A305800 -id:A305800 - cp's OEIS frontend

A305975 Filter sequence: All prime powers p^k, k >= 1, are allotted to distinct equivalence classes according to their exponent k, while all other numbers occur in singular equivalence classes of their own.

1, 2, 2, 3, 2, 4, 2, 5, 3, 6, 2, 7, 2, 8, 9, 10, 2, 11, 2, 12, 13, 14, 2, 15, 3, 16, 5, 17, 2, 18, 2, 19, 20, 21, 22, 23, 2, 24, 25, 26, 2, 27, 2, 28, 29, 30, 2, 31, 3, 32, 33, 34, 2, 35, 36, 37, 38, 39, 2, 40, 2, 41, 42, 43, 44, 45, 2, 46, 47, 48, 2, 49, 2, 50, 51, 52, 53, 54, 2, 55, 10, 56, 2, 57, 58, 59, 60, 61, 2, 62, 63, 64, 65, 66, 67, 68, 2, 69, 70, 71

Offset: 1

Antti Karttunen, Jul 02 2018

nonn

Restricted growth sequence transform of A305974.

For all i, j: A305800(i) = A305800(j) => a(i) = a(j) => A305976(i) = A305976(j).

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A000961 (A246655), A085970, A305800, A305974, A305976, A305979.

up_to = 100000;
partialsums(f,up_to) = { my(v = vector(up_to), s=0); for(i=1,up_to,s += f(i); v[i] = s); (v); }
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; };
v065515 = partialsums(n -> (omega(n)<=1), up_to);
A065515(n) = v065515[n];
A085970(n) = (n - A065515(n));
A305974(n) = if(1==n,n,my(e = isprimepower(n)); if(e,-e,1+A085970(n)));
v305975 = rgs_transform(vector(up_to,n,A305974(n)));
A305975(n) = v305975[n];

a(prime) = 2, a(prime^2) = 3, a(prime^3) = 5, a(prime^4) = 10, a(prime^5) = 19.

A305980 Filter sequence for a(Squarefree numbers > 1) = constant sequences.

Original entry on oeis.org

1, 2, 2, 3, 2, 2, 2, 4, 5, 2, 2, 6, 2, 2, 2, 7, 2, 8, 2, 9, 2, 2, 2, 10, 11, 2, 12, 13, 2, 2, 2, 14, 2, 2, 2, 15, 2, 2, 2, 16, 2, 2, 2, 17, 18, 2, 2, 19, 20, 21, 2, 22, 2, 23, 2, 24, 2, 2, 2, 25, 2, 2, 26, 27, 2, 2, 2, 28, 2, 2, 2, 29, 2, 2, 30, 31, 2, 2, 2, 32, 33, 2, 2, 34, 2, 2, 2, 35, 2, 36, 2, 37, 2, 2, 2, 38, 2, 39, 40, 41, 2, 2, 2, 42, 2, 2, 2, 43, 2, 2, 2

Offset: 1

Antti Karttunen, Jul 02 2018

nonn

For all i, j: A305800(i) = A305800(j) => a(i) = a(j).

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A005117, A008966, A057627, A305800.

up_to = 100000;
partialsums(f,up_to) = { my(v = vector(up_to), s=0); for(i=1,up_to,s += f(i); v[i] = s); (v); }
v057627 = partialsums(x -> !issquarefree(x),up_to);
A057627(n) = v057627[n];
A305980(n) = if(1==n,n,if(issquarefree(n),2,2+A057627(n)));

a(1) = 1; for n > 1, if A008966(n) = 1 [when n is in A005117], then a(n) = 2, otherwise a(n) = 2+A057627(n).

A317943 Filter sequence constructed from the coefficients of the Stern polynomials B(d,t) collected for each proper divisor d of n; Restricted growth sequence transform of A317942.

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, 76, 2, 77, 2, 78, 79, 80, 2, 81, 2, 82, 83, 84, 2, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 86

Offset: 1

Antti Karttunen, Aug 12 2018

nonn

For all i, j: a(i) = a(j) => A317837(i) = A317837(j).

			Proper divisors of 115 are 1, 5 and 23 and proper divisors of 125 are 1, 5 and 25. The divisors 1 and 5 occur in both, while for the Stern polynomials B(23,t) and B(25,t) (see A125184) the nonzero coefficients are {1, 2, 3, 1} and {1, 3, 2, 1}, that is, they are equal as multisets, thus A286378(23) = A286378(25). From this follows that a(115) = a(125).

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to Stern's sequences

Cf. A125184, A286378, A317837, A317942, A317945.
Cf. also A293217, A305793.
Differs from A305800 and A296073 for the first time at n=125, where a(125) = 86.

