A334261 Numbers m that are solutions of the arithmetic differential equation 2m" - m' - 4 = 0.

4, 9, 21, 25, 33, 49, 57, 69, 85, 93, 121, 129, 133, 145, 169, 177, 205, 213, 217, 237, 249, 253, 265, 289, 309, 361, 393, 417, 445, 469, 489, 493, 505, 517, 529, 553, 565, 573, 597, 633, 669, 685, 697, 753, 781, 793, 813, 817, 841, 865, 889, 913, 933, 949, 961, 973, 985, 993

Offset: 1

Author

Nathan Mabey, Apr 20 2020

Keywords

• easy
• nonn

Comments

Contains all the squared primes, plus additional terms.

At least some of these additional terms that are not divisible by 3 are given by the product of the smallest prime in a prime triplet, and the largest prime in that triplet, this certainly being true until the term 11413.

Examples

			4 is a term since 4'' = 4, 4' = 4, and 2*4 - 4 - 4 = 0.
		

Links

• Nathan Mabey, Table of n, a(n) for n = 1..20263

Crossrefs

Cf. A003415 (n'), A068346 (n"), A001248 (squared primes), A368701 (characteristic function).

Cf. also A189941.

Programs

PARI

d(n) = if (n>0, my(f=factor(n)); sum(i=1, #f~, n*f[i, 2]/f[i, 1]), 0);
isok(m) = -2*d(d(m)) + d(m) + 4 == 0; \\ Michel Marcus, Apr 21 2020

Python

from sympy import factorint
def d(n): return sum(n*e//p for p, e in factorint(n).items())
def ok(m): return m > 1 and 2*d(d(m)) - d(m) == 4
print([k for k in range(1, 999) if ok(k)]) # Michael S. Branicky, Aug 15 2022