A354186 -id:A354186 - cp's OEIS frontend

A348717 a(n) is the least k such that A003961^i(k) = n for some i >= 0 (where A003961^i denotes the i-th iterate of A003961).

1, 2, 2, 4, 2, 6, 2, 8, 4, 10, 2, 12, 2, 14, 6, 16, 2, 18, 2, 20, 10, 22, 2, 24, 4, 26, 8, 28, 2, 30, 2, 32, 14, 34, 6, 36, 2, 38, 22, 40, 2, 42, 2, 44, 12, 46, 2, 48, 4, 50, 26, 52, 2, 54, 10, 56, 34, 58, 2, 60, 2, 62, 20, 64, 14, 66, 2, 68, 38, 70, 2, 72, 2

Offset: 1

Rémy Sigrist, Oct 31 2021

nonn

All terms except a(1) = 1 are even.
To compute a(n) for n > 1:
- if n = Product_{j = 1..o} prime(p_j)^e_j (where prime(i) denotes the i-th prime number, p_1 < ... < p_o and e_1 > 0)
- then a(n) = Product_{j = 1..o} prime(p_j + 1 - p_1)^e_j.
This sequence has similarities with A304776: here we shift down prime indexes, there prime exponents.
Smallest number generated by uniformly decrementing the indices of the prime factors of n. Thus, for n > 1, the smallest m > 1 such that the first differences of the indices of the ordered prime factors (including repetitions) are the same for m and n. As a function, a(.) preserves properties such as prime signature. - Peter Munn, May 12 2022

Positions of particular values (see formula section): A000040, A001248, A006094, A030078, A030514, A046301, A050997, A090076, A090090, A166329, A251720.
Also see formula section for the relationship to: A000265, A003961, A004277, A005940, A020639, A046523, A055396, A071364, A122111, A156552, A243055, A243074, A297845, A322993.
Sequences with comparable definitions: A304776, A316437.
Cf. A246277 (terms halved), A305897 (restricted growth sequence transform), A354185 (Möbius transform), A354186 (Dirichlet inverse), A354187 (sum with it).

a[1] = 1; a[n_] := Module[{f = FactorInteger[n], d}, d = PrimePi[f[[1, 1]]] - 1; Times @@ ((Prime[PrimePi[#[[1]]] - d]^#[[2]]) & /@ f)]; Array[a, 100] (* Amiram Eldar, Oct 31 2021 *)

a(n) = { my (f=factor(n)); if (#f~>0, my (pi1=primepi(f[1,1])); for (k=1, #f~, f[k,1] = prime(primepi(f[k,1])-pi1+1))); factorback(f) }

a(n) = n iff n belongs to A004277.
A003961^(A055396(n)-1)(a(n)) = n for any n > 1.
a(n) = 2 iff n belongs to A000040 (prime numbers).
a(n) = 4 iff n belongs to A001248 (squares of prime numbers).
a(n) = 6 iff n belongs to A006094 (products of two successive prime numbers).
a(n) = 8 iff n belongs to A030078 (cubes of prime numbers).
a(n) = 10 iff n belongs to A090076.
a(n) = 12 iff n belongs to A251720.
a(n) = 14 iff n belongs to A090090.
a(n) = 16 iff n belongs to A030514.
a(n) = 30 iff n belongs to A046301.
a(n) = 32 iff n belongs to A050997.
a(n) = 36 iff n belongs to A166329.
a(1) = 1, for n > 1, a(n) = 2*A246277(n). - Antti Karttunen, Feb 23 2022
a(n) = A122111(A243074(A122111(n))). - Peter Munn, Feb 23 2022
From Peter Munn and Antti Karttunen, May 12 2022: (Start)
a(1) = 1; a(2n) = 2n; a(A003961(n)) = a(n). [complete definition]
a(n) = A005940(1+A322993(n)) = A005940(1+A000265(A156552(n))).
Equivalently, A156552(a(n)) = A000265(A156552(n)).
A297845(a(n), A020639(n)) = n.
A046523(a(n)) = A046523(n).
A071364(a(n)) = A071364(n).
a(n) >= A071364(n).
A243055(a(n)) = A243055(n).
(End)

A354351 Dirichlet inverse of A108951, primorial inflation of n.

