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 A324109 Numbers n such that A324108(n) = A324054(n-1). 8
 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 34, 36, 37, 38, 40, 41, 43, 44, 46, 47, 48, 49, 50, 52, 53, 54, 56, 58, 59, 61, 62, 64, 67, 68, 71, 72, 73, 74, 76, 79, 80, 81, 82, 83, 86, 87, 88, 89, 92, 94, 96, 97, 98, 100, 101, 103, 104, 106, 107, 108, 109, 112, 113, 116, 118, 121 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Numbers n such that A324054(n-1) is equal to A324108(n), which is a multiplicative function with A324108(p^e) = A324054((p^e)-1). Prime powers (A000961) is a subsequence by definition. Also A070776 is a subsequence. This follows because for every n of the form 2^i * p^j (where p is an odd prime, and i >= 0, j >= 0), we have A324108(2^i * p^j) = A324054(2^i - 1)*A324054(p^j - 1) = sigma(A005940(2^i)) * sigma(A005940(p^j)). Because A005940(1) = 1, and A005940(2n) = 2*A005940(n), the powers of two are among the fixed points of A005940 (cf. A029747), thus the left half of product is sigma(2^i), while on the other hand, we know that A005940(p^j) is odd (because A005940 also preserves parity), and thus the whole product is equal to sigma(2^i * A005940(p^j)) = sigma(A005940(2^i * p^j)) = A324054((2^i * p^j)-1). See subsequence A324111 for less regular solutions. LINKS Antti Karttunen, Table of n, a(n) for n = 1..10001 PROG (PARI) A324054(n) = { my(p=2, mp=p*p, m=1); while(n, if(!(n%2), p=nextprime(1+p); mp = p*p, if(3==(n%4), mp *= p, m *= (mp-1)/(p-1))); n>>=1); (m); }; A324108(n) = { my(f=factor(n)); prod(i=1, #f~, A324054((f[i, 1]^f[i, 2])-1)); }; isA324109(n) = (A324054(n-1)==A324108(n)); for(n=1, 121, if(isA324109(n), print1(n, ", "))); CROSSREFS Union of A070776 and A324111. Cf. A000961 (a subsequence), A029747, A324054, A324107, A324108, A324110 (complement). Sequence in context: A004829 A075592 A285801 * A070776 A320230 A076564 Adjacent sequences:  A324106 A324107 A324108 * A324110 A324111 A324112 KEYWORD nonn AUTHOR Antti Karttunen and David A. Corneth, Feb 15 2019 STATUS approved

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Last modified August 4 20:53 EDT 2021. Contains 346455 sequences. (Running on oeis4.)