A077462 -id:A077462 - cp's OEIS frontend

A353423 For even n, a(n) = -Sum_{d|n, dA064989(n)), with a(1) = 1.

1, -1, -1, 0, -1, -1, -1, 0, 0, -1, -1, -2, -1, -1, -1, 0, -1, 0, -1, -2, -1, -1, -1, -8, 0, -1, 0, -2, -1, -5, -1, 0, -1, -1, -1, 0, -1, -1, -1, -8, -1, -5, -1, -2, -2, -1, -1, -96, 0, 0, -1, -2, -1, 0, -1, -8, -1, -1, -1, -70, -1, -1, -2, 0, -1, -5, -1, -2, -1, -5, -1, 0, -1, -1, 0, -2, -1, -5, -1, -96, 0, -1, -1, -70

Offset: 1

Antti Karttunen, Apr 21 2022

sign

Apparently, for all i, j >= 1, A077462(i) = A077462(j) => a(i) = a(j).

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from indices in prime factorization

Cf. A003961, A064989, A077462, A348717.
Cf. A070003 (positions of 0's), A167171 (positions of -1's), A096156 (positions of -2's), A007304 (positions of -5's), A086975 (positions of -70's), all these are so far conjectural. Also a subsequence of A178739 seems to give the positions of -96's.
Cf. also A353454, A353457, A353458, A353467 for similar recurrences.

A000265(n) = (n>>valuation(n,2));
A064989(n) = { my(f=factor(A000265(n))); for(i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
memoA353423 = Map();
A353423(n) = if(1==n,1,my(v); if(mapisdefined(memoA353423,n,&v), v, if(!(n%2), v = -sumdiv(n,d,if(dA353423(n/2)*A353423(d),0)), v = A353423(A064989(n))); mapput(memoA353423,n,v); (v)));

a(p) = -1 for all primes p.

a(n) = a(A003961(n)) = a(A348717(n)), for all n >= 1.

A096153 Natural numbers (greater than 1) arranged in rows according to their ordered prime signature. Square array A(n,k) read by descending antidiagonals.

Original entry on oeis.org

2, 3, 4, 5, 9, 6, 7, 25, 10, 8, 11, 49, 14, 27, 12, 13, 121, 15, 125, 20, 16, 17, 169, 21, 343, 28, 81, 18, 19, 289, 22, 1331, 44, 625, 50, 24, 23, 361, 26, 2197, 45, 2401, 75, 40, 30, 29, 529, 33, 4913, 52, 14641, 98, 56, 42, 32, 31, 841, 34, 6859, 63, 28561, 147, 88, 66

Offset: 1

Alford Arnold, Jul 24 2004

The first row is A000040 (the prime numbers) and the first column is A055932 (the Quet prime signatures).

If we restrict the terms to those having ordered prime signatures that are not represented in A025487 (the least prime signature sequence), we get A096011.

			18 = 2^1 * 3^2, so has ordered prime signature (1,2) given by the exponents in the factorization shown. No earlier number has this prime signature, so 18 is placed at the start of the next empty row (row 7). Thus A(7,1) = 18.

For m >= 2, A077462/A335286 essentially give the row/column containing m.
Cf. A071364, A357126.
See the comments for the relationships with A000040, A025487, A055932, A096011.

Edited by Peter Munn, Oct 23 2023

cp's OEIS Frontend

A353423 For even n, a(n) = -Sum_{d|n, dA064989(n)), with a(1) = 1.

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