A322819 -id:A322819 - cp's OEIS frontend

A323173 Sum of divisors computed for conjugated prime factorization: a(n) = A000203(A122111(n)).

1, 3, 7, 4, 15, 12, 31, 6, 13, 28, 63, 18, 127, 60, 39, 8, 255, 24, 511, 42, 91, 124, 1023, 24, 40, 252, 31, 90, 2047, 72, 4095, 12, 195, 508, 120, 32, 8191, 1020, 403, 56, 16383, 168, 32767, 186, 93, 2044, 65535, 36, 121, 78, 819, 378, 131071, 48, 280, 120, 1651, 4092, 262143, 96, 524287, 8188, 217, 14, 600, 360, 1048575, 762, 3315, 234

Offset: 1

Antti Karttunen, Jan 10 2019

nonn

Cf. A000203, A038712, A122111, A322819, A323174.

Cf. also A323243.

A122111[n_] := Product[Prime[Sum[If[j < i, 0, 1], {j, #}]], {i, Max[#]}]&[ Flatten[Table[Table[PrimePi[f[[1]]], {f[[2]]}], {f, FactorInteger[n]}]]];
a[n_] := With[{k = A122111[n]}, DivisorSigma[1, k]];
Array[a, 70] (* Jean-François Alcover, Sep 23 2020 *)

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)};
A122111(n) = if(1==n,n,prime(bigomega(n))*A122111(A064989(n)));
A323173(n) = sigma(A122111(n));

a(n) = A000203(A122111(n)).
a(n) = 2*A122111(n) - A323174(n).
a(n) = A322819(n) * A038712(A122111(n)).

A322865 a(n) = A000265(A122111(n)).

Original entry on oeis.org

1, 1, 1, 3, 1, 3, 1, 5, 9, 3, 1, 5, 1, 3, 9, 7, 1, 15, 1, 5, 9, 3, 1, 7, 27, 3, 25, 5, 1, 15, 1, 11, 9, 3, 27, 21, 1, 3, 9, 7, 1, 15, 1, 5, 25, 3, 1, 11, 81, 45, 9, 5, 1, 35, 27, 7, 9, 3, 1, 21, 1, 3, 25, 13, 27, 15, 1, 5, 9, 45, 1, 33, 1, 3, 75, 5, 81, 15, 1, 11, 49, 3, 1, 21, 27, 3, 9, 7, 1, 35, 81, 5, 9, 3, 27, 13, 1, 135, 25, 63, 1, 15, 1, 7, 75

Offset: 1

Antti Karttunen, Dec 30 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences related to binary expansion of n
Index entries for sequences computed from indices in prime factorization

Cf. A000265, A122111, A322813, A322819, A322820, A322826, A322865, A322867.

Array[#/2^IntegerExponent[#, 2] &@ If[# < 3, 1, Block[{k = #, m = 0}, Times @@ Power @@@ Table[k -= m; k = DeleteCases[k, 0]; {Prime@ Length@ k, m = Min@ k}, Length@ Union@ k]] &@ Catenate[ConstantArray[PrimePi[#1], #2] & @@@ FactorInteger@ #]] &, 105] (* