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 1-6 of 6 results.

A359566 Primitive terms of A359565: terms of A359565 with no proper divisor in A359565.

Original entry on oeis.org

12, 40, 126, 364, 544, 546, 1026, 1100, 1170, 1650, 2812, 3591, 3608, 4095, 4100, 4214, 4218, 4510, 6138, 6150, 7564, 8658, 9394, 9548, 9990, 10804, 10850, 11096, 11132, 11346, 11368, 12078, 13286, 13870, 13950, 14210, 14322, 16206, 16376, 16698, 17082, 17290
Offset: 1

Views

Author

Amiram Eldar, Jan 06 2023

Keywords

Comments

If k is a term then all the positive multiples of k are terms of A359565.

Links

Crossrefs

Cf. A326835, A359563, A359564, A359565.

Programs

  • Mathematica
    q[n_] := Max[Tally[EulerPhi[Divisors[n]]][[;; , 2]]] > 2; primQ[n_] := q[n] && AllTrue[Divisors[n], # == n || !q[#] &]; Select[Range[17000], primQ]
  • PARI
    is1(k) = vecmax(matreduce(apply(x->eulerphi(x), divisors(k)))[,2]) > 2;
is(k) = fordiv(k, d, if(is1(d), return(d==k))); return(0);

A359563 Odd numbers that have at least two divisors with the same value of the Euler totient function (A000010).

Original entry on oeis.org

63, 189, 273, 315, 441, 513, 567, 585, 693, 819, 825, 945, 1071, 1197, 1323, 1365, 1449, 1539, 1575, 1701, 1755, 1827, 1911, 1953, 2079, 2107, 2109, 2205, 2255, 2331, 2457, 2475, 2565, 2583, 2709, 2835, 2925, 2961, 3003, 3069, 3075, 3087, 3213, 3339, 3465, 3549
Offset: 1

Views

Author

Amiram Eldar, Jan 06 2023

Keywords

Comments

The even numbers are excluded from this sequence since every even number has this property: it is divisible by 1 and 2, and phi(1) = phi(2) = 1.
If k is a term then all the odd multiples of k are terms. The primitive terms are in A359564.
The numbers of terms below 10^k, for k = 1, 2, ..., are 0, 1, 12, 140, 1402, 14193, 142606, 1427749, 14283236, 142855925, ... . Apparently, the asymptotic density of this sequence exists and equals 0.01428... .
The least term that is not divisible by 3 is a(26) = 2107.

Examples

			63 is a term since it is odd, 7 and 9 are both divisors of 63, and phi(7) = phi(9) = 6.

Links

Crossrefs

Complement of A326835 within the odd numbers.
Cf. A000010, A359564, A359565, A359566.

Programs

  • Mathematica
    Select[Range[1, 3500, 2], !UnsameQ @@ EulerPhi[Divisors[#]] &]
Select[Range[1,3601,2],Max[Tally[EulerPhi[Divisors[#]]][[;;,2]]]>1&] (* Harvey P. Dale, Mar 05 2025 *)
  • PARI
    is(k) = k>1 && k%2 && numdiv(k) > #Set(apply(x->eulerphi(x), divisors(k)));

A359564 Primitive terms of A359563: terms of A359563 with no proper divisor in A359563.

Original entry on oeis.org

63, 273, 513, 585, 825, 2107, 2109, 2255, 3069, 3075, 4329, 4697, 4995, 5425, 5673, 6039, 6643, 6935, 6975, 7105, 7161, 8103, 8349, 8541, 8645, 9855, 10235, 11543, 12625, 13725, 13869, 14497, 14841, 16385, 18639, 18915, 19825, 22165, 25025, 26169, 26533, 30225
Offset: 1

Views

Author

Amiram Eldar, Jan 06 2023

Keywords

Comments

Odd numbers that are not in A326835 but all of their proper divisors are in it.
If k is a term then all the odd multiples of k are terms of A359563.

Links

Crossrefs

Cf. A326835, A359563, A359565, A359566.

Programs

  • Mathematica
    q[n_] := !UnsameQ @@ EulerPhi[Divisors[n]]; primQ[n_] := q[n] && AllTrue[Divisors[n], # == n || !q[#] &]; Select[Range[1, 30000, 2], primQ]
  • PARI
    is1(k) = k>1 && k%2 && numdiv(k) > #Set(apply(x->eulerphi(x), divisors(k)));
is(k) = fordiv(k, d, if(is1(d), return(d==k))); return(0);

A373529 Numbers k such that k and k+1 both have at least three divisors with the same value of the Euler totient function (A000010).

