1, 11, 1, 7, 1, 211, 1, 11, 5, 7, 3, 31, 5, 31, 1, 13, 1, 353, 1, 1, 9, 31, 3, 71, 1, 5, 3, 19, 1, 317, 1, 5, 3, 1, 1, 31, 1, 167, 7, 7, 5
Offset: 1

Views

Author

Labos Elemer, May 27 2002

Keywords

Examples

			n=5: number of terms in invphi(10k) is 2,5,2,9,0,9,... for k=1,2,3,...; a(5)=5 because 0 appears at 5th position.

Links

Crossrefs

Cf. A000010, A005277, A002202, A071616.

Programs

  • Maple
    with(numtheory); a := proc(n) local m; for m from 1 do if (invphi(2*n*m)=[]) then return m end end end
  • Mathematica
    invphi[n_, plist_] := Module[{i, p, e, pe, val}, If[plist=={}, Return[If[n==1, {1}, {}]]]; val={}; p=Last[plist]; For[e=0; pe=1, e==0||Mod[n, (p-1)pe/p]==0, e++; pe*=p, val=Join[val, pe*invphi[If[e==0, n, n*p/pe/(p-1)], Drop[plist, -1]]]]; Sort[val]]; invphi[n_] := invphi[n, Select[1+Divisors[n], PrimeQ]]; a[n_] := For[m=1, True, m++, If[invphi[2n*m]=={}, Return[m]]] (* invphi[n, plist] is list of x with phi(x)=n and all prime divisors of x in plist. *)

Extensions

Edited and extended by Robert G. Wilson v, May 28 2002
Edited and extended by Dean Hickerson, Jun 04 2002

A329872 Nontotients (A005277) that are the product of two totients (A002202).

Original entry on oeis.org

484, 968, 1100, 2116, 3364, 4232, 6084, 6724, 6728, 8464, 10404, 11132, 11236, 13448, 16928, 19044, 22472, 26896, 27556, 29584, 31684, 36100, 44944, 51076, 53792, 55112, 56644, 59168, 63368, 65824, 67416, 68644, 72200, 79524, 80344, 89888, 96100, 99856, 102152, 107584
Offset: 1

Views

Author

Jianing Song, Nov 23 2019

Keywords

Comments

We can have a list of nontotients and their factorizations into two totients. A totient m is in A301587 if and only if m never occurs in this list as a divisor of the nontotients. Using the list, many totients (10, 22, 44, 46, ...) are ruled out of A301587. But generally it's hard to prove that a number is in A301587.

Examples

			484 is here, because 484 = 22*22, and 22 is a totient while 484 isn't. Similarly, if p == 3 (mod 4) is a prime such that (p-1)^2+1 is composite, then (p-1)^2 is here.

Links

Crossrefs

Cf. A002202, A005277, A301587.
Squares of terms of A281187 are terms of this sequence.

Programs

  • PARI
    isA329872(n) = if(!istotient(n), my(v=divisors(n)); for(i=1, (1+#v)\2, if(istotient(v[i])&&istotient(n/v[i]), return(1))); 0); \\ improved by Jinyuan Wang, Mar 25 2023

A333100 Even numbers k such that both k and k + 2 are nontotients (A005277).

Original entry on oeis.org

74, 122, 152, 186, 234, 242, 244, 246, 284, 302, 338, 362, 374, 402, 404, 410, 412, 426, 434, 470, 472, 482, 494, 514, 516, 530, 532, 548, 572, 594, 602, 608, 626, 666, 668, 678, 722, 728, 746, 752, 788, 802, 804, 842, 844, 866, 868, 870, 872, 890, 892, 914, 942
Offset: 1

Views

Author

Amiram Eldar, Mar 07 2020

Keywords

Examples

			74 is a term since both 74 and 76 are nontotients.

Links

Crossrefs

Cf. A005277, A063512, A231964, A306952, A333101.

Programs

  • PARI
    forstep(k=2, 100, 2, if(!istotient(k) && !istotient(k+2), print1(k,", ")))

A350086 a(n) is the smallest totient number k > 1 such that A005277(n)*k is a nontotient number, or 0 if no such number exists.

