A329044 -id:A329044 - cp's OEIS frontend

A329045 Lexicographically earliest infinite sequence such that a(i) = a(j) => A046523(A329044(i)) = A046523(A329044(j)) for all i, j.

1, 2, 2, 3, 2, 3, 2, 4, 4, 3, 2, 5, 2, 3, 6, 7, 2, 7, 2, 5, 6, 3, 2, 8, 9, 3, 4, 5, 2, 10, 2, 4, 6, 3, 11, 12, 2, 3, 6, 4, 2, 4, 2, 5, 10, 3, 2, 8, 13, 14, 6, 5, 2, 7, 9, 15, 6, 3, 2, 16, 2, 3, 16, 7, 17, 18, 2, 5, 6, 19, 2, 20, 2, 3, 21, 5, 22, 18, 2, 7, 13, 3, 2, 7, 23, 3, 6, 15, 2, 24, 25, 5, 6, 3, 26, 27, 2, 28, 24, 13, 2, 18, 2, 15, 29

Offset: 1

Antti Karttunen, Nov 08 2019

nonn

Restricted growth sequence transform of function f(n) = A046523(A329044(n)).
For all i, j:
  A305800(i) = A305800(j) => a(i) = a(j),
  a(i) = a(j) => A324888(i) = A324888(j),
  a(i) = a(j) => A329046(i) = A329046(j).

Cf. A034386, A064989, A108951, A276086, A305800, A324886, A324888, A329044, A329046.

Cf. also A278226, A286626.

up_to = 65537;
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; };
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));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523
v329045 = rgs_transform(vector(up_to, n, A046523(A329044(n))));
A329045(n) = v329045[n];

A329345 Lexicographically earliest infinite sequence such that a(i) = a(j) => A246277(A329044(i)) = A246277(A329044(j)) for all i, j.

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 2, 4, 4, 3, 2, 5, 2, 3, 6, 7, 2, 7, 2, 5, 6, 3, 2, 8, 9, 3, 10, 5, 2, 11, 2, 10, 6, 3, 12, 13, 2, 3, 6, 4, 2, 4, 2, 5, 14, 3, 2, 15, 16, 17, 6, 5, 2, 18, 19, 20, 6, 3, 2, 21, 2, 3, 21, 18, 22, 23, 2, 5, 6, 24, 2, 25, 2, 3, 26, 5, 27, 23, 2, 7, 28, 3, 2, 7, 29, 3, 6, 20, 2, 30, 31, 5, 6, 3, 32, 33, 2, 34, 35, 16, 2, 23, 2, 20, 36

Offset: 1

Antti Karttunen, Nov 11 2019

nonn

Restricted growth sequence transform of function f(n) = A246277(A329044(n)).
For all i, j:
  A305800(i) = A305800(j) => a(i) = a(j),
  a(i) = a(j) => A329045(i) = A329045(j),
  a(i) = a(j) => A329343(i) = A329343(j),
  a(i) = a(j) => A329348(i) = A329348(j),
  a(i) = a(j) => A329349(i) = A329349(j).

Cf. A034386, A064989, A108951, A246277, A276086, A305800, A324886, A329044, A329045, A329343, A329348, A329349.

Cf. also A329048.

up_to = 1024;
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; };
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));
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);
v329345 = rgs_transform(vector(up_to, n, A246277(A329044(n))));
A329345(n) = v329345[n];

A345941 a(n) = gcd(n, A329044(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 3, 7, 2, 3, 5, 11, 3, 13, 7, 5, 4, 17, 9, 19, 5, 7, 11, 23, 3, 25, 13, 3, 7, 29, 5, 31, 2, 11, 17, 7, 9, 37, 19, 13, 5, 41, 7, 43, 11, 5, 23, 47, 3, 7, 25, 17, 13, 53, 9, 11, 7, 19, 29, 59, 5, 61, 31, 7, 4, 13, 11, 67, 17, 23, 7, 71, 9, 73, 37, 25, 19, 11, 13, 79, 5, 3, 41, 83, 7, 17, 43, 29, 11, 89

Offset: 1

Antti Karttunen, Jul 03 2021

nonn

Only powers of primes (A000961) occur as terms. A346087 gives the exponents. - Antti Karttunen, Jul 07 2021

Cf. A000961, A006530, A020639, A071178, A108951, A324886, A329348, A329044, A345942, A345943, A346087, A346097.

A345941(n) = gcd(n,A329044(n)); \\ Rest of program given in A329044.

a(n) = gcd(n, A329044(n)).
a(n) = n / A345942(n).
a(n) = A329044(n) / A345943(n).
a(p) = p for all primes p.
From Antti Karttunen, Jul 07 2021: (Start)
a(n) = A006530(n)^A346087(n) = A006530(n)^min(A071178(n), A329348(n)).
a(n) = gcd(n, A346097(n)).
A006530(a(n)) = A020639(A329044(n)) = A006530(n).
(End)

A345942 a(n) = n / gcd(n, A329044(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 4, 3, 2, 1, 4, 1, 2, 3, 4, 1, 2, 1, 4, 3, 2, 1, 8, 1, 2, 9, 4, 1, 6, 1, 16, 3, 2, 5, 4, 1, 2, 3, 8, 1, 6, 1, 4, 9, 2, 1, 16, 7, 2, 3, 4, 1, 6, 5, 8, 3, 2, 1, 12, 1, 2, 9, 16, 5, 6, 1, 4, 3, 10, 1, 8, 1, 2, 3, 4, 7, 6, 1, 16, 27, 2, 1, 12, 5, 2, 3, 8, 1, 18, 7, 4, 3, 2, 5, 32, 1, 2, 9, 20, 1, 6

Offset: 1

Antti Karttunen, Jul 03 2021

nonn

Cf. A108951, A324886, A329044, A345941, A345943.

