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A324545
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An analog of sigma (A000203) for nonstandard factorization based on the sieve of Eratosthenes (A083221).
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6
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1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 14, 24, 24, 31, 18, 39, 20, 42, 40, 36, 24, 60, 31, 42, 32, 56, 30, 72, 32, 63, 78, 54, 48, 91, 38, 60, 48, 90, 42, 120, 44, 84, 121, 72, 48, 124, 57, 93, 124, 98, 54, 96, 156, 120, 104, 90, 60, 168, 62, 96, 56, 127, 72, 234, 68, 126, 240, 144, 72, 195, 74, 114, 72, 140, 96, 144, 80
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OFFSET
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1,2
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LINKS
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FORMULA
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a(1) = 1; for n > 1, let p =A020639(n) [the smallest prime factor of n], then a(n) = (((p^(1+A302045(n)))-1) / (p-1)) * a(A302044(n)).
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PROG
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(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); \\ FromA020639
v078898 = ordinal_transform(vector(up_to, n,A020639(n)));
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ FromA003961
(PARI)
\\ Or alternatively, using alsoA078898defined above:
A000265(n) = (n/2^valuation(n, 2));
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CROSSREFS
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Cf.A000203,A020639,A078898,A250246,A252754,A302044,A302045,A324054,A324535,A324544,A324546.
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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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