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.

A324334 Numbers d such that A324331(x)=d^2 not only for squarefree semiprimes x (A006881) but also for x in A324333.

Original entry on oeis.org

3, 5, 7, 8, 9, 10, 13, 15, 17, 19, 23, 26, 28, 31, 33, 37, 40, 46, 48, 49, 55, 61, 62, 63, 65, 78, 80, 82, 96, 99, 118, 122, 126, 127, 129, 142, 144, 145, 148, 157, 159, 163, 166, 172, 176, 179, 185, 226, 228, 230, 240, 242, 244, 246, 249, 255, 257, 258, 288, 296, 303, 320, 321, 328, 342, 354, 357, 358, 360
Offset: 1

Views

Author

Michel Marcus, Feb 23 2019

Keywords

Examples

			A324331(45) = 64, a square, even though 45 is not squarefree semiprime, so 8 is a term, and 45 is in A324332.

Links

Crossrefs

Cf. A000010, A000203, A006881.
Cf. A324331, A324332, A324333 (complement).

A324331 a(n) = (n-1)^2 - phi(n)*sigma(n), where phi is A000010 and sigma is A000203.

Original entry on oeis.org

-1, -2, -4, -5, -8, 1, -12, -11, -14, 9, -20, 9, -24, 25, 4, -23, -32, 55, -36, 25, 16, 81, -44, 49, -44, 121, -44, 57, -56, 265, -60, -47, 64, 225, 4, 133, -72, 289, 100, 81, -80, 529, -84, 169, 64, 441, -92, 225, -90, 541, 196, 249, -104, 649, 36, 145, 256, 729, -116, 793
Offset: 1

Views

Author

Michel Marcus, Feb 23 2019

Keywords

Comments

For squarefree semiprimes n = p*q a(n)=(p-q)^2 is a square. But the converse, a(n) is prime, can happen: see A324332.

Links

Crossrefs

Cf. A000010, A000203, A006881, A069249, A176881.
Cf. A324332, A324333, A324334.

Programs

  • Mathematica
    Table[(n-1)^2 - EulerPhi[n]*DivisorSigma[1, n], {n, 1, 60}] (* Vaclav Kotesovec, Feb 23 2019 *)
  • PARI
    a(n) = (n-1)^2 - eulerphi(n)*sigma(n);

Formula

a(A006881(n)) = A176881(n)^2.
a(n) = A069249(n) - 2*n + 1. - Amiram Eldar, Dec 04 2023

A324332 Numbers m such that A324331(m) = (m-1)^2 - phi(m)*sigma(m) is a square, even though they are not squarefree semiprimes (A006881).

Original entry on oeis.org

12, 20, 24, 40, 42, 44, 45, 48, 63, 72, 80, 96, 104, 105, 108, 132, 135, 160, 189, 190, 192, 200, 216, 275, 320, 342, 384, 385, 399, 405, 429, 452, 456, 465, 567, 575, 610, 637, 639, 640, 648, 693, 768, 783, 848, 969, 988, 1000, 1015, 1044, 1098, 1105, 1127, 1210, 1215
Offset: 1

Views

Author

Michel Marcus, Feb 23 2019

Keywords

Comments

If m is a squarefree semiprime, then A324331(m) is a square. But the converse is not always true.

Examples

			A324331(45) = 64, a square, even though 45 is not squarefree semiprime, so 45 is a term.

Links

Crossrefs

Cf. A000010, A000203, A006881.
Cf. A324331, A324333, A324334.

Programs

  • PARI
    f(n) = (n-1)^2 - eulerphi(n)*sigma(n); \\ A324331
isok(n) = !((bigomega(n) == 2) && issquarefree(n)) && issquare(f(n));
Showing 1-3 of 3 results.

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

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