\\ Needs also code from A286378:
up_to = 65537;
A317942(n) = { my(m=1); fordiv(n,d,if(dA286378(d)-1))); (m); };
v317943 = rgs_transform(vector(up_to, n, A317942(n)));
A317943(n) = v317943[n];

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];

A322826 Lexicographically earliest such sequence a that a(i) = a(j) => A052126(i) = A052126(j) for all i, j.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 4, 2, 1, 3, 1, 2, 4, 5, 1, 6, 1, 3, 4, 2, 1, 5, 7, 2, 8, 3, 1, 6, 1, 9, 4, 2, 7, 10, 1, 2, 4, 5, 1, 6, 1, 3, 8, 2, 1, 9, 11, 12, 4, 3, 1, 13, 7, 5, 4, 2, 1, 10, 1, 2, 8, 14, 7, 6, 1, 3, 4, 12, 1, 15, 1, 2, 16, 3, 11, 6, 1, 9, 17, 2, 1, 10, 7, 2, 4, 5, 1, 13, 11, 3, 4, 2, 7, 14, 1, 18, 8, 19, 1, 6, 1, 5, 16

Offset: 1

Antti Karttunen, Dec 27 2018

nonn

Restricted growth sequence transform of A052126, or equally, of A322820.
For all i, j:
  A300226(i) = A300226(j) => a(i) = a(j),
  a(i) = a(j) => A322813(i) = A322813(j),
  a(i) = a(j) => A322819(i) = A322819(j).
For all i, j > 1:
  a(i) = a(j) => A001222(i) = A001222(j).

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

Cf. A001222, A006530, A052126, A305800, A300226, A319994, A319996, A322813, A322819, A322820.

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; };
A006530(n) = if(n>1, vecmax(factor(n)[, 1]), 1);
A052126(n) = (n/A006530(n));
v322826 = rgs_transform(vector(up_to,n,A052126(n)));
A322826(n) = v322826[n];

A329351 Lexicographically earliest infinite sequence such that a(i) = a(j) => A329350(i) = A329350(j) for all i, j.

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, 4, 12, 9, 17, 2, 18, 2, 19, 20, 21, 22, 23, 2, 24, 25, 26, 2, 27, 2, 28, 29, 30, 2, 31, 32, 33, 34, 35, 2, 36, 37, 38, 39, 18, 2, 40, 2, 41, 42, 43, 44, 45, 2, 46, 47, 48, 2, 49, 2, 50, 51, 52, 44, 53, 2, 54, 55, 56, 2, 57, 58, 59, 60, 61, 2, 62, 63, 64, 65, 66, 29, 67, 2, 68, 69

Offset: 1

Antti Karttunen, Nov 12 2019

nonn

Restricted growth sequence transform of A329350.
For all i, j:
  A305800(i) = A305800(j) => a(i) = a(j),
  a(i) = a(j) => A069359(i) = A069359(j).

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

Cf. A069359, A305800, A329350.

Cf. also A329353, A329381.

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; };
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A329350(n) = { my(m=1); fordiv(n,d,if(isprime(n/d), m *= A276086(d))); (m); };
v329351 = rgs_transform(vector(up_to, n, A329350(n)));
A329351(n) = v329351[n];

A351235 Lexicographically earliest infinite sequence such that a(i) = a(j) => A046523(i) = A046523(j), A327858(i) = A327858(j) and A345000(i) = A345000(j) for all i, j >= 1.

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, 7, 2, 15, 16, 17, 18, 8, 2, 19, 2, 20, 21, 7, 22, 23, 2, 24, 10, 25, 2, 19, 2, 26, 27, 28, 2, 29, 30, 31, 14, 32, 2, 33, 10, 25, 10, 7, 2, 34, 2, 9, 27, 35, 36, 19, 2, 13, 10, 19, 2, 37, 2, 9, 38, 39, 36, 19, 2, 40, 41, 7, 2, 34, 10, 17, 10, 42, 2

Offset: 1

Antti Karttunen, Feb 06 2022

Restricted growth sequence transform of the triplet [A046523(n), A327858(n), A345000(n)].
For all i, j >= 1:
  A305800(i) = A305800(j) => a(i) = a(j) => A351085(i) = A351085(j).

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

Cf. A003415, A046523, A276086, A327858, A345000, A351086.

Cf. also A101296, A305800, A329620, A351085, A351236.

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; };
A003415(n) = {my(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1]))}; \\ From A003415
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A327858(n) = gcd(A003415(n),A276086(n));
A345000(n) = gcd(A003415(n),A003415(A276086(n)));
Aux351235(n) = [A046523(n), A327858(n), A345000(n)];
v351235 = rgs_transform(vector(up_to,n,Aux351235(n)));
A351235(n) = v351235[n];

A319693 Filter sequence combining sopfr(d) from all proper divisors d of n, where sopfr(d) is A001414(d) = sum of primes dividing d with repetition.

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, 76, 2, 77, 2, 78, 79, 80, 2, 73, 2, 81, 82, 83, 2, 84, 85

Offset: 1

Antti Karttunen, Oct 02 2018

nonn

Restricted growth sequence transform of A319692.

