cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A302051 An analog of A000005 for nonstandard factorization based on the sieve of Eratosthenes (A083221).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, 2, 4, 4, 5, 2, 6, 2, 6, 4, 4, 2, 8, 3, 4, 4, 6, 2, 8, 2, 6, 6, 4, 4, 9, 2, 4, 4, 8, 2, 8, 2, 6, 5, 4, 2, 10, 3, 6, 6, 6, 2, 8, 4, 8, 6, 4, 2, 12, 2, 4, 4, 7, 4, 12, 2, 6, 8, 8, 2, 12, 2, 4, 4, 6, 4, 8, 2, 10, 6, 4, 2, 12, 6, 4, 8, 8, 2, 10, 4, 6, 6, 4, 4, 12, 2, 6, 4, 9, 2, 12, 2, 8, 9
Offset: 1

Views

Author

Antti Karttunen, Apr 01 2018

Keywords

Comments

See A302042, A302044 and A302045 for a description of the factorization process.

Links

Crossrefs

Cf. A000005, A083221, A302042, A302044, A302045, A302052 (reduced modulo 2), A302053 (gives the positions of odd numbers).
Cf. also A253557, A302041, A302050, A302052, A302039, A302055 for other similar analogs.

Programs

  • PARI
    up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
A020639(n) = if(n>1, if(n>n=factor(n, 0)[1, 1], n, factor(n)[1, 1]), 1); \\ From A020639
v078898 = ordinal_transform(vector(up_to,n,A020639(n)));
A078898(n) = v078898[n];
A000265(n) = (n/2^valuation(n, 2));
A001511(n) = 1+valuation(n,2);
A302045(n) = A001511(A078898(n));
A302044(n) = { my(c = A000265(A078898(n))); if(1==c,1,my(p = prime(-1+primepi(A020639(n))+primepi(A020639(c))), d = A078898(c), k=0); while(d, k++; if((1==k)||(A020639(k)>=p),d -= 1)); (k*p)); };
A302051(n) = if(1==n,n,(A302045(n)+1)*A302051(A302044(n)));
  • PARI
    \\ Or, using also some of the code from above:
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A055396(n) = if(1==n,0,primepi(A020639(n)));
A250246(n) = if(1==n,n,my(k = 2*A250246(A078898(n)), r = A055396(n)); if(1==r, k, while(r>1, k = A003961(k); r--); (k)));
A302051(n) = numdiv(A250246(n));

Formula

a(1) = 1, for n > 1, a(n) = (A302045(n)+1) * a(A302044(n)).
a(n) = A000005(A250246(n)).
a(n) = A106737(A252754(n)).

A302032 Discard the least ludic factor of n: a(n) = A255127(A260738(c) + r - 1, A260739(c)), where r = A260738(n), c = A260739(n) are the row and the column index of n in the table A255127; a(n) = 1 if c = 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 4, 3, 5, 1, 6, 1, 7, 5, 8, 1, 9, 5, 10, 9, 11, 1, 12, 1, 13, 7, 14, 1, 15, 7, 16, 15, 17, 7, 18, 1, 19, 11, 20, 1, 21, 1, 22, 21, 23, 1, 24, 19, 25, 19, 26, 1, 27, 11, 28, 27, 29, 11, 30, 1, 31, 13, 32, 11, 33, 1, 34, 33, 35, 1, 36, 13, 37, 17, 38, 1, 39, 35, 40, 39, 41, 1, 42, 31, 43, 35, 44, 1, 45, 1, 46, 45, 47, 13, 48, 1, 49, 23, 50
Offset: 1

Views

Author

Antti Karttunen, Mar 31 2018

Keywords

Comments

Original definition: A032742 analog for a nonstandard factorization process based on the Ludic sieve (A255127); Discard a single instance of the Ludic factor A272565(n) from n.
Like [A020639(n), A032742(n)] or [A020639(n), A302042(n)], also ordered pair [A272565(n), a(n)] is unique for each n. Iterating n, a(n), a(a(n)), a(a(a(n))), ..., until 1 is reached, and taking the Ludic factor (A272565) of each term gives a multiset of Ludic numbers (A003309) in ascending order, unique for each natural number n >= 1. Permutation pair A302025/A302026 maps between this "Ludic factorization" and the ordinary prime factorization of n. See also comments in A302034.
The definition of "discard the least ludic factor" is based on the table A255127 of the ludic sieve, where row r lists the (r+1)-th ludic number k = A003309(r+1), determined at the r-th step of the sieve, followed by the numbers crossed out at this step, namely, every k-th of the numbers remaining so far after k. If the number n is in row r = A260738(n), column c = A260739(n) of that table, then its least ludic factor is A272565(n) = A003309(r+1), the 1st entry of the r-th row. To discard that factor means to consider the number which is r-1 rows below the number c in that table, whence a(n) = A255127(A260738(c)+r-1, A260739(c)) - unless n is a ludic number, in which case a(n) = 1. - M. F. Hasler, Nov 06 2024

