A353573 -id:A353573 - cp's OEIS frontend

A353572 Shifted variant of A342002: a(n) = A353571(A276086(n)), where A353571(x) = A003415(A003961(x)) / A003557(A003961(x)) and A276086 is the primorial base exp-function.

0, 1, 1, 8, 2, 11, 1, 10, 12, 71, 19, 92, 2, 13, 17, 86, 24, 107, 3, 16, 22, 101, 29, 122, 4, 19, 27, 116, 34, 137, 1, 14, 16, 103, 27, 136, 18, 131, 167, 886, 244, 1117, 29, 164, 222, 1051, 299, 1282, 40, 197, 277, 1216, 354, 1447, 51, 230, 332, 1381, 409, 1612, 2, 17, 21, 118, 32, 151, 25, 152, 202, 991, 279

Offset: 0

Antti Karttunen, Apr 27 2022

Cf. A003415, A003557, A003961, A276154, A342002, A349905, A353571, A353573 [= gcd(a(n), A342002(n))], A353574.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003557(n) = (n/factorback(factorint(n)[, 1]));
A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A353571(n) = { my(s=A003961(n)); (A003415(s)/A003557(s)); };
A353572(n) = A353571(A276086(n));

a(n) = A353571(A276086(n)).
a(n) = A342002(A276154(n)).
For all n >= 0, a(n) >= A342002(n).

A353574 a(n) = A342002(n) / gcd(A342002(n), A342002(A276154(n))).

Original entry on oeis.org

1, 1, 5, 1, 7, 1, 7, 2, 31, 13, 41, 1, 9, 11, 37, 2, 47, 1, 11, 7, 43, 19, 53, 1, 13, 17, 49, 11, 59, 1, 9, 5, 41, 17, 55, 2, 59, 71, 247, 53, 317, 19, 73, 46, 289, 127, 359, 13, 87, 113, 331, 74, 401, 11, 101, 67, 373, 169, 443, 1, 11, 13, 47, 5, 61, 17, 69, 43, 277, 121, 347, 2, 83, 107, 319, 71, 389, 31, 97, 16

Offset: 1

Antti Karttunen, Apr 29 2022

Compare the scatter plot to that of A342002.

Cf. A003415, A003557, A003961, A342001, A342002, A353572, A353573, A353575.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003557(n) = (n/factorback(factorint(n)[, 1]));
A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A342001(n) = (A003415(n) / A003557(n));
A353571(n) = A342001(A003961(n));
A353574(n) = { my(u=A342001(A276086(n))); u/gcd(u, A353571(A276086(n))); };

a(n) = A342002(n) / A353573(n) = A342002(n) / gcd(A342002(n), A353572(n)).

cp's OEIS Frontend

A353572 Shifted variant of A342002: a(n) = A353571(A276086(n)), where A353571(x) = A003415(A003961(x)) / A003557(A003961(x)) and A276086 is the primorial base exp-function.

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