A354089 -id:A354089 - 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).

A354205 a(n) = sigma(A354202(n)), where A354202 is fully multiplicative with a(p) = A354200(A000720(p)).

Original entry on oeis.org

1, 6, 8, 31, 14, 48, 12, 156, 57, 84, 20, 248, 18, 72, 112, 781, 30, 342, 24, 434, 96, 120, 32, 1248, 183, 108, 400, 372, 38, 672, 44, 3906, 160, 180, 168, 1767, 42, 144, 144, 2184, 54, 576, 48, 620, 798, 192, 60, 6248, 133, 1098, 240, 558, 62, 2400, 280, 1872, 192, 228, 68, 3472, 74, 264, 684, 19531, 252, 960, 72

Offset: 1

Antti Karttunen, May 23 2022

Cf. A000203, A000290 (positions of odd terms), A000720, A354200, A354202, A354204, A354206.

Cf. A003973, A354089, A354093 for variants.

A354200(n) = if(1==n,5,my(p=prime(n), m=p%4); forprime(q=1+p,,if(m==(q%4),return(q))));
A354205(n) = { my(f=factor(n)); for(k=1,#f~,f[k,1] = A354200(primepi(f[k,1]))); sigma(factorback(f)); };
\\ Alternatively:
A354205(n) = sumdiv(n,d,A354202(d));

Multiplicative with a(p^e) = (q^(e+1)-1)/(q-1) where q = A354200(A000720(p)).
a(n) = A000203(A354202(n)).
a(n) = Sum_{d|n} A354202(d).

A354088 Sum of divisors function conjugated by Pythagorean prime shift: a(n) = A348747(sigma(A348746(n))).

Original entry on oeis.org

1, 1, 2, 5, 7, 2, 1, 3, 31, 7, 2, 10, 4, 1, 14, 121, 6, 31, 3, 35, 2, 2, 2, 6, 106, 4, 10, 5, 19, 14, 1, 35, 4, 6, 7, 155, 14, 3, 8, 21, 8, 2, 11, 10, 217, 2, 2, 242, 38, 106, 12, 20, 31, 10, 14, 3, 6, 19, 6, 70, 29, 1, 31, 1069, 28, 4, 13, 30, 4, 7, 4, 93, 12, 14, 212, 15, 2, 8, 3, 847, 781, 8, 14, 10, 42, 11, 38

Offset: 1

Antti Karttunen, May 17 2022

This is variant of A326042, and like that sequence, also this one is multiplicative.

Cf. A000203, A326042, A348744, A348746, A348747, A354089.

Cf. also A326042, A354096 for variants.

A348746(n) = { my(f=factor(n)); for(k=1,#f~, if(2==f[k,1], f[k,1]=3, if(3==f[k,1], f[k,1]=5, if(1==(f[k,1]%4), for(i=1+primepi(f[k,1]),oo,if(1==(prime(i)%4), f[k,1]=prime(i); break)))))); factorback(f); };
A348747(n) = { my(f=factor(n)); for(k=1,#f~, if(f[k,1]<=3, f[k,1]--, if(5==f[k,1], f[k,1]=3, if(1==(f[k,1]%4), forstep(i=primepi(f[k,1])-1,0,-1,if(1==(prime(i)%4), f[k,1]=prime(i); break)))))); factorback(f); };
A354088(n) = A348747(sigma(A348746(n)));

Multiplicative with a(p^e) = A348747((q^(e+1)-1)/(q-1)), where q = A348744(A000720(p)).

a(n) = A348747(A354089(n)) = A348747(A000203(A348746(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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A354205 a(n) = sigma(A354202(n)), where A354202 is fully multiplicative with a(p) = A354200(A000720(p)).

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PARI

Formula

A354088 Sum of divisors function conjugated by Pythagorean prime shift: a(n) = A348747(sigma(A348746(n))).

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Programs

PARI

Formula