A324886 -id:A324886 - cp's OEIS frontend

A329044 a(n) = A064989(A324886(n)).

1, 2, 3, 4, 5, 9, 7, 6, 15, 25, 11, 81, 13, 49, 15625, 36, 17, 225, 19, 625, 117649, 121, 23, 135, 60025, 169, 21, 2401, 29, 21875, 31, 10, 1771561, 289, 697540921, 50625, 37, 361, 4826809, 35, 41, 77, 43, 14641, 84035, 529, 47, 375, 161212051, 3603000625, 24137569, 28561, 53, 441, 2474329, 5764801, 47045881, 841, 59, 42875, 61, 961

Offset: 1

Antti Karttunen, Nov 08 2019

nonn

Cf. A034386, A064989, A108951, A276086, A324886, A329045.

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));
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A329044(n) = A064989(A324886(n));

a(n) = A064989(A324886(n)).

a(A000040(n)) = A000040(n).

A346096 Numerator of the primorial deflation of A276086(A108951(n)): a(n) = A319626(A324886(n)).

Original entry on oeis.org

2, 3, 5, 9, 7, 25, 11, 5, 7, 49, 13, 625, 17, 121, 117649, 25, 19, 49, 23, 2401, 1771561, 169, 29, 175, 14641, 289, 55, 14641, 31, 26411, 37, 21, 4826809, 361, 299393809, 2401, 41, 529, 24137569, 11, 43, 13, 47, 28561, 161051, 841, 53, 343, 6311981, 214358881, 47045881, 83521, 59, 3025, 48841, 214358881, 148035889, 961

Offset: 1

Antti Karttunen, Jul 07 2021

Numerator of ratio A324886(n) / A329044(n).

Cf. A346097 (denominators).
Cf. A003961, A108951, A319626, A324886, A329044, A346095, A346098, A346106, A346108.
Cf. also A337376, A345941.

A064989(n) = { my(f = factor(n)); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
A319626(n) = (n / gcd(n, A064989(n)));
A346096(n) = A319626(A324886(n)); \\ Rest of program in A324886.

a(n) = A319626(A324886(n)).
a(n) = A324886(n) / A346095(n) = A324886(n) / gcd(A324886(n), A329044(n)).
For n >= 1, A108951(A346096(n)) / A108951(A346097(n)) = A324886(n).
For n > 1, a(n) = A003961(A346098(n)).

A346097 Denominator of the primorial deflation of A276086(A108951(n)): a(n) = A319627(A324886(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 9, 7, 2, 3, 25, 11, 81, 13, 49, 15625, 4, 17, 9, 19, 625, 117649, 121, 23, 27, 1225, 169, 21, 2401, 29, 3125, 31, 10, 1771561, 289, 5764801, 81, 37, 361, 4826809, 5, 41, 7, 43, 14641, 12005, 529, 47, 75, 1127357, 1500625, 24137569, 28561, 53, 441, 14641, 5764801, 47045881, 841, 59, 125, 61, 961, 343, 100, 302875106592253

Offset: 1

Antti Karttunen, Jul 07 2021

Denominator of ratio A324886(n) / A329044(n).

Cf. A346096 (numerators).
Cf. A006530, A020639, A108951, A276086, A319627, A324886, A329044, A345941 [= gcd(n, a(n))], A346095, A346098, A346107, A346109.
Cf. also A337377.

A064989(n) = { my(f = factor(n)); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
A319627(n) = (A064989(n) / gcd(n, A064989(n)));
A346097(n) = A319627(A324886(n)); \\ Rest of program given in A324886.

a(n) = A319627(A324886(n)).
a(n) = A329044(n) / A346095(n) = A329044(n) / gcd(A324886(n), A329044(n)).
A020639(a(n)) = A006530(n).
A108951(a(n)) = A346107(n).
A346105(a(n)) = A346109(n).

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];

A346095 a(n) = gcd(A324886(n), A064989(A324886(n))).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 5, 1, 1, 1, 1, 1, 1, 9, 1, 25, 1, 1, 1, 1, 1, 5, 49, 1, 1, 1, 1, 7, 1, 1, 1, 1, 121, 625, 1, 1, 1, 7, 1, 11, 1, 1, 7, 1, 1, 5, 143, 2401, 1, 1, 1, 1, 169, 1, 1, 1, 1, 343, 1, 1, 1331, 1, 17, 1, 1, 1, 1, 161051, 1, 175, 1, 1, 41503, 1, 169, 1, 1, 49, 35, 1, 1, 121, 19, 1, 1, 1, 1, 49, 24137569

Offset: 1

Antti Karttunen, Jul 07 2021

nonn

Cf. A006530, A324886, A329044, A330749, A346096, A346097.

A330749(n) = {my(f); f = factor(n); if((n>1 && f[1, 1]==2), f[1, 2] = 0); for (i=1, #f~, f[i, 1] = precprime(f[i, 1]-1)); gcd(n, factorback(f)); };
A346095(n) = A330749(A324886(n)); \\ Rest of program given in A324886.

a(n) = A330749(A324886(n)) = gcd(A324886(n), A329044(n)) = gcd(A324886(n), A064989(A324886(n))).
a(n) = A324886(n) / A346096(n).
a(n) = A329044(n) / A346097(n).
a(n) mod A006530(n) > 0, for all n > 1.

