A354088 -id:A354088 - cp's OEIS frontend

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

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.

A354206 a(n) = A354203(sigma(A354202(n))), where A354202 is fully multiplicative with a(p) = A354200(A000720(p)), and A354203 is its left inverse.

Original entry on oeis.org

1, 1, 1, 23, 3, 1, 1, 5, 11, 3, 2, 23, 1, 1, 3, 469, 2, 11, 1, 69, 1, 2, 1, 5, 53, 1, 4, 23, 11, 3, 7, 69, 2, 2, 3, 253, 3, 1, 1, 15, 1, 1, 1, 46, 33, 1, 2, 469, 33, 53, 2, 23, 23, 4, 6, 5, 1, 11, 13, 69, 29, 7, 11, 19507, 3, 2, 1, 46, 1, 3, 2, 55, 2, 3, 53, 23, 2, 1, 3, 1407, 2797, 1, 5, 23, 6, 1, 11, 10, 9, 33

Offset: 1

Antti Karttunen, May 23 2022

Cf. A000203, A000720, A354200, A354201, A354202, A354203, A354205, A354207.
Cf. A354361 (positions of 1's).
Cf. also A326042, A348750, A354088, A354096 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))));
A354201(n) = if(n<=3,(n+1)\2,my(m=prime(n)%4); forstep(i=n-1,0,-1,if(m==(prime(i)%4),return(prime(i)))));
A354202(n) = { my(f=factor(n)); for(k=1,#f~,f[k,1] = A354200(primepi(f[k,1]))); factorback(f); };
A354203(n) = { my(f=factor(n)); for(k=1,#f~,f[k,1] = A354201(primepi(f[k,1]))); factorback(f); };
A354206(n) = A354203(sigma(A354202(n)));

Multiplicative with a(p^e) = A354203((q^(e+1)-1)/(q-1)) where q = A354200(A000720(p)).
a(n) = A354203(A354205(n)) = A354203(sigma(A354202(n))).
a(n) = n - A354207(n).

A354089 Sum of divisors function applied to Pythagorean prime shift: a(n) = sigma(A348746(n)).

Original entry on oeis.org

1, 4, 6, 13, 14, 24, 8, 40, 31, 56, 12, 78, 18, 32, 84, 121, 30, 124, 20, 182, 48, 48, 24, 240, 183, 72, 156, 104, 38, 336, 32, 364, 72, 120, 112, 403, 42, 80, 108, 560, 54, 192, 44, 156, 434, 96, 48, 726, 57, 732, 180, 234, 62, 624, 168, 320, 120, 152, 60, 1092, 74, 128, 248, 1093, 252, 288, 68, 390, 144, 448, 72

Offset: 1

Antti Karttunen, May 17 2022

Inverse Möbius transform of A348746.
Cf. A000203, A000720, A348744, A354088.
Cf. A003973, A354093 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); };
A354089(n) = sigma(A348746(n));

Multiplicative with a(p^e) = (q^(e+1)-1)/(q-1) where q = A348744(A000720(p)).
a(n) = A000203(A348746(n)).
a(n) = Sum_{d|n} A348746(d).

cp's OEIS Frontend

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

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A354206 a(n) = A354203(sigma(A354202(n))), where A354202 is fully multiplicative with a(p) = A354200(A000720(p)), and A354203 is its left inverse.

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PARI

Formula

A354089 Sum of divisors function applied to Pythagorean prime shift: a(n) = sigma(A348746(n)).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula