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 A297113 a(1) = 0, a(2) = 1, after which, a(n) = a(n/2) if n is of the form 4k+2, and otherwise a(n) = 1+a(A252463(n)). 26
 0, 1, 2, 2, 3, 2, 4, 3, 3, 3, 5, 3, 6, 4, 3, 4, 7, 3, 8, 4, 4, 5, 9, 4, 4, 6, 4, 5, 10, 3, 11, 5, 5, 7, 4, 4, 12, 8, 6, 5, 13, 4, 14, 6, 4, 9, 15, 5, 5, 4, 7, 7, 16, 4, 5, 6, 8, 10, 17, 4, 18, 11, 5, 6, 6, 5, 19, 8, 9, 4, 20, 5, 21, 12, 4, 9, 5, 6, 22, 6, 5, 13, 23, 5, 7, 14, 10, 7, 24, 4, 6, 10, 11, 15, 8, 6, 25 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS From Gus Wiseman, Apr 06 2019: (Start) Also the number of squares in the Young diagram of the integer partition with Heinz number n that are graph-distance 1 from the lower-right boundary, where the Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). For example, the partition (6,5,5,3) with Heinz number 7865 has diagram   o o o o o o   o o o o o   o o o o o   o o o with inner rim             o           o         o o   o o o of size 7, so a(7867) = 7. (End) LINKS Antti Karttunen, Table of n, a(n) for n = 1..12721 FORMULA a(1) = 0, a(2) = 1, after which, a(n) = a(n/2) if n is of the form 4k+2, and otherwise a(n) = 1+a(A252463(n)) . For n > 1, a(n) = A001511(A297112(n)), where A297112(n) = Sum_{d|n} moebius(n/d)*A156552(d). a(n) = A252464(n) - A297155(n). For n > 1, a(n) = 1+A033265(A156552(n)) = 1+A297167(n) = A046660(n) + A061395(n). - Last two sums added by Antti Karttunen, Sep 02 2018 Other identities. For all n >= 1: a(A000040(n)) = n. [Each n occurs for the first time at the n-th prime.] MATHEMATICA Table[If[n==1, 0, PrimePi[FactorInteger[n][[-1, 1]]]+PrimeOmega[n]-PrimeNu[n]], {n, 100}] (* Gus Wiseman, Apr 06 2019 *) PROG (PARI) A064989(n) = {my(f); f = factor(n); if((n>1 && f[1, 1]==2), f[1, 2] = 0); for (i=1, #f~, f[i, 1] = precprime(f[i, 1]-1)); factorback(f)}; A297113(n) = if(n<=2, n-1, if(n%2, 1+A297113(A064989(n)), !(n%4)+A297113(n/2))); \\ More complex way, after Moebius transform: A156552(n) = if(1==n, 0, if(!(n%2), 1+(2*A156552(n/2)), 2*A156552(A064989(n)))); A297112(n) = sumdiv(n, d, moebius(n/d)*A156552(d)); A297113(n) = if(1==n, 0, 1+valuation(A297112(n), 2)); (Scheme, with memoization-macro definec) (definec (A297113 n) (cond ((<= n 2) (- n 1)) ((= 2 (modulo n 4)) (A297113 (/ n 2))) (else (+ 1 (A297113 (A252463 n)))))) CROSSREFS One more than A297167 (after the initial term). Cf. A001511, A008683, A033265, A064989, A156552, A252463, A252464, A297112, A297155. Cf. also A297157, A297161, A297162. Cf. A052126, A065770, A112798, A115994, A174090, A325166, A325167, A325169. Sequence in context: A155043 A337327 A065770 * A086375 A107324 A023522 Adjacent sequences:  A297110 A297111 A297112 * A297114 A297115 A297116 KEYWORD nonn AUTHOR Antti Karttunen, Dec 26 2017 STATUS approved

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Last modified September 17 14:07 EDT 2021. Contains 347478 sequences. (Running on oeis4.)