Original entry on oeis.org

32319, 111320, 175959, 179360, 191919, 212120, 246519, 254079, 254960, 279279, 319599, 355508, 357399, 398600, 436149, 463239, 512000, 520064, 524799, 542240, 580040, 606879, 657152, 678699, 685880, 701631, 718640, 726920, 739556, 750519, 759759, 775775, 787815
Offset: 1

Views

Author

Amiram Eldar, Jun 08 2024

Keywords

Comments

Numbers k such that k and k+1 are both in A359565.

Links

Crossrefs

Subsequence of A359565.
A373530 is a subsequence.
Cf. A000010, A102190.

Programs

  • Mathematica
    q[n_] := q[n] = Max[Tally[EulerPhi[Divisors[n]]][[;; , 2]]] > 2; Select[Range[3*10^6], q[#] && q[# + 1] &]
  • PARI
    is(k) = vecmax(matreduce(apply(x->eulerphi(x), divisors(k)))[,2]) > 2;
lista(kmax) = {my(q1 = 0, q2); for(k = 2, kmax, q2 = is(k); if(q1 && q2, print1(k-1, ", ")); q1 = q2);}

A373531 a(n) is the maximum number of divisors of n with an equal value of the Euler totient function (A000010).

Original entry on oeis.org

1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1
Offset: 1

Views

Author

Amiram Eldar, Jun 08 2024

Keywords

Comments

The sums of the first 10^k terms, for k = 1, 2, ..., are 15, 161, 1641, 16554, 166029, 1662306, 16630535, 166335597, 1663473941, 16635216306, ... . Apparently, this sequence has an asymptotic mean 1.663... .

Examples

			a(2) = 2 since 2 has 2 divisors, 1 and 2, and phi(1) = phi(2) = 1.
a(12) = 3 since 3 of the divisors of 12 (3, 4 and 6) have the same value of phi: phi(3) = phi(4) = phi(6) = 2.

Links

Crossrefs

Cf. A000010, A102190, A319696, A326835, A359563, A359565, A373532.

Programs

  • Mathematica
    a[n_] := Max[Tally[EulerPhi[Divisors[n]]][[;; , 2]]]; Array[a, 100]
  • PARI
    a(n) = vecmax(matreduce(apply(x->eulerphi(x), divisors(n)))[ , 2]);
  • Python
    from collections import Counter
from sympy import divisors, totient
def a(n):
    c = Counter(totient(d) for d in divisors(n, generator=True))
    return c.most_common(1)[0][1]
print([a(n) for n in range(1, 88)]) # Michael S. Branicky, Jun 08 2024

Formula

a(A326835(n)) = 1.
a(A359563(n)) >= 2.
a(A359565(n)) >= 3.
a(2*n) >= 2.
a(p) = 2 for an odd prime p.
a(m*n) >= a(n) for all m > 1.

A373530 Numbers k such that k, k+1 and k+2 all have at least three divisors with the same value of the Euler totient function (A000010).

Original entry on oeis.org

14052608, 83025998, 87703714, 93978520, 117345150, 163338174, 213589088, 218539880, 294321950, 369698434, 401177798, 463425920, 470217824, 497434040, 529524918, 539318438, 554556078, 559474838, 581302358, 584754848, 608842934, 612448640, 617445814, 625591966
Offset: 1

Views

Author

Amiram Eldar, Jun 08 2024

Keywords

Comments

Numbers k such that k, k+1 and k+2 are all in A359565.
There must be 3 or more divisors of k that have the same Euler totient value, and ditto for k+1 and k+2, but those values may differ as among k, k+1, and k+2. - Harvey P. Dale, Sep 01 2024

Crossrefs

Subsequence of A359565 and A373529.
Cf. A000010, A102190.

Programs

  • Mathematica
    q[n_] := Max[Tally[EulerPhi[Divisors[n]]][[;; , 2]]] > 2; seq[kmax_] := Module[{s = {}, q1 = 0, q2 = 0, q3}, Do[q3 = q[k]; If[q1 && q2 && q3, AppendTo[s, k-2]]; q1=q2; q2=q3, {k, 3, kmax}]; s]; seq[10^8]
SequencePosition[Table[If[Max[Tally[EulerPhi[Divisors[n]]][[;;,2]]]>2,1,0],{n,88*10^6}],{1,1,1}] [[;;,1]] (* The program generates the first 3 terms of the sequence. *) (* Harvey P. Dale, Sep 01 2024 *)
  • PARI
    is(k) = vecmax(matreduce(apply(x->eulerphi(x), divisors(k)))[,2]) > 2;
lista(kmax) = {my(q1 = 0, q2 = 0, q3); for(k = 3, kmax, q3 = is(k); if(q1 && q2 && q3, print1(k-2, ", ")); q1 = q2; q2 = q3);}
Showing 1-6 of 6 results.

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