Examples

			Frem _M. F. Hasler_, Nov 06 2024: (Start)
For ludic numbers 1, 2, 3, 5, 7, 11, 13, 17, 23, 25, 29, 37, ..., a(n) = 1.
For n = 4, an even number, we have r = A260738(4) = 1: It is listed in row 1 of the table A255127, which lists all numbers that were crossed out at the first step: namely, the ludic number k = 2 and every other larger number. Also, in this row 1, the number 4 is in column c = A260739(4) = 2. Therefore, we apply r-1 = 0 times the map A269379 to c = 2, whence a(4) = 2.
The number n = 6 is also even and therefore listed in row r = 1, now in column c = 3, whence a(6) = 3. Similarly, a(8) = 4 and a(2k) = k for all k >= 1.
The number n = 9 was crossed out at the 2nd step (so r = A260738(9) = 2), when k = 3 was added to the ludic numbers and every 3rd remaining number crossed out; 9 was the first of these (after k = 3) so it is in column c = A260739(9) = 2. Now we have to apply r-1 = 1 times the map A269379 to c. That map yields the number which is located just below the argument (here c = 2) in the table A255127. Since 2 is a ludic number, in the first column, we get the next larger ludic number, 3, whence a(9) = 3.
The number 15 was the (c = 3)rd number to be crossed out at the (r = 2)nd step. Hence a(15) = A269379^{r-1} (c) = A269379(3) = 5 (again, the next larger ludic number).
The number 19 was the (c = 2)nd number to be crossed out at the (r = 3)rd step (when k = 5, its least ludic factor, was added to the list of ludic numbers). Hence a(19) = A269379^2(2) = A269379(3) = 5 again (skipping twice to the next larger ludic number).
(End)
To illustrate how this sequence allows one to compute the complete "ludic factorization" of a number, we consider n = 100.
For n = 100, its Ludic factor A272565(100) is 2, and we have seen that a(n) = 100/2 = 50.
For n = 50, its Ludic factor A272565(50) is 2 again, and again a(50) = 50/2 = 25.
Since n = 25 = A003309(1+9) is a ludic number, it equals its Ludic factor A272565(25) = 25. Because it appeared at the A260738(25) = 9th step, we apply A269379 eight times to the column index A260739(25) = 1, a fixed point, so a(25) = A269379^8(1) = 1.
Collecting the Ludic factors given by A272565 we get the multiset of factors: [2, 2, 25] = [A003309(1+1), A003309(1+1), A003309(1+9)]. By definition, A302026(100) = prime(1)*prime(1)*prime(9) = 2*2*23 = 92, the product of the corresponding primes.
If we start from n = 100, iterating the map n -> A302034(n) [instead of A302032] and apply A272565 to each term obtained we get just a single instance of each Ludic factor: [2, 25]. Then by applying A302035 to the same terms we get the corresponding exponents (multiplicities) of those factors: [2, 1].

Links

Crossrefs

Cf. A003309, A255127, A269379, A269380, A272565, A260738, A260739, A269379, A269380, A302025, A302026, A302034, A302035.
Cf. the following analogs A302031 (omega), A302037 (bigomega).
Cf. also A032742, A302042.

Programs

  • PARI
    \\ Assuming A269379 and its inverse A269380 have been precomputed, then the following is reasonably fast:
A302032(n) = if(1==n,n,my(k=0); while((n%2), n = A269380(n); k++); n = n/2; while(k>0, n = A269379(n); k--); (n))

Formula

For n > 1, a(n) = A269379^r'(A260739(n)), where r' = A260738(n)-1 and A269379^r'(n) stands for applying r' times the map x -> A269379(x), starting from x = n.
a(n) = A302025(A032742(A302026(n))).
From M. F. Hasler, Nov 06 2024: (Start)
a(n) = 1 if n is a ludic number, i.e., in A003309. Otherwise:
a(n) = A255127(A260738(c) + r - 1, A260739(c)), with r = A260738(n), c = A260739(n).
In particular, a(2n) = n for all n. (End)

A302045 a(1) = 0, for n > 1, a(n) = A001511(A078898(n)); Number of instances of the smallest prime factor A020639(n) in nonstandard factorization of n that is based on the sieve of Eratosthenes (A083221).