A346098 a(n) = A064989(A346096(n)) = A064989(A319626(A324886(n))).

Original entry on oeis.org

1, 2, 3, 4, 5, 9, 7, 3, 5, 25, 11, 81, 13, 49, 15625, 9, 17, 25, 19, 625, 117649, 121, 23, 45, 2401, 169, 21, 2401, 29, 4375, 31, 10, 1771561, 289, 14235529, 625, 37, 361, 4826809, 7, 41, 11, 43, 14641, 16807, 529, 47, 125, 2093663, 5764801, 24137569, 28561, 53, 441, 20449, 5764801, 47045881, 841, 59, 343, 61, 961, 1331, 100, 396067447082177

Offset: 1

Antti Karttunen, Jul 07 2021

nonn

Cf. A064989, A319626, A324886, A346095, A346096, A346097, A346099 [= gcd(n, a(n))].

A064989(n) = { my(f = factor(n)); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
A319626(n) = (n / gcd(n, A064989(n)));
A346098(n) = A064989(A319626(A324886(n))); \\ Rest of program given in A324886.

a(n) = A064989(A346096(n)) = A064989(A319626(A324886(n))).

A329047 a(n) = A003415(A324886(n)).

Original entry on oeis.org

1, 1, 1, 6, 1, 10, 1, 8, 12, 14, 1, 500, 1, 22, 100842, 240, 1, 840, 1, 1372, 966306, 26, 1, 650, 465850, 34, 16, 5324, 1, 148862, 1, 10, 2227758, 38, 31919986098, 2058000, 1, 46, 8519142, 18, 1, 24, 1, 8788, 673486, 58, 1, 1078, 551741398, 668409965300, 14856594, 19652, 1, 1760, 3510806, 155897368, 38618058, 62, 1, 320166, 1, 74, 1472328, 420

Offset: 1

Antti Karttunen, Nov 08 2019

nonn

Cf. A003415, A034386, A108951, A276086, A324886, A327860.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
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));
A329047(n) = A003415(A324886(n));

a(n) = A003415(A324886(n)) = A003415(A276086(A108951(n))).

a(n) = A327860(A108951(n)).

A324896 Largest proper divisor of A324886(n).

Original entry on oeis.org

1, 1, 1, 3, 1, 5, 1, 5, 7, 7, 1, 125, 1, 11, 16807, 75, 1, 245, 1, 343, 161051, 13, 1, 175, 102487, 17, 11, 1331, 1, 26411, 1, 7, 371293, 19, 3293331899, 300125, 1, 23, 1419857, 11, 1, 13, 1, 2197, 161051, 29, 1, 343, 82055753, 73525096183, 2476099, 4913, 1, 605, 634933, 19487171, 6436343, 31, 1, 65219, 1, 37, 265837, 147

Offset: 1

Antti Karttunen, Mar 30 2019

nonn

Cf. A032742, A108951, A276086, A324886, A324895.

A034386(n) = prod(i=1, primepi(n), prime(i));
A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) };  \\ From A108951
A276086(n) = { my(i=0,m=1,pr=1,nextpr); while((n>0),i=i+1; nextpr = prime(i)*pr; if((n%nextpr),m*=(prime(i)^((n%nextpr)/pr));n-=(n%nextpr));pr=nextpr); m; };
A324886(n) = A276086(A108951(n));
A324896(n) = A032742(A324886(n));

a(n) = A032742(A324886(n)) = A324895(A108951(n)).

A329046 a(n) = A000005(A324886(n)).

Original entry on oeis.org

2, 2, 2, 3, 2, 3, 2, 4, 4, 3, 2, 5, 2, 3, 7, 9, 2, 9, 2, 5, 7, 3, 2, 8, 15, 3, 4, 5, 2, 12, 2, 4, 7, 3, 27, 25, 2, 3, 7, 4, 2, 4, 2, 5, 12, 3, 2, 8, 28, 45, 7, 5, 2, 9, 15, 9, 7, 3, 2, 16, 2, 3, 16, 9, 28, 13, 2, 5, 7, 36, 2, 24, 2, 3, 72, 5, 51, 13, 2, 9, 28, 3, 2, 9, 24, 3, 7, 9, 2, 33, 91, 5, 7, 3, 16, 21, 2, 117, 33, 28, 2, 13, 2, 9, 40

Offset: 1

Antti Karttunen, Nov 08 2019

nonn

Cf. A000005, A034386, A064989, A108951, A276086, A324655, A324886, A329045.

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));
A329046(n) = numdiv(A324886(n));

a(n) = A000005(A324886(n)) = A000005(A276086(A108951(n))).

a(n) = A324655(A108951(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))).

cp's OEIS Frontend

A329044 a(n) = A064989(A324886(n)).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A346096 Numerator of the primorial deflation of A276086(A108951(n)): a(n) = A319626(A324886(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A346097 Denominator of the primorial deflation of A276086(A108951(n)): a(n) = A319627(A324886(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A346095 a(n) = gcd(A324886(n), A064989(A324886(n))).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A346098 a(n) = A064989(A346096(n)) = A064989(A319626(A324886(n))).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A329047 a(n) = A003415(A324886(n)).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A324896 Largest proper divisor of A324886(n).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A329046 a(n) = A000005(A324886(n)).

Views

Author

Keywords

Links