A348045 -id:A348045 - cp's OEIS frontend

A348046 Positions of 2's in A348045.

5, 8, 9, 10, 12, 14, 15, 26, 38, 62, 86, 122, 146, 206, 218, 278, 302, 362, 386, 398, 458, 482, 542, 566, 626, 698, 842, 866, 926, 1046, 1142, 1202, 1238, 1286, 1322, 1622, 1646, 1658, 1718, 1766, 2042, 2066, 2102, 2126, 2186, 2306, 2462, 2558, 2582, 2606, 2642, 2858, 2906, 2966, 2978, 3218, 3242, 3338, 3398, 3446

Offset: 1

Antti Karttunen, Oct 12 2021

nonn

The only odd terms in this sequence are 5, 9 and 15.

Cf. A176810 (subsequence), A348045.

PARI
```
isA348046(n) = (2==A348045(n));
```

A285702 a(n) = A000010(A064216(n)).

Original entry on oeis.org

1, 1, 2, 4, 2, 6, 10, 2, 12, 16, 4, 18, 6, 4, 22, 28, 6, 8, 30, 10, 36, 40, 4, 42, 20, 12, 46, 12, 16, 52, 58, 8, 20, 60, 18, 66, 70, 6, 24, 72, 8, 78, 24, 22, 82, 40, 28, 32, 88, 12, 96, 100, 8, 102, 106, 30, 108, 36, 20, 48, 42, 36, 18, 112, 40, 126, 64, 8, 130, 136, 42, 60, 44, 20, 138, 148, 24, 56, 150, 46, 72, 156, 12, 162, 110, 32, 166, 24, 52, 172, 178

Offset: 1

Antti Karttunen, Apr 26 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000010, A064216, A064989, A099774, A151799, A285703.

Odd bisection of the following sequences: A347115, A348045, A349127, A349128.

Table[EulerPhi@ If[n == 1, 1, Apply[Times, FactorInteger[2 n - 1] /. {p_, e_} /; p > 2 :> NextPrime[p, -1]^e]], {n, 91}] (* Michael De Vlieger, Apr 26 2017 *)

(define (A285702 n) (A000010 (A064216 n)))

a(n) = A000010(A064216(n)).

Sum_{k=1..n} a(k) ~ c * n^2, where c = Product_{p prime} (p^3/((p+1)*(p^2-q(p)))) = 0.5366875995..., where q(p) = prevprime(p) (A151799) if p > 2 and q(2) = 1. - Amiram Eldar, Dec 21 2023

A347115 Möbius transform of A341515.

Original entry on oeis.org

1, 4, 1, 10, 2, 5, 4, 30, 2, 118, 6, 12, 10, 236, 2, 90, 12, 64, 16, 240, 4, 594, 18, 36, 6, 830, 4, 480, 22, -116, 28, 270, 6, 1428, 8, 132, 30, 1784, 10, 720, 36, 1076, 40, 1200, 4, 2622, 42, 108, 20, 144, 12, 1680, 46, 458, 12, 1440, 16, 4178, 52, -228, 58, 4772, 8, 810, 20, 1242, 60, 2880, 18, 2752, 66, 396

Offset: 1

Antti Karttunen, Aug 20 2021

sign

Cf. A008683, A064989, A329603, A341515.
Cf. A285702 (odd bisection), A347116 (even bisection).
Cf. also A332463, A347117, A347118, A348045.

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
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)};
A156552(n) = { my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res };
A329603(n) = A005940(2+(3*A156552(n)));
A341515(n) = if(n%2, A064989(n), A329603(n));
A347115(n) = sumdiv(n,d,moebius(n/d)*A341515(d));

a(n) = A008683(n/d) * A341515(d).

