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 A304277 Numbers equal to the sum of their aliquot parts, each of them increased by 4. 8
 5, 182, 230, 344, 1072, 3424, 11456, 12844, 321470, 2182144, 33959936, 1084153472, 8598519808, 15381952750, 36113287330004992 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Searched up to n = 10^12. a(16) > 10^18. - Hiroaki Yamanouchi, Aug 28 2018 From Giovanni Resta, May 11 2018: (Start) If p = 2^(1+t) + (1+2*t)*k - 1 is a prime, for some t > 0 and k even, then x = 2^t*p is in the sequence where k is the value by which the sum of aliquot parts is increased. In this sequence k = 4; for t = 64 we get 680564733841876936426822412823955505152, which is a term greater than 15381952750, but this does not exclude the existence of other intermediate terms following a different solution pattern. In fact, there could be also sporadic solutions of the type x = 2^t*r*q, where r and q are prime and for which no closed form is known. E.g., for k = 4 we have x = 2^17*500069*550959. To find them, since d(n) = 4*(t+1) and sigma(n) = (2^(t+1)-1)*(1+r)*(1+q), the relation 2*n = sigma(n) + k*(d(n)-1) becomes 2^(t+1)*r*q = (2^(t+1)-1)*(1+r)*(1+q) + k*(4*t+3), which, for fixed t and k, is a quadratic Diophantine equation in r and q that could admit solutions with r and q prime. (End) Terms using odd values of k seem very hard to find. Up to n = 10^12, only three such terms are known: 2, 98, and 8450, for k = 1, 5, and -7, respectively. LINKS EXAMPLE Aliquot part of 5 is 1 and 1+4 = 5. Aliquot parts of 182 are 1, 2, 7, 13, 14, 26, 91 and (1+4) + (2+4) + (7+4) + (13+4) + (14+4) + (26+4) + (91+4) = 182. MAPLE with(numtheory): P:=proc(q, k) local n; for n from 1 to q do if 2*n=sigma(n)+k*(tau(n)-1) then print(n); fi; od; end: P(10^12, 4); MATHEMATICA With[{k = 4}, Select[Range[10^6], DivisorSum[#, # + k &] - (# + k) == # &] ] (* Michael De Vlieger, May 14 2018 *) PROG (PARI) isok(n) = sumdiv(n, d, if (d < n, d+4)) == n; \\ Michel Marcus, May 14 2018 CROSSREFS Cf. A000005, A000203, A000396, A304276, A304278, A304279, A304280, A304281, A304282, A304283, A304284. Sequence in context: A050239 A218690 A256286 * A208403 A280797 A094081 Adjacent sequences:  A304274 A304275 A304276 * A304278 A304279 A304280 KEYWORD nonn,hard,more AUTHOR Paolo P. Lava, Giovanni Resta, May 11 2018 EXTENSIONS a(15) from Hiroaki Yamanouchi, Aug 28 2018 STATUS approved

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Last modified January 29 04:57 EST 2020. Contains 331335 sequences. (Running on oeis4.)