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 3, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 5, 2, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 4, 1, 1, 4, 2, 1, 1, 2, 1, 1, 3, 3, 2, 1, 1, 2, 1, 1, 1, 6, 1, 1, 1, 2, 3, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 4, 2, 1, 1, 2, 2, 1, 1, 3, 1, 1, 3, 2, 5, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 3, 2
Offset: 1

Views

Author

Antti Karttunen, Mar 31 2018

Keywords

Comments

Iterating the map n -> A302044(n) until 1 is reached, and taking the smallest prime factor (A020639) of each term gives a sequence of distinct primes in ascending order, while applying this function (A302045) to those terms gives the corresponding "exponents" of those primes, that is, the count of consecutive occurrences of each prime when iterating the map n -> A302042(n), which gives the same primes with repetitions. Permutation pair A250245/A250246 maps between this non-standard prime factorization of n and the ordinary factorization of n. See also comments and examples in A302042.

Links

Crossrefs

Cf. A001511, A067029, A078898, A250245, A250246, A268674, A302042, A302044.
Cf. also A302035, A302050, A302051.

Programs

Formula

a(1) = 0, for n > 1, a(n) = A001511(A078898(n)).
For n > 1, a(n) = A250245(A067029(A250246(n))).

A302055 An arithmetic derivative analog for nonstandard factorization process based on the sieve of Eratosthenes (A083221).

Original entry on oeis.org

0, 0, 1, 1, 4, 1, 5, 1, 12, 6, 7, 1, 16, 1, 9, 8, 32, 1, 21, 1, 24, 27, 13, 1, 44, 10, 15, 10, 32, 1, 31, 1, 80, 39, 19, 12, 60, 1, 21, 14, 68, 1, 75, 1, 48, 102, 25, 1, 112, 14, 45, 55, 56, 1, 47, 75, 92, 57, 31, 1, 92, 1, 33, 16, 192, 16, 111, 1, 72, 150, 59, 1, 156, 1, 39, 20, 80, 18, 67, 1, 176, 81, 43, 1, 192, 95, 45, 71, 140, 1, 249, 147, 96
Offset: 0

Views

Author

Antti Karttunen, Mar 31 2018

Keywords

Comments

The formula is analogous to Reinhard Zumkeller's May 09 2011 formula in A003415, with A032742 replaced by A302042. See the comments in the latter sequence.
Note that this cannot be computed just as f(n) = A003415(A250246(n)), in contrast to many other such analogs, like A253557, A302039, A302041, A302050, A302051 and A302052.

Links

Crossrefs

Cf. A003415, A020639, A078898, A083221, A302042.

Programs

  • PARI
    up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
A020639(n) = if(n>1, if(n>n=factor(n, 0)[1, 1], n, factor(n)[1, 1]), 1); \\ From A020639.
v078898 = ordinal_transform(vector(up_to,n,A020639(n)));
A078898(n) = v078898[n];
A302042(n) = if((1==n)||isprime(n),1,my(c = A078898(n), p = prime(-1+primepi(A020639(n))+primepi(A020639(c))), d = A078898(c), k=0); while(d, k++; if((1==k)||(A020639(k)>=p),d -= 1)); (k*p));
A302055(n) = if(n<2,0,my(k=A302042(n)); (A020639(n)*A302055(k))+k);

Formula

a(0) = a(1) = 0; for n > 1, a(n) = (A020639(n)*a(A302042(n))) + A302042(n).

A302039 Analog of A056239 for nonstandard factorization based on the sieve of Eratosthenes (A083221).

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 4, 3, 4, 4, 5, 4, 6, 5, 5, 4, 7, 5, 8, 5, 6, 6, 9, 5, 6, 7, 6, 6, 10, 6, 11, 5, 7, 8, 7, 6, 12, 9, 7, 6, 13, 7, 14, 7, 8, 10, 15, 6, 8, 7, 8, 8, 16, 7, 9, 7, 8, 11, 17, 7, 18, 12, 8, 6, 8, 8, 19, 9, 9, 8, 20, 7, 21, 13, 9, 10, 9, 8, 22, 7, 9, 14, 23, 8, 10, 15, 9, 8, 24, 9, 12, 11, 10, 16, 9, 7, 25, 9, 10, 8, 26, 9, 27, 9, 10
Offset: 1

Views

Author

Antti Karttunen, Mar 31 2018

Keywords

Comments

Each n occurs A000041(n) times in total.

