A324334 -id:A324334 - cp's OEIS frontend

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

-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

Michel Marcus, Feb 23 2019

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.

Brian Alspach, Research problems, Problem 18, Discrete Math 40 (1982), page 126.

Cf. A000010, A000203, A006881, A069249, A176881.

Cf. A324332, A324333, A324334.

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);
```

a(A006881(n)) = A176881(n)^2.

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

A324333 Numbers d such that A324331(x) = d^2 only when x is a squarefree semiprime (A006881).

Original entry on oeis.org

1, 2, 4, 6, 11, 12, 14, 16, 18, 20, 21, 22, 24, 25, 27, 29, 30, 32, 34, 35, 36, 38, 39, 41, 42, 43, 44, 45, 47, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 64, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 79, 81, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 97, 98, 100

Offset: 1

Michel Marcus, Feb 23 2019

nonn

Problem posed by Albert A. Mullin, who asks if this sequence is infinite.

B. Alspach, Research problems, Problem 18, Discrete Math 40 (1982), page 126.

Cf. A000010, A000203, A006881.

Cf. A324331, A324334 (complement).

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

Michel Marcus, Feb 23 2019

nonn

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

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

B. Alspach, Research problems, Problem 18, Discrete Math 40 (1982), page 126.

Cf. A000010, A000203, A006881.

Cf. A324331, A324333, A324334.

f(n) = (n-1)^2 - eulerphi(n)*sigma(n); \\ A324331
isok(n) = !((bigomega(n) == 2) && issquarefree(n)) && issquare(f(n));

cp's OEIS Frontend

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

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A324333 Numbers d such that A324331(x) = d^2 only when x is a squarefree semiprime (A006881).

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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).

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