A349127 Möbius transform of A064989, where A064989 is multiplicative with a(2^e) = 1 and a(p^e) = prevprime(p)^e for odd primes p.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 4, 0, 2, 0, 6, 0, 10, 0, 2, 0, 12, 0, 16, 0, 4, 0, 18, 0, 6, 0, 4, 0, 22, 0, 28, 0, 6, 0, 8, 0, 30, 0, 10, 0, 36, 0, 40, 0, 4, 0, 42, 0, 20, 0, 12, 0, 46, 0, 12, 0, 16, 0, 52, 0, 58, 0, 8, 0, 20, 0, 60, 0, 18, 0, 66, 0, 70, 0, 6, 0, 24, 0, 72, 0, 8, 0, 78, 0, 24, 0, 22, 0, 82, 0, 40, 0, 28, 0, 32

Offset: 1

Antti Karttunen, Nov 13 2021

The multiplicative definition of this sequence ("Möbius transform of prime shift towards lesser primes") differs from otherwise similarly defined A349128 (Euler phi applied to A064989) only in that here a(2^e) = 0, while A349128(2^e) = 1.

Compare the situation with A003961 ("prime shift towards larger primes"), where A003972(n) = A000010(A003961(n)) is also the Möbius transform of A003961.

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

Agrees with A347115, A348045 and A349128 on odd numbers.
Cf. A000004, A285702 (even and odd bisection).
Cf. A000010, A008683, A064989, A151799.
Cf. also A003961, A003972, A349125, A349136.

f[p_, e_] := ((q = NextPrime[p, -1]) - 1)*q^(e - 1); a[1] = 1; a[n_] := If[EvenQ[n], 0, Times @@ f @@@ FactorInteger[n]]; Array[a, 100] (* Amiram Eldar, Nov 13 2021 *)

A349127(n) = if(!(n%2),0, my(f = factor(n), q); prod(i=1, #f~, q = precprime(f[i,1]-1); (q-1)*(q^(f[i,2]-1))));

A064989(n) = { my(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); };
A349127(n) = if(n%2, eulerphi(A064989(n)), 0);

A349127(n) = sumdiv(n,d,moebius(n/d)*A064989(d));

Multiplicative with a(2^e) = 0, and for odd primes p, a(p^e) = (q-1)*q^(e-1), where q = prevprime(p), where prevprime is A151799.
If n is odd, then a(n) = A000010(A064989(n)), and if n is even, then a(n) = 0.
a(n) = Sum_{d|n} A008683(d) * A064989(n/d).
For all n >= 1, a(2n-1) = A347115(2n-1) = A348045(2n-1) = A349128(2n-1) = A285702(n).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (16/Pi^4) / Product_{p prime > 2} (1+1/p-q(p)/p^2-q(p)/p^3) = 0.1341718..., where q(p) = prevprime(p) = A151799(p). - Amiram Eldar, Dec 24 2022

A349128 a(n) = phi(A064989(n)), where A064989 is multiplicative with a(2^e) = 1 and a(p^e) = prevprime(p)^e for odd primes p, and phi is Euler totient function.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 4, 1, 2, 2, 6, 1, 10, 4, 2, 1, 12, 2, 16, 2, 4, 6, 18, 1, 6, 10, 4, 4, 22, 2, 28, 1, 6, 12, 8, 2, 30, 16, 10, 2, 36, 4, 40, 6, 4, 18, 42, 1, 20, 6, 12, 10, 46, 4, 12, 4, 16, 22, 52, 2, 58, 28, 8, 1, 20, 6, 60, 12, 18, 8, 66, 2, 70, 30, 6, 16, 24, 10, 72, 2, 8, 36, 78, 4, 24, 40, 22, 6, 82, 4, 40

Offset: 1

Antti Karttunen, Nov 13 2021

See comments in A349127.