Original entry on oeis.org

22, 22, 2, 2, 22, 2, 10, 10, 2, 6, 106, 2, 22, 46, 2, 2, 2, 6, 2, 10, 2, 2, 6, 2, 78, 2, 18, 2, 6, 2, 2, 2, 2, 46, 58, 2, 2, 2, 58, 2, 6, 2, 2, 2, 10, 10, 2, 46, 2, 2, 2, 82, 2, 30, 2, 6, 2, 10, 2, 10, 46, 2, 2, 2, 2, 2, 6, 78, 2, 10, 2, 10, 46, 10, 2, 46, 2
Offset: 1

Views

Author

Jianing Song, Dec 12 2021

Keywords

Comments

Subsequence of A350085.
Conjecture: a(n) != 0 for all n.
Records: 22 (A005277(n) = 14), 106 (A005277(n) = 90), 2010 (A005277(n) = 450), ...
By definition, a totient number N > 1 is a term if and only if there exists an even nontotient r such that: (i) k*r is a totient for totient numbers 2 <= k < N; (ii) N*r is a nontotient. No term can be of the form m*m', where m > 1 is a totient and m' > 1 is in A301587 (otherwise m*r is a totient implies m*m'*r is a totient).
Conjecture: every totient number > 1 which is not of the form m*m', where m > 1 is a totient and m' > 1 is in A301587, appears in this sequence. For example, the numbers 2, 6, 10, 18, 22, 28, 30 first appears when A007617(n) = 7, 15, 5, 33, 11, 902, 3.

Examples

			A005277(11) = 90. N = 106 is a totient number > 1 such that 90*k is a totient for totient numbers 2 <= k < N, and 90*N is a nontotient, so a(11) = 106.
A005277(83) = 450. N = 2010 is a totient number > 1 such that 450*k is a totient for totient numbers 2 <= k < N, and 450*N is a nontotient, so a(83) = 2010.
A005277(187) = 902. N = 28 is a totient number > 1 such that 902*k is a totient for totient numbers 2 <= k < N, and 902*N is a nontotient, so a(187) = 28.
A005277(73991) = 241010. N = 100 is a totient number > 1 such that 241010*k is a totient for totient numbers 2 <= k < N, and 241010*N is a nontotient, so a(73991) = 100. Note that although 100 = 10*10 is a product of 2 totient number > 1, neither factor is in A301587, so nothing prevents that 100 is a term of this sequence.

Links

Crossrefs

Cf. A005277, A350085, A301587.

Programs

  • PARI
    b(n) = if(!istotient(n), for(k=2, oo, if(istotient(k) && !istotient(n*k), return(k))))
list(lim) = my(v=[]); forstep(n=2, lim, 2, if(!istotient(n), v=concat(v,b(n)))); v \\ gives a(n) for A005277(n) <= lim

A071616 Smallest even number divisible by 2n which is nontotient, i.e., in A005277.

Original entry on oeis.org

14, 68, 90, 152, 50, 516, 14, 304, 90, 340, 154, 4008, 26, 308, 90, 608, 34, 2412, 38, 680, 714, 308, 230, 10128, 50, 364, 594, 728, 174, 8340, 62, 1984, 594, 68, 350, 7848, 74, 76, 234, 6800, 246, 5124, 86, 968, 90, 644, 94, 20256, 98, 1100, 510, 728, 318
Offset: 1

Views

Author

Labos Elemer, May 27 2002

Keywords

Comments

a(n) = 2n*A071615(n).

Examples

			n=4: 2n=8 and number of terms in invphi(8k) is 5, 6, 10, 7, 9, 11, 3, 8, 17, 10, 6, 17, 3, 6, 17, 9, 9, 21, 0, 12, ... for k=1,2,...,20,...; zero appears first at k=19, so a(4) = 8k = 152.

Links

Crossrefs

Cf. A000010, A002202, A005277, A071615.

Programs

  • Mathematica
    invphi[n_, plist_] := Module[{i, p, e, pe, val}, If[plist=={}, Return[If[n==1, {1}, {}]]]; val={}; p=Last[plist]; For[e=0; pe=1, e==0||Mod[n, (p-1)pe/p]==0, e++; pe*=p, val=Join[val, pe*invphi[If[e==0, n, n*p/pe/(p-1)], Drop[plist, -1]]]]; Sort[val]]; invphi[n_] := invphi[n, Select[1+Divisors[n], PrimeQ]]; a[n_] := For[m=1, True, m++, If[invphi[2n*m]=={}, Return[2n*m]]] (* invphi[n, plist] is list of x with phi(x)=n and all prime divisors of x in plist. *)

Extensions

Edited and extended by Robert G. Wilson v, May 28 2002 and by Dean Hickerson, Jun 04 2002

A079697 Odd part of A005277(n).

Original entry on oeis.org

7, 13, 17, 19, 25, 31, 17, 37, 19, 43, 45, 47, 49, 57, 59, 61, 31, 67, 71, 73, 19, 77, 79, 85, 87, 91, 93, 47, 97, 101, 103, 107, 109, 115, 117, 59, 121, 61, 123, 31, 127, 129, 133, 137, 139, 71, 143, 145, 149, 151, 19, 77, 157, 159
Offset: 1

Views

Author

Jon Perry, Jan 31 2003

Keywords

Comments

If A057192(k)=-1 then prime(k) appears an infinite number of times in this sequence, otherwise it occurs A057192(k)-1 times. - T. D. Noe, Sep 13 2007

Examples

			A005277(7)=68, therefore a(7)=17

Links

Crossrefs

Cf. A005277, A079698.

