A329618 -id:A329618 - cp's OEIS frontend

A329619 Difference between the maximal digit value used when A108951(n) is written in primorial base and its 2-adic valuation.

1, 0, 0, 0, 0, 0, 0, -2, -1, 0, 0, 1, 0, 0, 4, -2, 0, -1, 0, 1, 4, 0, 0, -1, 2, 0, -2, 1, 0, 2, 0, -4, 4, 0, 6, 0, 0, 0, 4, -3, 0, -2, 0, 1, 2, 0, 0, -2, 4, 5, 4, 1, 0, -2, 2, 4, 4, 0, 0, -1, 0, 0, 0, -4, 11, 9, 0, 1, 4, 2, 0, -2, 0, 0, 2, 1, 14, 9, 0, -3, 2, 0, 0, -2, 9, 0, 4, 4, 0, 6, 10, 1, 4, 0, 5, 0, 0, 9, 7, 2, 0, 9, 0, 4, 1

Offset: 1

Antti Karttunen, Nov 18 2019

sign

Cf. A001222, A007814, A034386, A108951, A276086, A324886, A328114, A329344, A329618, A329620, A329622.

With[{b = Reverse@ Prime@ Range@ 120}, Array[Max@ IntegerDigits[#, MixedRadix[b]] &@ Apply[Times, Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Times @@ Prime@ Range@ PrimePi@ p, e}]] - PrimeOmega[#] &, 105] ] (* Michael De Vlieger, Nov 18 2019 *)

A034386(n) = prod(i=1, primepi(n), prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) };  \\ From A108951
A328114(n) = { my(s=0, p=2); while(n, s = max(s,(n%p)); n = n\p; p = nextprime(1+p)); (s); };
A329344(n) = A328114(A108951(n));
A329619(n) = (A329344(n) - bigomega(n));

a(n) = A329344(n) - A001222(n).
a(n) = A328114(A108951(n)) - A007814(A108951(n)).
a(p) = 0 for all primes p.

A329620 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A046523(n), A246277(A324886(n))].

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 6, 4, 2, 7, 2, 4, 8, 9, 2, 10, 2, 7, 8, 4, 2, 11, 12, 4, 13, 7, 2, 14, 2, 15, 8, 4, 16, 17, 2, 4, 8, 18, 2, 19, 2, 7, 20, 4, 2, 21, 22, 23, 8, 7, 2, 24, 25, 26, 8, 4, 2, 27, 2, 4, 28, 29, 30, 31, 2, 7, 8, 32, 2, 33, 2, 4, 34, 7, 35, 31, 2, 36, 37, 4, 2, 38, 39, 4, 8, 26, 2, 40, 41, 7, 8, 4, 42, 43, 2, 44, 45, 46, 2, 31, 2, 26, 47

Offset: 1

Antti Karttunen, Nov 18 2019

nonn

Restricted growth sequence transform of the ordered pair [A046523(n), A246277(A324886(n))].
For all i, j:
  A305800(i) = A305800(j) => a(i) = a(j),
  a(i) = a(j) => A101296(i) = A101296(j),
  a(i) = a(j) => A329345(i) = A329345(j),
  a(i) = a(j) => A329618(i) = A329618(j),
  a(i) = a(j) => A329619(i) = A329619(j).

Cf. A046523, A034386, A108951, A246277, A276086, A305800, A324886, A329345, A329618, A329619.

up_to = 8192;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523
A034386(n) = prod(i=1, primepi(n), prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) };  \\ From A108951
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A324886(n) = A276086(A108951(n));
A246277(n) = if(1==n, 0, my(f = factor(n), k = primepi(f[1,1])-1); for (i=1, #f~, f[i,1] = prime(primepi(f[i,1])-k)); factorback(f)/2);
Aux329620(n) = [A046523(n), A246277(A324886(n))];
v329620 = rgs_transform(vector(up_to, n, Aux329620(n)));
A329620(n) = v329620[n];

A329621 a(n) = gcd(A056239(n), A324888(n)) = gcd(A001222(A108951(n)), A001222(A324886(n))).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 4, 1, 1, 1, 4, 1, 1, 1, 1, 6, 2, 1, 1, 6, 1, 2, 2, 1, 6, 1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 2, 8, 1, 3, 4, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 2, 2, 1, 4, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 9, 3, 1, 1, 8, 2, 1, 4, 2, 1, 6, 8, 1, 4, 2, 1, 1, 2, 1, 1, 1, 1, 3, 8, 1, 2, 1, 1, 1

Offset: 1

Antti Karttunen, Nov 18 2019

nonn

Cf. A001222, A034386, A056239, A108951, A276086, A324886, A324888, A329618, A329622.

With[{b = MixedRadix[Reverse@ Prime@ Range@ 500]}, Array[GCD @@ PrimeOmega@ {#, Function[k, Times @@ Power @@@ # &@ Transpose@ {Prime@ Range@ Length@ k, Reverse@ k}]@ IntegerDigits[#, b]} &@ Apply[Times, Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Times @@ Prime@ Range@ PrimePi@ p, e}]] &, 105]] (* Michael De Vlieger, Nov 18 2019 *)

A034386(n) = prod(i=1, primepi(n), prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) };  \\ From A108951
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A329621(n) = { my(u=A108951(n)); gcd(bigomega(u), bigomega(A276086(u))); };

a(n) = gcd(A056239(n), A324888(n)) = gcd(A001222(A108951(n)), A001222(A324886(n))).

A331290 a(n) = gcd(A001222(n), A329348(n)).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 4, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 2, 4, 2, 2, 1, 1, 1, 2, 3, 2, 1, 3, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 2, 1, 2, 1, 2, 2, 4, 1, 2, 2, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 3, 1, 4, 1

Offset: 1

Antti Karttunen, Jan 15 2020

Records occur at n = 1, 4, 36, 112, 352, 1088, 2016, 2688, 8064, 63360, ...

Cf. A001222, A002110, A117366, A329348, A331188, A331289, A331291.

Cf. also A329618, A329621.

A002110(n) = prod(i=1, n, prime(i));
A331290(n) = if(1==n, 1, my(f=factor(n), p=nextprime(1+vecmax(f[, 1]))); gcd((prod(i=1, #f~, A002110(primepi(f[i, 1]))^(f[i, 2]-(#f~==i)))%p),bigomega(n)));

a(n) = gcd(A001222(n), A329348(n)) = gcd(A001222(n), A331188(n) mod A117366(n)).

a(n) = gcd(A001222(n), A331289(n)).

cp's OEIS Frontend

A329619 Difference between the maximal digit value used when A108951(n) is written in primorial base and its 2-adic valuation.

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A329620 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A046523(n), A246277(A324886(n))].

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A329621 a(n) = gcd(A056239(n), A324888(n)) = gcd(A001222(A108951(n)), A001222(A324886(n))).

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A331290 a(n) = gcd(A001222(n), A329348(n)).

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