Search: a302040 -id:a302040
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A302041
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An omega analog for a nonstandard factorization based on the sieve of Eratosthenes (A083221).
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+10
14
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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
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OFFSET
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1,6
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LINKS
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FORMULA
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a(1) = 0; for n > 1, a(n) = 1 + a(A302044(n)).
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PROG
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(PARI)
\\ 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));
(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)));
A000265(n) = (n/2^valuation(n, 2));
(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
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CROSSREFS
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Cf. A302040 (positions of terms < 2).
Differs from A302031 for the first time at n=59, where a(59) = 1, while A302031(59) = 2.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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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
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OFFSET
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1,6
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COMMENTS
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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.
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LINKS
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FORMULA
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PROG
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(PARI)
\\ 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));
(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; };
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)));
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CROSSREFS
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Cf. A000265, A020639, A055396, A078898, A028234, A083221, A250245, A250246, A250469, A302034, A302042, A302045.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A302036
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Ludic powers: numbers k such that A302031(k) < 2; numbers k such that A260739(k) is a power of 2.
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+10
9
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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
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OFFSET
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1,2
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COMMENTS
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An analog of A000961 for factorization process based on the Ludic sieve (A255127).
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LINKS
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PROG
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(PARI) for(n=1, 257, if(A302031(n)<2, print1(n, ", "))); \\ See also code in A302031.
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A302053
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Squares (A000290) analog for nonstandard factorization process based on the sieve of Eratosthenes (A083221).
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+10
4
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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
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OFFSET
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0,3
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COMMENTS
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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).
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LINKS
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PROG
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(PARI) for(n=0, 4096, if(1==A302052(n), print1(n, ", ")));
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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