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Restricted growth sequence transform of A300834, product_{d|n, d<n} A019565( A003714(d)); Filter sequence related to Zeckendorf-representations of proper divisors of n.
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6
1, 2, 2, 3, 2, 4, 2, 5, 6, 7, 2, 8, 2, 9, 10, 11, 2, 12, 2, 13, 14, 15, 2, 16, 17, 18, 19, 20, 2, 21, 2, 22, 23, 24, 25, 26, 2, 27, 28, 29, 2, 30, 2, 31, 32, 33, 2, 34, 7, 35, 36, 37, 2, 38, 39, 40, 41, 42, 2, 43, 2, 44, 45, 46, 47, 48, 2, 49, 50, 51, 2, 52, 2, 53, 54, 55, 56, 57, 2, 58, 59, 60, 2, 61, 41, 62, 63, 64, 2, 65, 66, 67, 68, 69
EXAMPLE
For cases n=10 and 49, we see that 10 has proper divisors 1, 2 and 5 and these have Zeckendorf-representations ( A014417) 1, 10 and 1000, while 49 has proper divisors 1 and 7 and these have Zeckendorf-representations 1 and 1010. When these Zeckendorf-representations are summed (columnwise without carries), result in both cases is 1011, thus a(10) = a(49).
PROG
(PARI)
up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om, invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om, invec[i], i); outvec[i] = u; u++ )); outvec; };
write_to_bfile(start_offset, vec, bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A072649(n) = { my(m); if(n<1, 0, m=0; until(fibonacci(m)>n, m++); m-2); }; \\ From A072649
A003714(n) = { my(s=0, w); while(n>2, w = A072649(n); s += 2^(w-1); n -= fibonacci(w+1)); (s+n); }
A019565(n) = {my(j, v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
write_to_bfile(1, rgs_transform(vector(up_to, n, A300834(n))), "b300835.txt");
a(n) = Product_{d|n, d<n} A019565(d).
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25
1, 2, 2, 6, 2, 36, 2, 30, 12, 60, 2, 2700, 2, 180, 120, 210, 2, 7560, 2, 6300, 360, 252, 2, 661500, 20, 420, 168, 94500, 2, 23814000, 2, 2310, 504, 132, 600, 43659000, 2, 396, 840, 2425500, 2, 187110000, 2, 207900, 352800, 1980, 2, 560290500, 60, 194040, 264, 485100, 2, 115259760, 840, 254677500, 792, 4620, 2, 264737261250000, 2, 13860
FORMULA
a(n) = Product_{d|n, d<n} A019565(d). Other identities.
For n >= 1:
PROG
(PARI)
A019565(n) = {my(j, v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A293214(n) = { my(m=1); fordiv(n, d, if(d < n, m *= A019565(d))); m; };
CROSSREFS
Cf. also A293216, A293221, A293222, A293225, A293231, A293442, A300830, A300831, A300832, A300833, A300834.
a(n) is the total number of terms (1-digits) in Zeckendorf representation of all proper divisors of n.
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0, 1, 1, 2, 1, 3, 1, 4, 2, 3, 1, 7, 1, 4, 3, 5, 1, 7, 1, 7, 4, 4, 1, 11, 2, 3, 4, 8, 1, 10, 1, 7, 4, 5, 4, 14, 1, 5, 3, 11, 1, 10, 1, 8, 7, 4, 1, 15, 3, 8, 5, 7, 1, 12, 4, 12, 5, 4, 1, 21, 1, 5, 7, 10, 3, 13, 1, 8, 4, 11, 1, 19, 1, 4, 8, 10, 5, 10, 1, 16, 7, 5, 1, 20, 5, 5, 4, 12, 1, 20, 4, 10, 5, 4, 5, 21, 1, 9, 10, 16, 1, 13, 1, 11, 10
EXAMPLE
For n=12, its proper divisors are 1, 2, 3, 4 and 6. Zeckendorf-representations ( A014417) of these numbers are 1, 10, 100, 101 and 1001. Total number of 1's present is 7, thus a(12) = 7.
PROG
(PARI)
A072649(n) = { my(m); if(n<1, 0, m=0; until(fibonacci(m)>n, m++); m-2); }; \\ From A072649
A007895(n) = { my(s=0); while(n>0, s++; n -= fibonacci(1+ A072649(n))); (s); }
a(n) = Product_{d|n, d<n} A019565(phi(d)), where phi is the Euler totient function A000010.
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7
1, 2, 2, 4, 2, 12, 2, 12, 6, 20, 2, 108, 2, 60, 30, 60, 2, 540, 2, 300, 90, 84, 2, 2700, 10, 140, 90, 2700, 2, 6300, 2, 420, 126, 44, 150, 121500, 2, 132, 210, 10500, 2, 283500, 2, 5292, 3150, 660, 2, 132300, 30, 5500, 66, 14700, 2, 267300, 210, 472500, 198, 1540, 2, 4630500, 2, 4620, 47250, 4620, 350, 873180, 2, 1452, 990
PROG
(PARI)
A019565(n) = {my(j, v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A318834(n) = { my(m=1); fordiv(n, d, if(d < n, m *= A019565(eulerphi(d)))); m; };
1, 2, 2, 3, 2, 4, 2, 5, 3, 4, 2, 6, 2, 4, 4, 10, 2, 6, 2, 6, 4, 4, 2, 10, 3, 4, 5, 6, 2, 8, 2, 7, 4, 4, 4, 9, 2, 4, 4, 10, 2, 8, 2, 6, 6, 4, 2, 20, 3, 6, 4, 6, 2, 10, 4, 10, 4, 4, 2, 12, 2, 4, 6, 14, 4, 8, 2, 6, 4, 8, 2, 15, 2, 4, 6, 6, 4, 8, 2, 20, 10, 4, 2, 12, 4, 4, 4, 10, 2, 12, 4, 6, 4, 4, 4, 14, 2, 6, 6, 9, 2, 8, 2, 10, 8
PROG
(PARI)
A003714(n) = { my(s=0, w); while(n>2, w = A072649(n); s += 2^(w-1); n -= fibonacci(w+1)); (s+n); }
A072649(n) = { my(m); if(n<1, 0, m=0; until(fibonacci(m)>n, m++); m-2); }; \\ From A072649
A019565(n) = {my(j, v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
a(n) = Product_{d|n, d<n} A276086(d).
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4
1, 2, 2, 6, 2, 36, 2, 54, 12, 108, 2, 1620, 2, 60, 216, 810, 2, 5400, 2, 43740, 120, 540, 2, 607500, 36, 300, 360, 40500, 2, 21870000, 2, 182250, 1080, 2700, 360, 151875000, 2, 1500, 600, 246037500, 2, 101250000, 2, 5467500, 972000, 13500, 2, 85429687500, 20, 6075000, 5400, 5062500, 2, 2531250000, 3240, 3417187500, 3000, 67500, 2
FORMULA
a(n) = Product_{d|n, d<n} A276086(d). For all n >= 1:
PROG
(PARI)
A276086(n) = { my(i=0, m=1, pr=1, nextpr); while((n>0), i=i+1; nextpr = prime(i)*pr; if((n%nextpr), m*=(prime(i)^((n%nextpr)/pr)); n-=(n%nextpr)); pr=nextpr); m; };
A319708(n) = { my(m=1); fordiv(n, d, if(d<n, m *= A276086(d))); (m); };
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