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 A227478 Numbers k such that both the sum of the semiprime divisors of k and the sum of the prime divisors of k are squares. 0
 1146, 2874, 9870, 33220, 34353, 43140, 50694, 52290, 66440, 86280, 94350, 100804, 101097, 103059, 106140, 121540, 125070, 127897, 132880, 139908, 156870, 172560, 183475, 191140, 193410, 201608, 208692, 212280, 243080, 248378, 265760, 276094, 279816, 303291 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS The sequence is infinite: if a number of the form p(1) * p(2) * ... * p(i)^2 * p(i+1) * ... * p(m) is in the sequence where p(1), ..., p(m) are primes, then the numbers p(1) * p(2) * ... * p(i)^q * p(i+1) * ... * p(m) are also in the sequence for q = 3, 4, ... For example, the infinite subsequence 33220, 66440, 132880, ... contains the numbers of the form 2^q * 5 * 11 * 151 for q = 2, 3, 4, ... where 2+5+11+151 = 169 = 13^2 and 2*2 + 2*5 + 2*11 + 2*151 + 5*11 + 5*151 + 11*151 = 2809 = 53^2. In this sequence, the corresponding pairs of squares are (961, 196), (2401, 484), (900, 64), (2809, 169), (4900, 361), (7225, 729), (2304, 100), (1521, 100), (2809, 169), (7225, 729), (1225, 64), (3721, 121), (12100, 289), (4900, 361), (2704, 100), (7225, 169), (8100, 400), (2916, 169), (2809, 169), (12769, 225), (1521, 100), (7225, 729), (8464, 225), (13225, 529), (5329, 121), (3721, 121), (1369, 64), (2704, 100), (7225, 169), (13689, 289), (2809, 169), (3364, 100), (12769, 225), (12100, 289), ... LINKS EXAMPLE 1146 = 2*3*191 is in the sequence because the divisors are {1, 2, 3, 6, 191, 382, 573, 1146}, so the sum of the semiprime divisors is 6 + 382 + 573 = 961 = 31^2 and the sum of the prime divisors is 2 + 3 + 191 = 196 = 14^2. MAPLE with(numtheory):for n from 2 to 310000 do:x:=divisors(n):n1:=nops(x): y:=factorset(n):n2:=nops(y):s1:=0:s2:=0:for i from 1 to n1 do: if bigomega(x[i])=2 then s1:=s1+x[i]:else fi:od: s2:=sum('y[i]', 'i'=1..n2):if sqrt(s1)=floor(sqrt(s1)) and sqrt(s2)=floor(sqrt(s2)) then printf(`%d, `, n):else fi:od: MATHEMATICA Rest@ Select[Range[3*10^5], AllTrue[{DivisorSum[#, # &, PrimeOmega@ # == 2 &], DivisorSum[#, # &, PrimeQ]}, IntegerQ@ Sqrt@ # &] &] (* Michael De Vlieger, Sep 15 2017 *) CROSSREFS Cf. A001358, A008472, A076920, A164722, A227476. Sequence in context: A202312 A123697 A260977 * A190924 A196779 A180344 Adjacent sequences:  A227475 A227476 A227477 * A227479 A227480 A227481 KEYWORD nonn AUTHOR Michel Lagneau, Jul 13 2013 STATUS approved

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Last modified April 22 11:46 EDT 2019. Contains 322330 sequences. (Running on oeis4.)