For all i, j: a(i) = a(j) => A305611(i) = A305611(j).

			The proper divisors of  96 are 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, while
the proper divisors of 108 are 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54.
It happens that sopfr(8) = sopfr(9), sopfr(16) = sopfr(18), sopfr(24) = sopfr(27), sopfr(32) = sopfr(36) and sopfr(48) = sopfr(54), and the rest of proper divisors (1, 2, 3, 4, 6, 12) are shared by both numbers, from which follows that by taking product of sopfr over proper divisors gives an identical result for both, thus a(96) = a(108). Here sopfr = A001414.

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

Cf. A001414, A319692.
Cf. also A319353.
Differs from A305800, A296073 and A317943 for the first time at n=108, as here a(108) = 73.

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; };
A001414(n) = ((n=factor(n))[, 1]~*n[, 2]); \\ From A001414.
A319692(n) = { my(m=1); fordiv(n, d, if(dA001414(d)))); (m); };
v319693 = rgs_transform(vector(up_to,n,A319692(n)));
A319693(n) = v319693[n];

A319706 Filter sequence which for primes p records the prime signature of 2p+1, and for all other numbers assigns a unique number.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 5, 6, 7, 8, 2, 9, 10, 11, 12, 13, 5, 14, 5, 15, 16, 17, 2, 18, 19, 20, 21, 22, 2, 23, 24, 25, 26, 27, 28, 29, 24, 30, 31, 32, 2, 33, 5, 34, 35, 36, 5, 37, 38, 39, 40, 41, 2, 42, 43, 44, 45, 46, 5, 47, 5, 48, 49, 50, 51, 52, 53, 54, 55, 56, 5, 57, 24, 58, 59, 60, 61, 62, 5, 63, 64, 65, 2, 66, 67, 68, 69, 70, 2, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 5

Offset: 1

Antti Karttunen, Sep 26 2018

nonn

Restricted growth sequence transform of function f defined as f(n) = A046523(2n+1) when n is a prime, otherwise -n.
For all i, j:
  A305810(i) = A305810(j) => a(i) = a(j),
and
  a(i) = a(j) => A305800(i) = A305800(j),
  a(i) = a(j) => A305978(i) = A305978(j),
  a(i) = a(j) => A305985(i) = A305985(j).

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf- A046523, A305800, A305810, A305900, A305901, A305978, A305985.

Cf. A005384 (positions of 2's), A234095 (positions of 5's).

up_to = 100000;
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; };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A319706aux(n) = if(isprime(n),A046523(n+n+1),-n);
v319706 = rgs_transform(vector(up_to,n,A319706aux(n)));
A319706(n) = v319706[n];

A329381 Lexicographically earliest infinite sequence such that a(i) = a(j) => A329380(i) = A329380(j) for all i, j.

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, 16, 31, 32, 2, 33, 34, 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, 47, 56, 2, 57, 58, 59, 2, 60, 61, 62, 63, 64, 2, 65, 66, 67, 68, 69, 70, 71, 2, 72, 73

Offset: 1

Antti Karttunen, Nov 12 2019

nonn

Restricted growth sequence transform of A329380.
For all i, j:
  A305800(i) = A305800(j) => a(i) = a(j),
  a(i) = a(j) => A323599(i) = A323599(j).

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

Cf. A001221, A276086, A305800, A323599, A329380.

Cf. also A329351.

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; };
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A329380(n) = { my(m=1); fordiv(n,d,m *= A276086(d)^omega(n/d)); (m); };
v329381 = rgs_transform(vector(up_to, n, A329380(n)));
A329381(n) = v329381[n];

cp's OEIS Frontend

A305975 Filter sequence: All prime powers p^k, k >= 1, are allotted to distinct equivalence classes according to their exponent k, while all other numbers occur in singular equivalence classes of their own.

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A305980 Filter sequence for a(Squarefree numbers > 1) = constant sequences.

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A317943 Filter sequence constructed from the coefficients of the Stern polynomials B(d,t) collected for each proper divisor d of n; Restricted growth sequence transform of A317942.

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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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A322826 Lexicographically earliest such sequence a that a(i) = a(j) => A052126(i) = A052126(j) for all i, j.

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A329351 Lexicographically earliest infinite sequence such that a(i) = a(j) => A329350(i) = A329350(j) for all i, j.

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A351235 Lexicographically earliest infinite sequence such that a(i) = a(j) => A046523(i) = A046523(j), A327858(i) = A327858(j) and A345000(i) = A345000(j) for all i, j >= 1.

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PARI

A319693 Filter sequence combining sopfr(d) from all proper divisors d of n, where sopfr(d) is A001414(d) = sum of primes dividing d with repetition.

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PARI

A319706 Filter sequence which for primes p records the prime signature of 2p+1, and for all other numbers assigns a unique number.

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