Links

Crossrefs

Cf. A000041, A056239, A250246, A302042, A302044, A302045.
Cf. also A253557, A302041, A302050, A302051, A302052, A302055 for other similar analogs.

Programs

  • PARI
    up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
A020639(n) = if(n>1, if(n>n=factor(n, 0)[1, 1], n, factor(n)[1, 1]), 1); \\ From A020639, by Hasler.
A055396(n) = if(1==n,0,primepi(A020639(n)));
v078898 = ordinal_transform(vector(up_to,n,A020639(n)));
A078898(n) = v078898[n];
A302042(n) = if((1==n)||isprime(n),1,my(c = A078898(n), p = prime(-1+primepi(A020639(n))+primepi(A020639(c))), d = A078898(c), k=0); while(d, k++; if((1==k)||(A020639(k)>=p),d -= 1)); (k*p));
A302039(n) = if(1==n,0,A055396(n) + A302039(A302042(n)));

Formula

a(1) = 0; for n > 1, a(n) = A055396(n) + a(A302042(n)).
a(1) = 0; for n > 1, a(n) = (A055396(n)*A302045(n)) + a(A302044(n)).
a(n) = A056239(A250246(n)).

A302040 Numbers k such that A078898(k) is a power of 2; an analog for A000961 based on factorization-kind of process involving the sieve of Eratosthenes (A083221).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 21, 23, 25, 29, 31, 32, 37, 41, 43, 45, 47, 49, 53, 55, 59, 61, 64, 67, 71, 73, 79, 83, 89, 91, 93, 97, 101, 103, 107, 109, 113, 115, 121, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 187, 189, 191, 193, 197, 199, 203, 211, 223, 227, 229, 233, 235, 239, 241, 247, 251, 256, 257
Offset: 1

Views

Author

Antti Karttunen, Apr 02 2018

Keywords

Comments

Numbers k for which A302041(k) < 2, or equally, for which A302044(k) = 1.
Sequence A250245(A000961(k)) sorted into ascending order, or in other words, numbers k such that A250246(k) is a prime power (in A000961).
Numbers k such that all terms in iteration sequence k, A302042(k), A302042(A302042(k)), A302042(A302042(A302042(k))), ..., have an equal smallest prime factor (A020639) before the sequence settles to 1, in other words, that they all stay on the same row of A083221. This also forces the column position of each (A078898) to be a power of 2 (A000079).

Examples

			For k = 21 = 3*7, the smallest prime factor is 3. A302042(21) = 9, and A302042(9) = 3, both (9 and 3) which also have 3 as their smallest prime factor, and after that the sequence settles to 1, as A302042(3) = 1, thus 21 is included in this sequence.
For k = 27 = 3*3*3, the smallest prime factor is 3. However, A302042(27) = 7, thus 27 is not included in this sequence.

Links

Crossrefs

Cf. A000961, A078898, A302036, A302041, A302044, A302053.
Cf. A000040, A000079, A001248 (subsequences).

Programs

  • PARI
    for(n=1,257,if(2>A302041(n),print1(n,","))); \\ Other code as in A302041.

A326075 Difference between the number of prime divisors in a nonstandard factorization process based on the sieve of Eratosthenes vs. their number in the ordinary factorization of n (when counted with multiplicity): a(n) = A253557(n) - A001222(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, -1, 1, 0, 1, 0, 0, 0, 0, 0, -1, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, -1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 3, 0, 0, 0, 0, 0, -1, 0, 0, 1, 0, 0, 1
Offset: 1

Views

Author

Antti Karttunen, Aug 23 2019

Keywords

Examples

			A001222(21) = 2 because A032742(21) = 7, and A032742(7) = 1, while A253557(21) = 3 because A302042(21) = 9, A302042(9) = 3, and A302042(3) = 1. Thus a(21) = 3-2 = 1.
A001222(27) = 3 because A032742(27) = 9, A032742(9) = 3 and A032742(3) = 1, while A253557(27) = 2 because A302042(27) = 7 and A302042(7) = 1. Thus a(27) = 2-3 = -1.

Links

Crossrefs

Cf. A001222, A032742, A250246, A253557, A302042, A323888, A326139, A326189, A326190, A326191.

