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 1-5 of 5 results.

A383484 Integers k such that there exists an integer 0

Original entry on oeis.org

3, 33, 69, 77, 133, 161, 235, 267, 287, 321, 385, 699, 715, 1235, 1379, 1437, 1529, 1595, 1653, 1719, 2047, 2233, 2241, 2569, 2727, 2829, 3237, 3269, 3999, 4585, 4683, 4911, 5075, 5163, 5215, 5497, 5667, 5691, 7085, 7089, 7587, 7761, 7797, 7945, 8259, 9159, 9659, 10653
Offset: 1

Views

Author

S. I. Dimitrov, Apr 28 2025

Keywords

Comments

From David A. Corneth, May 04 2025: (Start)
If (t, u) is a divisor pair of sigma(k)^2 then m = (t + u - 2*k)/2, sigma(m) = m + k - t.
Proof:
Since sigma(m)^2 + sigma(k)^2 = (m+k)^2 we have sigma(k)^2 = (m+k)^2 - sigma(m)^2 = (m + k - sigma(m)) * (m + k + sigma(m)) = t * u where t, u | sigma(k)^2.
This gives the system (m + k - sigma(m)) = t and (m + k + sigma(m)) = u. Solving gives
m = (t + u - 2*k)/2, sigma(m) = m + k - t. For every pair (t, u) of divisors of sigma(k)^2 we can test if the given values of m and sigma(m) hold. If at least one of them holds then k is in the sequence. Q. E. D.
Are there any even terms? There are none in the first 1006 terms. (End)

Examples

			(2, 3) is such a pair because sigma^2(2)+sigma^2(3) = 3^2+4^2 = (2+3)^2.
33 is in the sequence. As sigma(33)^2 = 2304 and for the divisor pair (32, 72) we have m = (32 + 72 - 2*33)/2 = 19 and sigma(m) = m + k - 32 = 19 + 33 - 32 = 20 and indeed sigma(19) = 20. - _David A. Corneth_, May 04 2025

Links

Crossrefs

Cf. A063990, A259180, A383239.

Programs

  • PARI
    isok(k) = for (m=1, k, if (sigma(m)^2+sigma(k)^2==(m+k)^2, return(1))); \\ Michel Marcus, Apr 28 2025
  • PARI
    is(n) = {my(sn = sigma(n)^2, d = divisors(sn)); for(i = 1, #d / 2, k = (d[i] + d[#d + 1 - i] - 2*n) / 2; if(denominator(k) == 1, sk = n + k - d[i]; if(k < n && sigma(k) == sk, return(1)))); 0} \\ David A. Corneth, May 04 2025

Extensions

More terms from Michel Marcus, Apr 28 2025

A383483 Numbers k such that k = sigma(m)-m where m = sigma(3*k)-3*k.

Original entry on oeis.org

3, 15, 5919, 118719, 179871, 33750303
Offset: 1

Views

Author

S. I. Dimitrov, Apr 28 2025

Keywords

Comments

S. I. Dimitrov introduced the notion of (alpha, beta)-amicable pairs.

Examples

			For alpha=1, beta=3 we have (3, 4), (15, 33), (5919, 7905).
Here (3, 4) is such a pair because 3=sigma(4)-4 and 4=sigma(3*3)-3*3.

Links

Crossrefs

Cf. A063990, A259180, A383239.

Programs

  • PARI
    isok(k) = my(m = sigma(3*k) - 3*k); if (m>0, sigma(m) - m == k); \\ Michel Marcus, Apr 28 2025

Formula

We say that the numbers m and n form an (alpha, beta)-amicable pair if sigma(alpha*n)-alpha*n=m and sigma(beta*m)-beta*m=n, where alpha and beta are positive integers, and sigma(n) is the sum of the divisors of n.

Extensions

a(4)-a(6) from Michel Marcus, Apr 28 2025

A383932 Integers k such that there exists an integer 0

Original entry on oeis.org

84, 102, 160, 186, 276, 284, 330, 582, 624, 762, 868, 1164, 1210, 1372, 1404, 1446, 1488, 1540, 1988, 2156, 2640, 2716, 2898, 2924, 3556, 3708, 3882, 4074, 4228, 4536, 5382, 5564, 5610, 5802, 6018, 6282, 6368, 6392, 6486, 6612, 6748, 7140, 7452, 7494, 7960, 8358, 8432, 9222, 9834
Offset: 1

Views

Author

S. I. Dimitrov, May 15 2025

Keywords

Comments

The numbers m and k form a GM-amicable pair. See Dimitrov link.

Examples

			For k=2 we have (28, 84), (42, 102), (60, 276), (92, 160).

Links

Crossrefs

Cf. A063990, A259180, A383239, A383483, A383484.

Programs

  • PARI
    isok(k) = for (m=1, k-1, if (sigma(m)*sigma(k) == (m+k)^2, return(m))); \\ Michel Marcus, May 15 2025

Extensions

More terms from Michel Marcus, May 15 2025

A383964 Integers k such that there exists an integer 0

Original entry on oeis.org

168, 1320, 3792, 4968, 7176, 8184, 14364, 15240, 20076, 29904, 30672, 41952, 48312, 48768, 54264, 56856, 57960, 60144, 64296, 72996, 73344, 83328, 90552, 91512, 99828, 106020, 110952, 113280, 114156, 119016, 128592, 149292, 150024, 151272, 157608, 168588, 175584, 183240
Offset: 1

Views

Author

S. I. Dimitrov, May 16 2025

Keywords

Comments

The numbers m and k form a HM(2,1)-amicable pair (HM = harmonic mean). See Dimitrov link.

Examples

			(120, 168) is such a pair because (1/sigma(120)^2 + 1/sigma(168)^2)*(120+168)^2 = 1.

Links

Crossrefs

Cf. A063990, A259180, A383239, A383483, A383484.

Programs

  • PARI
    isok(k) = for(m=1, k-1, if((1/sigma(m)^2 + 1/sigma(k)^2)*(m+k)^2 == 1, return(m))); \\ Michel Marcus, May 16 2025

Extensions

a(7) and a(9)-a(25) from Michel Marcus, May 16 2025
More terms from David A. Corneth, Jun 21 2025

A383714 Integers k such that there exists an integer 0

Original entry on oeis.org

21, 231, 284, 1210, 2499, 2924, 5564, 6368, 10856, 14595, 18416, 66992, 71145, 76084, 87633, 88730
Offset: 1

Views

Author

S. I. Dimitrov, May 14 2025

Keywords

Comments

The numbers m and k form a WPM(2)-amicable pair (WPM = weighted power mean). See Dimitrov link.

Examples

			(7, 21) is such a pair because 7*sigma(7)^2 + 21*sigma(21)^2 = 7*8^2 + 21*32^2 = (7+21)^3.

Links

Crossrefs

Cf. A002046 (a subsequence), A063990, A259180, A383239, A383483, A383484.

Programs

  • PARI
    isok(k)= for (m=1, k-1, if (m*sigma(m)^2 + k*sigma(k)^2 == (m+k)^3, return(m))); \\ Michel Marcus, May 15 2025

Extensions

a(3)-a(16) from Michel Marcus, May 15 2025
Showing 1-5 of 5 results.

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