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

Agrees with A347115, A348045 and A349127 on odd numbers.
Cf. A285702 (odd bisection).
Cf. A000010, A064989, A151799, A349122 (inverse Möbius transform).
Cf. also A003972.

f[p_, e_] := If[p == 2, 1, Module[{q = NextPrime[p, -1]}, (q - 1)*q^(e - 1)]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 24 2022 *)

A349128(n) = { my(f = factor(n), q); prod(i=1, #f~, if(2==f[i,1], 1, q = precprime(f[i,1]-1); (q-1)*(q^(f[i,2]-1)))); };

Multiplicative with a(2^e) = 1, and for odd primes p, a(p^e) = (q - 1)*q^(e-1), where q = prevprime(p), where prevprime is A151799.
For odd n, a(n) = A349127(n), for even n, a(n) = a(n/2).
For all n >= 1, a(n) = a(2*n) = a(A000265(n)).
For all n >= 1, a(A000040(1+n)) = A006093(n) = A000040(n)-1.
Sum_{k=1..n} a(k) ~ c * n^2, where c = (64/(3*Pi^4)) / Product_{p prime > 2} (1+1/p-q(p)/p^2-q(p)/p^3) = 0.17889586..., where q(p) = prevprime(p) = A151799(p). - Amiram Eldar, Dec 24 2022

A349348 Dirichlet inverse of A252463, where A252463 shifts the prime factorization of odd numbers one step towards smaller primes and divides even numbers by two.

Original entry on oeis.org

1, -1, -2, -1, -3, 1, -5, -1, 0, 1, -7, 2, -11, 3, 6, -1, -13, -1, -17, 3, 10, 3, -19, 3, 0, 9, 0, 5, -23, -1, -29, -1, 14, 9, 15, 1, -31, 15, 22, 5, -37, -3, -41, 7, 0, 15, -43, 4, 0, -4, 26, 11, -47, -3, 21, 7, 34, 17, -53, -2, -59, 27, 0, -1, 33, -3, -61, 13, 38, -3, -67, 2, -71, 25, 0, 17, 35, -9, -73, 7, 0, 33

Offset: 1

Antti Karttunen, Nov 15 2021

sign

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

Coincides with A349125 on odd numbers.
Cf. A064989, A252463, A349349.
Cf. also A348045, A349437, A349438.

up_to = 20000;
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA064989(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)};
A252463(n) = if(!(n%2),n/2,A064989(n));
v349348 = DirInverseCorrect(vector(up_to,n,A252463(n)));
A349348(n) = v349348[n];

a(1) = 1; a(n) = -Sum_{d|n, d < n} A252463(n/d) * a(d).
a(n) = A349349(n) - A252463(n).
For all n >= 1, a(2n-1) = A349125(2n-1).

A349437 Dirichlet convolution of A252463 with A055615 (Dirichlet inverse of n), where A252463 shifts the prime factorization of odd numbers one step towards smaller primes and divides even numbers by two.

Original entry on oeis.org

1, -1, -1, 0, -2, 2, -2, 0, -2, 4, -4, 0, -2, 4, 2, 0, -4, 4, -2, 0, 2, 8, -4, 0, -6, 4, -4, 0, -6, -4, -2, 0, 4, 8, 4, 0, -6, 4, 2, 0, -4, -4, -2, 0, 4, 8, -4, 0, -10, 12, 4, 0, -6, 8, 8, 0, 2, 12, -6, 0, -2, 4, 4, 0, 4, -8, -6, 0, 4, -8, -4, 0, -2, 12, 6, 0, 8, -4, -6, 0, -8, 8, -4, 0, 8, 4, 6, 0, -6, -8, 4, 0, 2

Offset: 1

Antti Karttunen, Nov 18 2021

sign

Dirichlet convolution of this sequence with Euler phi (A000010) is A348045.

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

Cf. A055615, A064989, A252463, A349438 (Dirichlet inverse), A349439 (sum with it).

