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A359778
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Number of factorizations of n into factors not divisible by p^p for any prime p (terms of A048103).
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2
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1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 2, 1, 1, 4, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 5, 1, 1, 2, 2, 2, 5, 1, 2, 2, 2, 1, 5, 1, 2, 4, 2, 1, 2, 2, 4, 2, 2, 1, 5, 2, 2, 2, 2, 1, 6, 1, 2, 4, 1, 2, 5, 1, 2, 2, 5, 1, 5, 1, 2, 4, 2, 2, 5, 1, 2, 3, 2, 1, 6, 2, 2, 2, 2, 1, 11, 2, 2, 2, 2, 2, 2, 1, 4, 4, 5, 1, 5, 1, 2, 5, 2, 1, 7
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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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EXAMPLE
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108 has in total 16 = A001055(108) factorizations:
Factors Are there any factors that are divisible by p^p,
where p is any prime?
-------------------------------------------------------------------
[3, 3, 3, 2, 2] No
[4, 3, 3, 3] Yes (4, divisible by 2^2)
[6, 3, 3, 2] No
[6, 6, 3] No
[9, 3, 2, 2] No
[9, 4, 3] Yes (4)
[9, 6, 2] No
[12, 3, 3] Yes (12, divisible by 2^2)
[12, 9] Yes (12)
[18, 3, 2] No
[18, 6] No
[27, 2, 2] Yes (27, divisible by 3^3)
[27, 4] Yes (both 27 and 4)
[36, 3] Yes (36)
[54, 2] Yes (54, divisible by 3^3)
[108] Yes (108 = 2^2 * 3^3)
Thus only seven of the factorizations satisfy the criterion, and a(108) = 7.
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PROG
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(PARI)
A359550(n) = { my(f = factor(n)); prod(k=1, #f~, (f[k, 1]>f[k, 2])); };
A359778(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1) && (d<=m) && A359550(d), s += A359778(n/d, d))); (s));
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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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