A348747 -id:A348747 - cp's OEIS frontend

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

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))).

A348746 Fully multiplicative with a(2) = 3, a(3) = 5, a(A002144(n)) = A002144(1+n) and a(A002145(1+n)) = a(A002145(1+n)) for all n >= 1, where A002144 and A002145 give the primes of the form 4k+1 and 4k+3 respectively.

Original entry on oeis.org

1, 3, 5, 9, 13, 15, 7, 27, 25, 39, 11, 45, 17, 21, 65, 81, 29, 75, 19, 117, 35, 33, 23, 135, 169, 51, 125, 63, 37, 195, 31, 243, 55, 87, 91, 225, 41, 57, 85, 351, 53, 105, 43, 99, 325, 69, 47, 405, 49, 507, 145, 153, 61, 375, 143, 189, 95, 111, 59, 585, 73, 93, 175, 729, 221, 165, 67, 261, 115, 273, 71, 675, 89, 123

Offset: 1

Antti Karttunen, Nov 02 2021

Permutation of odd numbers. Preserves the prime signature.

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

Cf. A000720, A002144, A002145, A348744, A348747 (left inverse).

Cf. also A003961, A332818 for similar maps.

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); };

Fully multiplicative with a(p) = A348744(A000720(p)), where A348744 is the lexicographically earliest bijection from primes to odd primes where each prime of the form 4k+1 is mapped to the next larger prime of the same form.

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.

A348745 Bijection from primes to {1} U primes, a left inverse of A348744.

Original entry on oeis.org

1, 2, 3, 7, 11, 5, 13, 19, 23, 17, 31, 29, 37, 43, 47, 41, 59, 53, 67, 71, 61, 79, 83, 73, 89, 97, 103, 107, 101, 109, 127, 131, 113, 139, 137, 151, 149, 163, 167, 157, 179, 173, 191, 181, 193, 199, 211, 223, 227, 197, 229, 239, 233, 251, 241, 263, 257, 271, 269, 277, 283, 281, 307, 311, 293, 313, 331, 317, 347, 337

Offset: 1

Antti Karttunen, Nov 02 2021

nonn

Cf. A002144, A002145, A348744, A348747.

up_to = 10000;
A348745list(up_to) = { my(lista=List([]), xs=Map(), i=2, p, q, u); mapput(xs,3,1); for(n=2,up_to, p = prime(n); if(1==(p%4), for(k=1+n,oo,q=prime(k);if((1==(q%4))&&!mapisdefined(xs,q),break)), while(mapisdefined(xs,prime(i)), i++); q = prime(i)); mapput(xs,q,n)); listput(lista,1); for(i=2,oo,if(!mapisdefined(xs,prime(i),&u),return(Vec(lista)),listput(lista,prime(u)))); };
v348745 = A348745list(up_to);
A348745(n) = v348745[n];

cp's OEIS Frontend

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

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A348746 Fully multiplicative with a(2) = 3, a(3) = 5, a(A002144(n)) = A002144(1+n) and a(A002145(1+n)) = a(A002145(1+n)) for all n >= 1, where A002144 and A002145 give the primes of the form 4k+1 and 4k+3 respectively.

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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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A348745 Bijection from primes to {1} U primes, a left inverse of A348744.

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