Cf. also A000010, A348045.

f[p_, e_] := NextPrime[p, -1]^e; s[1] = 1; s[n_] := If[EvenQ[n], n/2, Times @@ f @@@ FactorInteger[n]]; a[n_] := DivisorSum[n, # * MoebiusMu[#] * s[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 18 2021 *)

A055615(n) = (n*moebius(n));
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)};
A252463(n) = if(!(n%2),n/2,A064989(n));
A349437(n) = sumdiv(n,d,A055615(n/d)*A252463(d));

a(n) = Sum_{d|n} A055615(n/d) * A252463(d).

A349438 Dirichlet convolution of A000027 with A349348 (Dirichlet inverse of A252463), where A252463 shifts the prime factorization of odd numbers one step towards smaller primes and divides even numbers by two.

Original entry on oeis.org

1, 1, 1, 1, 2, 0, 2, 1, 3, 0, 4, -1, 2, 0, 2, 1, 4, -1, 2, -2, 2, 0, 4, -2, 10, 0, 9, -2, 6, -4, 2, 1, 4, 0, 4, -4, 6, 0, 2, -4, 4, -4, 2, -4, 6, 0, 4, -3, 14, -4, 4, -2, 6, -6, 8, -4, 2, 0, 6, -6, 2, 0, 6, 1, 4, -8, 6, -4, 4, -8, 4, -6, 2, 0, 10, -2, 8, -4, 6, -6, 27, 0, 4, -6, 8, 0, 6, -8, 6, -16, 4, -4, 2, 0, 4

Offset: 1

Antti Karttunen, Nov 18 2021

sign

Convolving this sequence with A348045 gives Euler phi, A000010.

It might first seem that A000265(a(p^k)) = p^(k-1) for all odd primes and all exponents k >= 1, but this does not hold for prime 37. However, with p=37, identity A065330(A349438(37^k)) = 37^(k-1) seems to hold for all exponents k >= 1. - Antti Karttunen, Nov 20 2021

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

Cf. A000027, A064989, A252463, A349348, A349437 (Dirichlet inverse), A349439 (sum with it).

Cf. also A000010, A000265, A065330, A348045.

f[p_, e_] := NextPrime[p, -1]^e; s[1] = 1; s[n_] := If[EvenQ[n], n/2, Times @@ f @@@ FactorInteger[n]]; sinv[1] = 1; sinv[n_] := sinv[n] = -DivisorSum[n, sinv[#] * s[n/#] &, # < n &]; a[n_] := DivisorSum[n, # * sinv[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 18 2021 *)

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)};
A252463(n) = if(!(n%2),n/2,A064989(n));
memoA349348 = Map();
A349348(n) = if(1==n,1,my(v); if(mapisdefined(memoA349348,n,&v), v, v = -sumdiv(n,d,if(dA252463(n/d)*A349348(d),0)); mapput(memoA349348,n,v); (v)));
A349438(n) = sumdiv(n,d,d*A349348(n/d));

a(n) = Sum_{d|n} d * A349348(n/d).

cp's OEIS Frontend

A348046 Positions of 2's in A348045.

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A285702 a(n) = A000010(A064216(n)).

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A347115 Möbius transform of A341515.

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A349127 Möbius transform of A064989, where A064989 is multiplicative with a(2^e) = 1 and a(p^e) = prevprime(p)^e for odd primes p.

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A349128 a(n) = phi(A064989(n)), where A064989 is multiplicative with a(2^e) = 1 and a(p^e) = prevprime(p)^e for odd primes p, and phi is Euler totient function.

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A349348 Dirichlet inverse of A252463, where A252463 shifts the prime factorization of odd numbers one step towards smaller primes and divides even numbers by two.

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A349437 Dirichlet convolution of A252463 with A055615 (Dirichlet inverse of n), where A252463 shifts the prime factorization of odd numbers one step towards smaller primes and divides even numbers by two.

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A349438 Dirichlet convolution of A000027 with A349348 (Dirichlet inverse of A252463), where A252463 shifts the prime factorization of odd numbers one step towards smaller primes and divides even numbers by two.

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