A293214 -id:A293214 - cp's OEIS frontend

A305792 a(n) = Product_{d|n, dA286622(d)-1).

1, 2, 2, 4, 2, 12, 2, 8, 6, 20, 2, 72, 2, 28, 30, 16, 2, 180, 2, 200, 42, 44, 2, 432, 10, 44, 30, 392, 2, 11700, 2, 32, 66, 20, 70, 5400, 2, 44, 66, 2000, 2, 29988, 2, 968, 1950, 76, 2, 2592, 14, 1100, 30, 968, 2, 20700, 110, 5488, 66, 76, 2, 4563000, 2, 116, 3570, 64, 110, 21780, 2, 200, 114, 53900, 2, 162000, 2, 68, 4290, 968, 154, 82764, 2, 20000

Offset: 1

Antti Karttunen, Jun 11 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8192

Cf. A008578, A278222, A286622, A305793 (rgs-transform), A305794.

Cf. also A293214, A304102.

A305792(n) = { my(m=1); fordiv(n,d,if(dA286622(d)-1))); (m); }; \\ Needs also code from A286622.

a(n) = Product_{d|n, dA008578(A286622(d)).

For all k >= 0, a(2^k) = 2^k.

A320014 Filter sequence combining the binary expansions of proper divisors of n, grouped by their residue classes mod 3.

Original entry on oeis.org

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, 35, 36, 37, 38, 2, 39, 40, 41, 42, 43, 2, 44, 2, 45, 46, 47, 48, 49, 2, 50, 51, 52, 2, 53, 2, 54, 55, 56, 57, 58, 2, 59, 60, 61, 2, 62, 63, 64, 65, 66, 2, 67, 68, 69, 70, 71, 72, 73, 2, 74, 75

Offset: 1

Antti Karttunen, Oct 03 2018

nonn

Restricted growth sequence transform of triple [A319990(n), A319991(n), A319992(n)], or equally, of ordered pair [A320010(n), A320013(n)].
Apart from trivial cases of primes, all other duplicates in range 1 .. 65537 seem to be squarefree semiprimes of the form 3k+1, i.e., both prime factors are either of the form 3k+1 or of the form 3k+2. Question: Is there any reason that more complicated cases would not occur later?
For all i, j: a(i) = a(j) => A293215(i) = A293215(j).
Differs from A319693 first for n = 108. - Georg Fischer, Oct 16 2018

			The first set of numbers that forms a nontrivial equivalence class is [295, 583, 799, 943] = [5*59, 11*53, 17*47, 23*41]. The prime factors in these are all of the form 3k+2, and when the binary expansions of the factors (like e.g., "101" for 5 and "111011" for 59 or "10111" for 23 and "101001" for 41) are overlaid, the resulting bit vector is always [1, 1, 1, 1, 1, 1^2], with the least significant bit-position containing 2 copies of 1's. Thus we have a(295) = a(583) = a(799) = a(943).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A019565, A319990, A319991, A319992, A320010, A320011, A320012, A320013.
Cf. also A293214, A293215, A293226, A300833.
Differs from A305800 for the first time at n=583, where a(583) = 234, while A305800(478).

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; };
A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A319990(n) = { my(m=1); fordiv(n,d,if((dA019565(d))); m; };
A319991(n) = { my(m=1); fordiv(n,d,if((dA019565(d))); m; };
A319992(n) = { my(m=1); fordiv(n,d,if((dA019565(d))); m; };
v320014 = rgs_transform(vector(up_to,n,[A319990(n),A319991(n),A319992(n)]));
A320014(n) = v320014[n];

A319708 a(n) = Product_{d|n, dA276086(d).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Oct 03 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..2310
Index entries for sequences related to primorial base

Cf. A276085, A276086, A319709 (rgs-transform).

Cf. A293214, A293221, A293222, A300834 for similar constructions for other bases.

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(dA276086(d))); (m); };

a(n) = Product_{d|n, dA276086(d).
For all n >= 1:
A276085(a(n)) = A001065(n).
A001222(a(n)) = A319713(n).

A318445 a(n) = Sum_{d|n, dA005187(d).

Original entry on oeis.org

0, 1, 1, 4, 1, 8, 1, 11, 5, 12, 1, 25, 1, 15, 13, 26, 1, 34, 1, 37, 16, 23, 1, 62, 9, 27, 21, 47, 1, 70, 1, 57, 24, 36, 20, 97, 1, 39, 28, 90, 1, 93, 1, 71, 55, 46, 1, 139, 12, 77, 37, 83, 1, 118, 28, 115, 40, 58, 1, 193, 1, 61, 71, 120, 32, 142, 1, 109, 47, 133, 1, 228, 1, 75, 86, 119, 31, 164, 1, 199, 71, 83, 1, 256, 41

Offset: 1

Antti Karttunen, Aug 26 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A005187, A293214, A293447, A318446, A318447.

A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A318445(n) = sumdiv(n,d,(dA005187(d));

a(n) = Sum_{d|n, dA005187(d).
a(n) = A318446(n) - A005187(n).
a(n) = A211779(n) + A318447(n).
a(n) = A293447(A293214(n)).

cp's OEIS Frontend

A305792 a(n) = Product_{d|n, dA286622(d)-1).

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A320014 Filter sequence combining the binary expansions of proper divisors of n, grouped by their residue classes mod 3.

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A319708 a(n) = Product_{d|n, dA276086(d).

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A318445 a(n) = Sum_{d|n, dA005187(d).

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