A345942(n) = (n/gcd(n,A329044(n))); \\ Rest of program given in A329044.

a(n) = n / A345941(n).

A345943 a(n) = A329044(n) / gcd(n, A329044(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 3, 5, 5, 1, 27, 1, 7, 3125, 9, 1, 25, 1, 125, 16807, 11, 1, 45, 2401, 13, 7, 343, 1, 4375, 1, 5, 161051, 17, 99648703, 5625, 1, 19, 371293, 7, 1, 11, 1, 1331, 16807, 23, 1, 125, 23030293, 144120025, 1419857, 2197, 1, 49, 224939, 823543, 2476099, 29, 1, 8575, 1, 31, 65219, 25, 396067447082177, 285311670611

Offset: 1

Antti Karttunen, Jul 04 2021

nonn

Cf. A108951, A324886, A329044, A345941, A345942.

A345943(n) = { my(u=A329044(n)); (u / gcd(n, u)); }; \\ The rest of the program given in A329044.

a(n) = A329044(n) / A345941(n) = A329044(n) / gcd(n, A329044(n)).

A351949 Lexicographically earliest infinite sequence such that a(i) = a(j) => A246277(A329044(i)) = A246277(A329044(j)) and A003557(i) = A003557(j), for all i, j >= 1.

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, 5, 2, 18, 2, 7, 19, 4, 2, 20, 21, 22, 8, 7, 2, 23, 24, 25, 8, 4, 2, 26, 2, 4, 27, 28, 29, 30, 2, 7, 8, 31, 2, 32, 2, 4, 33, 7, 34, 30, 2, 9, 35, 4, 2, 36, 37, 4, 8, 25, 2, 38, 39, 7, 8, 4, 40

Offset: 1

Antti Karttunen, Apr 05 2022

nonn

Restricted growth sequence transform of the ordered pair [A003557(n), A329345(n)].

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A003557, A305800, A329044, A329345, A329620.

up_to = 65537;
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; };
A003557(n) = (n/factorback(factorint(n)[, 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]) };
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));
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);
v351949 = rgs_transform(vector(up_to, n, [A003557(n), A246277(A329044(n))]));
A351949(n) = v351949[n];

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

Original entry on oeis.org

2, 3, 5, 9, 7, 25, 11, 15, 35, 49, 13, 625, 17, 121, 117649, 225, 19, 1225, 23, 2401, 1771561, 169, 29, 875, 717409, 289, 55, 14641, 31, 184877, 37, 21, 4826809, 361, 36226650889, 1500625, 41, 529, 24137569, 77, 43, 143, 47, 28561, 1127357, 841, 53, 1715, 902613283, 514675673281, 47045881, 83521, 59, 3025, 8254129, 214358881, 148035889, 961, 61

Offset: 1

Antti Karttunen, Mar 30 2019

nonn

Cf. A034386, A108951, A117366, A276086, A324887, A324888, A324896, A329047, A342920, A344592, A346106, A346107.
Permutation of A324576.
Cf. also A346105.

With[{b = MixedRadix[Reverse@ Prime@ Range@ 120]}, Array[Function[k, Times @@ Power @@@ # &@ Transpose@ {Prime@ Range@ Length@ k, Reverse@ k}]@ IntegerDigits[Apply[Times, Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Times @@ Prime@ Range@ PrimePi@ p, e}]], b] &, 58]] (* Michael De Vlieger, Nov 18 2019 *)
A276086[n0_] := Module[{m = 1, i = 1, n = n0, p}, While[n > 0, p = Prime[i]; m *= p^Mod[n, p]; n = Quotient[n, p]; i++]; m];
(* b is A108951 *)
b[n_] := b[n] = Module[{pe = FactorInteger[n], p, e}, If[Length[pe] > 1, Times @@ b /@ Power @@@ pe, {{p, e}} = pe; Times @@ (Prime[Range[ PrimePi[p]]]^e)]]; b[1] = 1;
a[n_] := A276086[b[n]];
Array[a, 100] (* Jean-François Alcover, Dec 01 2021, after _Antti Karttunen in A296086 *)

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(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));

a(n) = A276086(A108951(n)).
a(n) = A117366(n) * A324896(n).
A001222(a(n)) = A324888(n).
A020639(a(n)) = A117366(n).
A032742(a(n)) = A324896(n).
a(A000040(n)) = A000040(1+n).
From Antti Karttunen, Jul 09 2021: (Start)
For n > 1, a(n) = A003961(A329044(n)).
a(n) = A346091(n) * A344592(n).
a(n) = A346106(n) /  A346107(n).
A003415(a(n)) = A329047(n).
A003557(a(n)) = A344592(n).
A342001(a(n)) = A342920(n) = A329047(n) / A344592(n).
(End)

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).

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.

cp's OEIS Frontend

A329045 Lexicographically earliest infinite sequence such that a(i) = a(j) => A046523(A329044(i)) = A046523(A329044(j)) for all i, j.

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A329345 Lexicographically earliest infinite sequence such that a(i) = a(j) => A246277(A329044(i)) = A246277(A329044(j)) for all i, j.

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A345941 a(n) = gcd(n, A329044(n)).

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A345942 a(n) = n / gcd(n, A329044(n)).

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A345943 a(n) = A329044(n) / gcd(n, A329044(n)).

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A351949 Lexicographically earliest infinite sequence such that a(i) = a(j) => A246277(A329044(i)) = A246277(A329044(j)) and A003557(i) = A003557(j), for all i, j >= 1.

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

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

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

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