A354091 -id:A354091 - cp's OEIS frontend

A354093 a(n) = sigma(A354091(n)), where A354091 is fully multiplicative prime shift which replaces the primes of the form 3k+2 by the next larger such prime, while other primes stay as they are, and sigma is the sum of divisors function.

1, 6, 4, 31, 12, 24, 8, 156, 13, 72, 18, 124, 14, 48, 48, 781, 24, 78, 20, 372, 32, 108, 30, 624, 133, 84, 40, 248, 42, 288, 32, 3906, 72, 144, 96, 403, 38, 120, 56, 1872, 48, 192, 44, 558, 156, 180, 54, 3124, 57, 798, 96, 434, 60, 240, 216, 1248, 80, 252, 72, 1488, 62, 192, 104, 19531, 168, 432, 68, 744, 120, 576

Offset: 1

Antti Karttunen, May 17 2022

Inverse Möbius transform of A354091.
Cf. A000203, A003627, A010872, A074941, A354095.
Cf. A003973, A354089 for variants.

A354093(n) = { my(f=factor(n)); for(k=1,#f~, if(2==(f[k,1]%3), for(i=1+primepi(f[k,1]),oo,if(2==(prime(i)%3), f[k,1]=prime(i); break)))); sigma(factorback(f)); };

Multiplicative with a(p^e) = (q^(e+1)-1)/(q-1) where q = A003627(1+n) if p = A003627(n), otherwise q = p.
a(n) = Sum_{d|n} A354091(d).
For all n >= 1, A010872(a(n)) = A010872(A000203(n)) = A074941(n).

A354096 a(n) = A354092(sigma(A354091(n))).

Original entry on oeis.org

1, 3, 1, 31, 3, 3, 1, 39, 13, 9, 9, 31, 7, 3, 3, 295, 3, 39, 2, 93, 1, 27, 6, 39, 133, 21, 2, 31, 21, 9, 1, 1953, 9, 9, 3, 403, 19, 6, 7, 117, 3, 3, 5, 279, 39, 18, 27, 295, 57, 399, 3, 217, 6, 6, 27, 39, 2, 63, 9, 93, 31, 3, 13, 19531, 21, 27, 11, 93, 6, 9, 21, 507, 37, 57, 133, 62, 9, 21, 2, 885, 25, 9, 18, 31, 9

Offset: 1

Antti Karttunen, May 17 2022

Cf. A000203, A003627, A007949, A010872, A329963, A354091, A354092, A354093, A354095.

Cf. also A326042, A354088 for variants.

A354091(n) = { my(f=factor(n)); for(k=1,#f~, if(2==(f[k,1]%3), for(i=1+primepi(f[k,1]),oo,if(2==(prime(i)%3), f[k,1]=prime(i); break)))); factorback(f); };
A354092(n) = { my(f=factor(n)); for(k=1,#f~, if(2==(f[k,1]%3), if(2==f[k,1], f[k,1]--, forstep(i=primepi(f[k,1])-1,0,-1,if(2==(prime(i)%3), f[k,1]=prime(i); break))))); factorback(f); };
A354096(n) = A354092(sigma(A354091(n)));

Multiplicative with a(p^e) = A354092((q^(e+1)-1)/(q-1)), where q = A003627(1+n) if p = A003627(n), otherwise q = p.
a(n) = A354092(A354093(n)) = A354095(A354091(n)) = A354092(A000203(A354091(n))).
For all n >= 1, A010872(a(n)) = A010872(A354095(n)).
For all k in A329963, A007949(a(k)) = A007949(sigma(k)) = A354100(k) = 0.

A354094 a(n) = phi(A354091(n)), where A354091 is fully multiplicative prime shift which replaces the primes of the form 3k+2 by the next larger such prime, while other primes stay as they are, and phi is Euler totient function.

Original entry on oeis.org

1, 4, 2, 20, 10, 8, 6, 100, 6, 40, 16, 40, 12, 24, 20, 500, 22, 24, 18, 200, 12, 64, 28, 200, 110, 48, 18, 120, 40, 80, 30, 2500, 32, 88, 60, 120, 36, 72, 24, 1000, 46, 48, 42, 320, 60, 112, 52, 1000, 42, 440, 44, 240, 58, 72, 160, 600, 36, 160, 70, 400, 60, 120, 36, 12500, 120, 128, 66, 440, 56, 240, 82, 600, 72

Offset: 1

Antti Karttunen, May 17 2022

Index entries for sequences computed from indices in prime factorization

Möbius transform of A354091.
Cf. A000010, A003627, A007949, A008683, A010872, A074942, A354099.
Cf. also A003972.