Original entry on oeis.org

1, -2, -6, 0, -30, 12, -210, 0, 0, 60, -2310, 0, -30030, 420, 180, 0, -510510, 0, -9699690, 0, 1260, 4620, -223092870, 0, 0, 60060, 0, 0, -6469693230, -360, -200560490130, 0, 13860, 1021020, 6300, 0, -7420738134810, 19399380, 180180, 0, -304250263527210, -2520, -13082761331670030, 0, 0, 446185740, -614889782588491410

Offset: 1

Antti Karttunen, Jun 05 2022

Multiplicative with a(p^e) = 0 if e > 1, and -A034386(p) otherwise.

Cf. A002110, A008683, A013929 (positions of 0's), A034386, A108951, A354352.

Cf. also A347379, A354186, A354349, A354359, A354365, A354366.

A002110(n) = prod(i=1,n,prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A002110(primepi(f[i, 1]))^f[i, 2]) }; \\ From A108951
A354351(n) = (moebius(n)*A108951(n));

a(n) = A008683(n) * A108951(n).
a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d < n} A108951(n/d) * a(d).
a(n) = A354352(n) - A108951(n).

A354359 Dirichlet inverse of A124859.

Original entry on oeis.org

1, -2, -2, -2, -2, 4, -2, -14, -2, 4, -2, 4, -2, 4, 4, -110, -2, 4, -2, 4, 4, 4, -2, 28, -2, 4, -14, 4, -2, -8, -2, -1526, 4, 4, 4, 4, -2, 4, 4, 28, -2, -8, -2, 4, 4, 4, -2, 220, -2, 4, 4, 4, -2, 28, 4, 28, 4, 4, -2, -8, -2, 4, 4, -20858, 4, -8, -2, 4, 4, -8, -2, 28, -2, 4, 4, 4, 4, -8, -2, 220, -110, 4, -2, -8

Offset: 1

Antti Karttunen, Jun 05 2022

Multiplicative because A124859 is.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A002110, A124859.

Cf. also A354186, A354349, A354351, A354358.

A124859(n) = { my(f=factor(n)); for(k=1, #f~, f[k, 1] = prod(j=1, f[k, 2], prime(j)); f[k, 2] = 1); factorback(f); }; \\ From A124859
memoA354359 = Map();
A354359(n) = if(1==n,1,my(v); if(mapisdefined(memoA354359,n,&v), v, v = -sumdiv(n,d,if(dA124859(n/d)*A354359(d),0)); mapput(memoA354359,n,v); (v)));

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d < n} A124859(n/d) * a(d).

A354349 Dirichlet inverse of A181819, prime shadow of n.

Original entry on oeis.org

1, -2, -2, 1, -2, 4, -2, -1, 1, 4, -2, -2, -2, 4, 4, 2, -2, -2, -2, -2, 4, 4, -2, 2, 1, 4, -1, -2, -2, -8, -2, -3, 4, 4, 4, 1, -2, 4, 4, 2, -2, -8, -2, -2, -2, 4, -2, -4, 1, -2, 4, -2, -2, 2, 4, 2, 4, 4, -2, 4, -2, 4, -2, 7, 4, -8, -2, -2, 4, -8, -2, -1, -2, 4, -2, -2, 4, -8, -2, -4, 2, 4, -2, 4, 4, 4, 4, 2, -2, 4, 4

Offset: 1

Antti Karttunen, Jun 05 2022

Multiplicative because A181819 is.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A181819.

Cf. also A354186, A354351, A354359.

