A302040 -id:A302040 - cp's OEIS frontend

A302041 An omega analog for a nonstandard factorization based on the sieve of Eratosthenes (A083221).

0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 1, 2, 3, 1, 2, 2, 3, 1, 2, 1, 2, 2, 2, 2, 3, 1, 2, 2, 2, 1, 2, 2, 2, 3, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2

Offset: 1

Antti Karttunen, Mar 31 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences generated by sieves

Cf. A001221, A069010, A250246, A252754, A302044.
Cf. A302040 (positions of terms < 2).
Cf. A253557 (a similar analog for bigomega), A302050, A302051, A302052, A302039, A302055 (other analogs).
Differs from A302031 for the first time at n=59, where a(59) = 1, while A302031(59) = 2.

\\ Assuming A250469 and its inverse A268674 have been precomputed, then the following is reasonably fast:
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)));

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));
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)); };
A302041(n) = if(1==n, 0,1+A302041(A302044(n)));

\\ 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)));
A302041(n) = omega(A250246(n));

a(1) = 0; for n > 1, a(n) = 1 + a(A302044(n)).
a(n) = A001221(A250246(n)).
a(n) = A069010(A252754(n)).

A302044 A028234 analog for factorization process based on the sieve of Eratosthenes (A083221).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 1, 3, 1, 7, 5, 1, 1, 9, 1, 5, 1, 11, 1, 3, 1, 13, 7, 7, 1, 15, 1, 1, 5, 17, 7, 9, 1, 19, 11, 5, 1, 21, 1, 11, 1, 23, 1, 3, 1, 25, 25, 13, 1, 27, 1, 7, 7, 29, 1, 15, 1, 31, 13, 1, 11, 33, 1, 17, 5, 35, 1, 9, 1, 37, 17, 19, 11, 39, 1, 5, 11, 41, 1, 21, 7, 43, 35, 11, 1, 45, 1, 23, 1, 47, 13, 3, 1, 49, 19, 25, 1, 51, 1, 13, 25

Offset: 1

Antti Karttunen, Mar 31 2018

nonn

Iterating n, a(n), a(a(n)), a(a(a(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 A302045 to the same terms gives the corresponding exponents (multiplicities) of those primes. Permutation pair A250245/A250246 maps between this non-standard prime factorization and the ordinary factorization of n. See also comments and examples in A302042.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences generated by sieves

Cf. A000265, A020639, A055396, A078898, A028234, A083221, A250245, A250246, A250469, A302034, A302042, A302045.

Cf. A302040 (positions of 1's).

\\ Assuming A250469 and its inverse A268674 have been precomputed, then the following is fast enough:
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));

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; };
A000265(n) = (n/2^valuation(n, 2));
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];
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)); };

For n > 1, a(n) = A250469^(r)(A000265(A078898(n))), where r = A055396(n)-1 and A250469^(r)(n) stands for applying r times the map x -> A250469(x), starting from x = n.

a(n) = A250245(A028234(A250246(n))).

A302036 Ludic powers: numbers k such that A302031(k) < 2; numbers k such that A260739(k) is a power of 2.

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, 61, 64, 67, 71, 73, 77, 83, 85, 89, 91, 93, 97, 101, 107, 109, 115, 119, 121, 127, 128, 131, 143, 145, 149, 151, 157, 161, 167, 173, 175, 179, 181, 189, 191, 193, 197, 205, 209, 211, 221, 223, 227, 229, 233, 235, 239, 247, 253, 256, 257

Offset: 1

Antti Karttunen, Apr 02 2018

nonn

An analog of A000961 for factorization process based on the Ludic sieve (A255127).

Numbers k for which A302031(k) < 2, or equally, for which A302034(k) = 1, or equally, for which A209229(A260739(k)) = 1.

Antti Karttunen, Table of n, a(n) for n = 1..3600
Index entries for sequences generated by sieves

Cf. A260739, A302031, A302034.
Cf. A000079, A003309, A254100 (subsequences).
Cf. also A000961, A302038, A302040.

for(n=1,257,if(A302031(n)<2,print1(n,","))); \\ See also code in A302031.

A302053 Squares (A000290) analog for nonstandard factorization process based on the sieve of Eratosthenes (A083221).

Original entry on oeis.org

0, 1, 4, 9, 16, 25, 36, 45, 49, 64, 100, 105, 115, 121, 144, 169, 180, 189, 196, 203, 256, 265, 289, 297, 341, 361, 400, 420, 429, 460, 469, 475, 481, 484, 529, 537, 576, 585, 676, 697, 720, 745, 756, 765, 784, 803, 812, 817, 833, 841, 961, 1024, 1027, 1060, 1075, 1081, 1156, 1188, 1197, 1257, 1309, 1345, 1364, 1369, 1377, 1411, 1444

Offset: 0

Antti Karttunen, Mar 31 2018

nonn

Indexing starts with zero, with a(0) = 0, to match with the indexing of A000290.
After initial zero, gives the positions of odd terms in A302051.
After initial zero, contains values obtained with A250245(n^2) sorted into ascending order, or in other words, numbers n such that A250246(n) is a square (in A000290).
Numbers n such that for all terms in iteration sequence n, A302044(n), A302044(A302044(n)), A302044(A302044(A302044(n))), ..., applying A302045(n) gives an even number before the sequence settles to 1.

Antti Karttunen, Table of n, a(n) for n = 0..3283
Index entries for sequences generated by sieves

Cf. A000290, A250245, A302051, A302040, A302044, A302045, A302052 (characteristic function).

Cf. A000302, A001248 (subsequences).

for(n=0,4096,if(1==A302052(n),print1(n,",")));

cp's OEIS Frontend

A302041 An omega analog for a nonstandard factorization based on the sieve of Eratosthenes (A083221).

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A302044 A028234 analog for factorization process based on the sieve of Eratosthenes (A083221).

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A302036 Ludic powers: numbers k such that A302031(k) < 2; numbers k such that A260739(k) is a power of 2.

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A302053 Squares (A000290) analog for nonstandard factorization process based on the sieve of Eratosthenes (A083221).

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Author

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Programs

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