A354094(n) = { my(f=factor(n)); for(k=1,#f~, if(2==(f[k,1]%3), for(i=1+primepi(f[k,1]),oo,if(2==(prime(i)%3), f[k,1]=prime(i); break)))); eulerphi(factorback(f)); };

Multiplicative with a(p^e) = (q-1) * q^(e-1) where q = A003627(1+n) if p = A003627(n), otherwise q = p.
a(n) = A000010(A354091(n)).
a(n) = Sum_{d|n} A008683(n/d) * A354091(d).
For all n >= 1, A010872(a(n)) = A010872(A000010(n)) = A074942(n).
For all n >= 1, A007949(a(n)) = A007949(A000010(n)) = A354099(n).

A354202 Fully multiplicative with a(p^e) = A354200(A000720(p))^e.

Original entry on oeis.org

1, 5, 7, 25, 13, 35, 11, 125, 49, 65, 19, 175, 17, 55, 91, 625, 29, 245, 23, 325, 77, 95, 31, 875, 169, 85, 343, 275, 37, 455, 43, 3125, 133, 145, 143, 1225, 41, 115, 119, 1625, 53, 385, 47, 475, 637, 155, 59, 4375, 121, 845, 203, 425, 61, 1715, 247, 1375, 161, 185, 67, 2275, 73, 215, 539, 15625, 221, 665, 71, 725

Offset: 1

Antti Karttunen, May 23 2022

Permutation of A007310. Preserves the prime signature.

Index entries for sequences computed from indices in prime factorization

Cf. A007310 (terms sorted into ascending order), A354200, A354203 (left inverse), A354204 (Möbius transform), A354205 (inverse Möbius transform).

Cf. also A003961, A108548, A267099, A332818, A348746, A354091 for similar constructions.

A354200(n) = if(1==n,5,my(p=prime(n), m=p%4); forprime(q=1+p,,if(m==(q%4),return(q))));
A354202(n) = { my(f=factor(n)); for(k=1,#f~,f[k,1] = A354200(primepi(f[k,1]))); factorback(f); };

A354092 Fully multiplicative prime shift where the primes of the form 3k+2 are replaced by the previous such prime (with 2 -> 1), and primes of the form 3k and 3k+1 stay as they are.

Original entry on oeis.org

1, 1, 3, 1, 2, 3, 7, 1, 9, 2, 5, 3, 13, 7, 6, 1, 11, 9, 19, 2, 21, 5, 17, 3, 4, 13, 27, 7, 23, 6, 31, 1, 15, 11, 14, 9, 37, 19, 39, 2, 29, 21, 43, 5, 18, 17, 41, 3, 49, 4, 33, 13, 47, 27, 10, 7, 57, 23, 53, 6, 61, 31, 63, 1, 26, 15, 67, 11, 51, 14, 59, 9, 73, 37, 12, 19, 35, 39, 79, 2, 81, 29, 71, 21, 22, 43, 69, 5, 83

Offset: 1

Antti Karttunen, May 17 2022

Index entries for sequences computed from indices in prime factorization

Left inverse of A354091.
Cf. A003627, A007645.
Cf. A064989, A348747 (variants).

A354092(n) = { my(f=factor(n)); for(k=1,#f~, if(2==(f[k,1]%3), if(2==f[k,1], f[k,1]--, forstep(i=primepi(f[k,1])-1,0,-1,if(2==(prime(i)%3), f[k,1]=prime(i); break))))); factorback(f); };

Fully multiplicative with a(2) = 1, a(A003627(1+n)) = A003627(n), a(A007645(n)) = A007645(n).

For all n >= 1, a(A354091(n)) = n.

cp's OEIS Frontend

A354093 a(n) = sigma(A354091(n)), where A354091 is fully multiplicative prime shift which replaces the primes of the form 3k+2 by the next larger such prime, while other primes stay as they are, and sigma is the sum of divisors function.

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A354096 a(n) = A354092(sigma(A354091(n))).

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A354094 a(n) = phi(A354091(n)), where A354091 is fully multiplicative prime shift which replaces the primes of the form 3k+2 by the next larger such prime, while other primes stay as they are, and phi is Euler totient function.

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A354202 Fully multiplicative with a(p^e) = A354200(A000720(p))^e.

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A354092 Fully multiplicative prime shift where the primes of the form 3k+2 are replaced by the previous such prime (with 2 -> 1), and primes of the form 3k and 3k+1 stay as they are.

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