A181819(n) = factorback(apply(e->prime(e),(factor(n)[,2])));
memoA354349 = Map();
A354349(n) = if(1==n,1,my(v); if(mapisdefined(memoA354349,n,&v), v, v = -sumdiv(n,d,if(dA181819(n/d)*A354349(d),0)); mapput(memoA354349,n,v); (v)));

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d < n} A181819(n/d) * a(d).

A354866 Dirichlet inverse of A122111.

Original entry on oeis.org

1, -2, -4, 1, -8, 10, -16, -1, 7, 20, -32, -10, -64, 40, 46, 2, -128, -27, -256, -20, 92, 80, -512, 14, 37, 160, -17, -40, -1024, -150, -2048, -3, 184, 320, 202, 53, -4096, 640, 368, 28, -8192, -300, -16384, -80, -146, 1280, -32768, -26, 175, -129, 736, -160, -65536, 85, 404, 56, 1472, 2560, -131072, 242, -262144

Offset: 1

Antti Karttunen, Jun 09 2022

sign

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

Cf. A122111, A354867, A354868 (parity), A354869 (positions of odd terms).

Cf. also A354186, A354349, A354351, A354359, A354826.

A122111(n) = if(1==n,n,my(f=factor(n), es=Vecrev(f[,2]),is=concat(apply(primepi,Vecrev(f[,1])),[0]),pri=0,m=1); for(i=1, #es, pri += es[i]; m *= prime(pri)^(is[i]-is[1+i])); (m));
memoA354866 = Map();
A354866(n) = if(1==n,1,my(v); if(mapisdefined(memoA354866,n,&v), v, v = -sumdiv(n,d,if(dA122111(n/d)*A354866(d),0)); mapput(memoA354866,n,v); (v)));

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA122111(n/d) * a(d).

a(n) = A354867(n) - A122111(n).

A354187 Sum of A348717 and its Dirichlet inverse.

Original entry on oeis.org

2, 0, 0, 4, 0, 8, 0, 8, 4, 8, 0, 16, 0, 8, 8, 16, 0, 16, 0, 32, 8, 8, 0, 24, 4, 8, 8, 48, 0, 40, 0, 32, 8, 8, 8, 44, 0, 8, 8, 40, 0, 72, 0, 80, 16, 8, 0, 48, 4, 32, 8, 96, 0, 48, 8, 56, 8, 8, 0, 56, 0, 8, 32, 64, 8, 120, 0, 128, 8, 72, 0, 64, 0, 8, 16, 144, 8, 168, 0, 80, 16, 8, 0, 72, 8, 8, 8, 88, 0, 80, 8, 176

Offset: 1

Antti Karttunen, May 19 2022

sign

The first negative term is a(520) = -8.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences computed from indices in prime factorization

Cf. A348717, A354186.

A348717(n) = if(1==n, 1, my(f = factor(n), k = primepi(f[1, 1])-1); for (i=1, #f~, f[i, 1] = prime(primepi(f[i, 1])-k)); factorback(f));
memoA354186 = Map();
A354186(n) = if(1==n,1,my(v); if(mapisdefined(memoA354186,n,&v), v, v = -sumdiv(n,d,if(dA348717(n/d)*A354186(d),0)); mapput(memoA354186,n,v); (v)));
A354187(n) = (A348717(n)+A354186(n));

a(n) = A348717(n) + A354186(n).

a(1) = 2, and for n > 1, a(n) = -Sum_{d|n, 1A348717(d) * A354186(n/d).

cp's OEIS Frontend

A348717 a(n) is the least k such that A003961^i(k) = n for some i >= 0 (where A003961^i denotes the i-th iterate of A003961).

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A354351 Dirichlet inverse of A108951, primorial inflation of n.

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A354359 Dirichlet inverse of A124859.

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A354349 Dirichlet inverse of A181819, prime shadow of n.

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A354866 Dirichlet inverse of A122111.

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A354187 Sum of A348717 and its Dirichlet inverse.

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