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 1-3 of 3 results.

A249396 Composite numbers whose Euler totient divides the sum of the Euler totients of the numbers less than n that are relatively prime to n.

Original entry on oeis.org

9, 75, 135, 265, 1383, 6585, 17251, 165615, 503111, 856195
Offset: 1

Views

Author

Paolo P. Lava, Oct 27 2014

Keywords

Comments

No more terms < 10^6.

Examples

			Numbers coprime to 9 are 1, 2, 4, 5, 7, 8 and phi(9) = 6. Then, phi(1) + phi(2) + phi(4) + phi(5) + phi(7) + phi(8) = 1 + 1 + 2 + 4 + 6 + 4 = 18 and 18/6 = 3.

Crossrefs

Cf. A000010, A053570, A249397.

Programs

  • Maple
    with(numtheory): P:=proc(q) local a,k,n; for n from 2 to q do
if not isprime(n) then a:=0;
for k from 1 to n do if gcd(k,n)=1 then a:=a+phi(k); fi; od;
if type(a/phi(n),integer) then print(n); fi; fi; od; end: P(10^9);
  • PARI
    isok(n) = (n!=1) && !isprime(n) && (sum(k=1, n-1, if (gcd(k, n) == 1, eulerphi(k), 0)) % eulerphi(n) == 0); \\ Michel Marcus, Oct 29 2014

Extensions

a(8) from Ray Chandler, Nov 07 2014
a(9)-a(10) from Ray Chandler, Nov 11 2014

A249108 Composite numbers whose sum of aliquot parts divides the sum of aliquot parts of the numbers less than or equal to n and relatively prime to n.

Original entry on oeis.org

133, 667, 961, 1007, 2013, 3986, 5662, 15979, 17453, 33233, 46943, 51101, 94870, 101444, 119045, 134298, 136957, 179567, 188897, 213511, 226203, 246149, 279749, 299139, 306667, 310157, 434531, 449087, 449183, 518459, 519203
Offset: 1

Views

Author

Paolo P. Lava, Oct 21 2014

Keywords

Examples

			Numbers coprime to 133 are 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 20, 22, 23, 24, 25, 26, 27, 29, 30, 31, 32, 33, 34, 36, 37, 39, 40, 41, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 71, 72, 73, 74, 75, 78, 79, 80, 81, 82, 83, 85, 86, 87, 88, 89, 90, 92, 93, 94, 96, 97, 99, 100, 101, 102, 103, 104, 106, 107, 108, 109, 110, 111, 113, 115, 116, 117, 118, 120, 121, 122, 123, 124, 125, 127, 128, 129, 130, 131, 132. The sum of their aliquot parts is 4401; sigma(133) - 133 = 27 and 4401 / 27 = 163.

Links

Crossrefs

Cf. A001065, A249109, A249396, A249397.

Programs

  • Maple
    with(numtheory): P:=proc(q) local a,k,n; for n from 2 to q do
if not isprime(n) then a:=0;
for k from 1 to n do if gcd(k,n)=1 then a:=a+sigma(k)-k; fi; od;
if type(a/(sigma(n)-n),integer) then print(n); fi; fi; od; end: P(10^9);
  • PARI
    lista(nn) = {forcomposite(n=1, nn, s = 0; for (i=1, n, if (gcd(n, i) == 1, s += sigma(i)-i);); if ((s % (sigma(n)-n)) == 0, print1(n, ", ")););} \\ Michel Marcus, Nov 07 2014

Extensions

a(10)-a(13) from Michel Marcus, Nov 07 2014
a(14)-a(31) from Ray Chandler, Nov 12 2014

A249109 Composite numbers whose sum of aliquot parts divides the sum of the aliquot parts of the numbers less than or equal to n and not relatively prime to n.

Original entry on oeis.org

15, 26, 27, 38, 76, 194, 531, 1445, 1501, 2923, 2988, 4427, 4499, 4769, 5817, 7831, 9523, 10602, 12412, 14963, 16117, 24863, 26768, 29041, 29329, 30229, 36577, 45246, 49817, 58483, 58823, 71165, 75469, 76273, 79799, 83429, 86941, 94037
Offset: 1

Views

Author

Paolo P. Lava, Oct 21 2014

Keywords

Examples

			Numbers not coprime to 15 are 3, 5, 6, 9, 10, 12, 15. Then, sigma(3) - 3 = 1, sigma(5) - 5 = 1, sigma(6) - 6 = 6, sigma(9) - 9 = 4, sigma(10) - 10 = 8, sigma(12) - 12 = 16, sigma(15) - 15 = 9; their sum is 1 + 1 + 6 + 4 + 8 + 16 + 9 = 45 and 45 / 9 = 5.

Links

Crossrefs

Cf. A001065, A249108, A249396, A249397.

Programs

  • Maple
    with(numtheory): P:=proc(q) local a,k,n; for n from 2 to q do
if not isprime(n) then a:=0;
for k from 1 to n do if gcd(k,n)>1 then a:=a+sigma(k)-k; fi; od;
if type(a/(sigma(n)-n),integer) then print(n); fi; fi; od; end: P(10^9);
  • PARI
    lista(nn) = {forcomposite(n=1, nn, if (sum(k=1, n, if (gcd(k,n) !=1, sigma(k)-k)) % (sigma(n) - n) == 0, print1(n, ", ")););} \\ Michel Marcus, Nov 09 2014

Extensions

a(22)-a(38) from Michel Marcus, Nov 09 2014
Showing 1-3 of 3 results.

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