A079698 Values of the odd part of A005277(n).

Original entry on oeis.org

7, 13, 17, 19, 25, 31, 37, 43, 45, 47, 49, 57, 59, 61, 67, 71, 73, 77, 79, 85, 87, 91, 93, 97, 101, 103, 107, 109, 115, 117, 121, 123, 127, 129, 133, 137, 139, 143, 145, 149, 151, 157, 159, 161, 163, 167, 169, 175, 177, 181, 185, 187, 193, 195, 197, 199, 201
Offset: 0

Views

Author

Jon Perry, Jan 31 2003

Keywords

Crossrefs

A079697 sorted and duplicates removed. Cf. A005277(n).

Extensions

More terms from Sean A. Irvine, Aug 24 2025

A361686 a(n) is the least totient divisor of A329872(n), where A329872 are nontotients (A005277) that are the product of two totients (A002202).

Original entry on oeis.org

22, 22, 10, 46, 58, 46, 78, 82, 58, 46, 102, 22, 106, 82, 46, 138, 106, 82, 166, 172, 178, 190, 106, 226, 82, 166, 238, 172, 178, 22, 106, 262, 190, 282, 22, 106, 310, 316, 226, 82, 166, 238, 172, 346, 46, 178, 22, 358, 22, 10, 366, 106, 262, 382, 82, 388, 58, 22, 22, 46, 418
Offset: 1

Views

Author

Michel Marcus, Mar 29 2023

Keywords

Comments

Let k be the least instance a(k) = m, then A329872(k) = m*A361058(m). For instance a(3)=10, and A329872(3) = 1100 = 10*110 = 10*A361058(10).
Can we get a(k)=30 or a(k)=52 (see A361058)?

Examples

			a(3)=10 because A329872(3)=1100 which can be expressed as 1*1100, 2*550, 4*275, 5*220, 10*110, ... where 10*110 is the first case where both factors are nontotients.

Links

Crossrefs

Cf. A002202, A005277, A329872, A361058.

Programs

  • PARI
    is(n) = if(!istotient(n), my(v=divisors(n)); for(i=1, (1+#v)\2, if(istotient(v[i])&&istotient(n/v[i]), return(1))); 0); \\ A329872
lista(nn) = for (n=1, nn, if (is(n), my(d=divisors(n)); for (i=1, (1+#d)\2, if (istotient(d[i]) && istotient(n/d[i]), print1(d[i], ", "); break););););

A302495 a(n) is the least k such that k * A005277(n) is a term of A002202.

Original entry on oeis.org

2, 2, 3, 5, 2, 5, 2, 2, 3, 2, 2, 3, 2, 2, 6, 3, 3, 2, 4, 2, 3, 3, 2, 6, 2, 3, 2, 3, 2, 3, 3, 3, 5, 2, 2, 3, 3, 3, 2, 3, 2, 4, 6, 3, 2, 2, 3, 2, 4, 3, 3, 2, 5, 2, 3, 2, 7, 2, 3, 2, 2, 3, 3, 4, 3, 3, 2, 2, 9, 2, 4, 2, 2, 2, 3, 2, 5, 2, 2, 3, 5, 6, 2, 12, 6, 2, 3, 3, 3, 3, 2, 3, 3, 2, 3, 2, 5, 3, 2, 3
Offset: 1

Views

Author

Torlach Rush, Jun 10 2018

Keywords

Comments

This sequence maps non-totients to totients, a(n) * A005277(n) is not a term of A005277, rather it is a term of A002202.
Conjecture: Every k > 1 eventually appears in the sequence.

Examples

			a(1) = 2 because 2*A005277(1) = 2*14 = 28 is not a term of A005277.
a(3) = 3 because 3*A005277(3) = 3*34 = 102 is not a term of A005277.

Crossrefs

Cf. A000010, A002202, A005277, A000027.

Programs

  • PARI
    lista(nn) = {forstep(n=2, nn, 2, if (! istotient(n), my(k = 1); while (! istotient(k*n), k++); print1(k, ", ");););} \\ Michel Marcus, Jul 19 2018

Formula

a(n) = A067005(A005277(n)). - Michel Marcus, Jul 25 2018

Extensions

More terms from Michel Marcus, Jul 19 2018
Showing 1-10 of 92 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.