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A190273
Numbers n such that n' = m+1, with n and m semiprimes and gcd(m,n)>1, where n' is the arithmetic derivative of n.
5
6, 10, 21, 26, 39, 55, 57, 74, 93, 111, 122, 146, 155, 201, 203, 253, 301, 305, 314, 327, 381, 386, 417, 471, 497, 543, 554, 597, 626, 633, 689, 737, 755, 791, 794, 842, 889, 905, 914, 921, 1011, 1027, 1055, 1081, 1082, 1137, 1226, 1227, 1322, 1346, 1379, 1461, 1466, 1477, 1497, 1514, 1623, 1655, 1703, 1711, 1713, 1731, 1751, 1754, 1893, 1967, 1994
OFFSET
1,1
COMMENTS
The sequence is related to the Rassias Conjecture ("for any prime p there are two primes p1 and p2 such that p*p1=p1+p2+1, p>2, p2>p1", see A190272-A190275), because n = p1*p2, m=p1*p -> p1*p = p1+p2-1. The sequence includes the cases with p=p1 (or p2). Generalization can be achieved by removing semiprimarity condition or accepting gcd(n,m)=1. The differential equation in its general form n'=m+1 includes Giuga Numbers, i.e., n'=b*n+1, or n'=n+1 (A007850).
These are semiprimes n = p*q such that 1/p + 1/q - 1/n = P/Q, where P <> Q are primes. Cf. A326690. - Amiram Eldar and Thomas Ordowski, Jul 25 2019
LINKS
For Rassias conjecture: Preda Mihăilescu, Review of Problem Solving and Selected Topics in Number Theory, Newsletter of the European Mathematical Society, March 2011, p. 46.
EXAMPLE
n=6, 6'=5, m=5+1=6, gcd(6,6)=6 -> a(1)=6
MAPLE
der:=n->n*add(op(2, p)/op(1, p), p=ifactors(n)[2]);
seq(`if`(bigomega(i)=2 and bigomega(der(i)-1)=2 and gcd(i, der(i)-1)>1, i, NULL), i=1..2000);
CROSSREFS
Cf. A001358 (semiprimes), A003415 (arithmetic derivative), A007850 (Giuga numbers), A190272 (n'=m-1), A190273, A190274, A190275.
Sequence in context: A085274 A175648 A178661 * A030007 A323729 A372295
KEYWORD
nonn
AUTHOR
Giorgio Balzarotti, May 07 2011
STATUS
approved