Programs

  • PARI
    up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
A020639(n) = if(n>1, if(n>n=factor(n, 0)[1, 1], n, factor(n)[1, 1]), 1); \\ From A020639
v078898 = ordinal_transform(vector(up_to,n,A020639(n)));
A078898(n) = v078898[n];
A302042(n) = if((1==n)||isprime(n),1,my(c = A078898(n), p = prime(-1+primepi(A020639(n))+primepi(A020639(c))), d = A078898(c), k=0); while(d, k++; if((1==k)||(A020639(k)>=p),d -= 1)); (k*p));
A253557(n) = if(1==n, 0,1+A253557(A302042(n)));
A326075(n) = (A253557(n)-bigomega(n));

Formula

a(n) = A253557(n) - A001222(n) = A001222(A250246(n)) - A001222(n).
a(p) = 0 for all primes p.

A302046 A filter sequence analogous to A101296 for nonstandard factorization based on the sieve of Eratosthenes (A083221).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 3, 4, 2, 6, 2, 4, 4, 7, 2, 6, 2, 6, 5, 4, 2, 8, 3, 4, 4, 6, 2, 9, 2, 10, 6, 4, 4, 11, 2, 4, 4, 8, 2, 8, 2, 6, 7, 4, 2, 12, 3, 6, 6, 6, 2, 9, 5, 8, 6, 4, 2, 13, 2, 4, 4, 14, 4, 13, 2, 6, 8, 9, 2, 15, 2, 4, 4, 6, 4, 9, 2, 12, 6, 4, 2, 15, 6, 4, 9, 8, 2, 12, 5, 6, 10, 4, 4, 16, 2, 6, 4, 11, 2, 13, 2, 8, 11
Offset: 1

Views

Author

Antti Karttunen, Mar 31 2018

Keywords

Comments

Restricted growth sequence transform of A278524.
See A302042 for the description of the nonstandard factorization employed here.
For all i, j:
a(i) = a(j) => A253557(i) = A253557(j).
a(i) = a(j) => A302041(i) = A302041(j).
a(i) = a(j) => A302050(i) = A302050(j).
a(i) = a(j) => A302051(i) = A302051(j) => A302052(i) = A302052(j).

Links

Crossrefs

Cf. A101296, A250246, A278524, A302042, A302044, A302045.

Programs

  • PARI
    up_to = 32769;
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])); }
A020639(n) = { if(1==n,n,vecmin(factor(n)[, 1])); };
A078898(n) = { if(n<=1,n, my(spf=A020639(n),k=1,m=n/spf); while(m>1,if(A020639(m)>=spf,k++); m--); (k)); };
A001511(n) = 1+valuation(n,2);
A302045(n) = A001511(A078898(n));
A302044(n) = if(1==n,n,my(k=0); while((n%2), n = A268674(n); k++); n = (n/2^valuation(n, 2)); while(k>0, n = A250469(n); k--); (n));
A302041(n) = if(1==n, 0,1+A302041(A302044(n)));
Aux302046(n) = if(1==n,n, my(k=A302041(n), v = vector(k),i=1); while(n>1,v[i] = A302045(n); n = A302044(n); i++); vecsort(v));
write_to_bfile(1,rgs_transform(vector(up_to,n,Aux302046(n))),"b302046.txt");

A322870 Ordinal transform of A302043.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 3, 1, 1, 2, 1, 1, 1, 1, 3, 2, 1, 1, 3, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 2, 3, 1, 1, 3, 3, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 2, 4, 3, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 2, 1, 1, 2, 3, 1, 1, 4, 2, 1, 2, 1, 1, 1, 5, 2, 1, 1, 2, 2, 1, 1, 1, 3, 1, 1, 1, 3, 2
Offset: 1

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Author

Antti Karttunen, Dec 29 2018

Keywords

Links

Crossrefs

Cf. A078898, A302042, A302043, A322871.

Programs

  • PARI
    up_to = 1024;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
A020639(n) = if(n>1, if(n>n=factor(n, 0)[1, 1], n, factor(n)[1, 1]), 1); \\ From A020639
v078898 = ordinal_transform(vector(up_to,n,A020639(n)));
A078898(n) = v078898[n];
A302042(n) = if((1==n)||isprime(n),1,my(c = A078898(n), p = prime(-1+primepi(A020639(n))+primepi(A020639(c))), d = A078898(c), k=0); while(d, k++; if((1==k)||(A020639(k)>=p),d -= 1)); (k*p));
A302043(n) = (n - A302042(n));
v322870 = ordinal_transform(vector(up_to,n,A302043(n)));
A322870(n) = v322870[n];
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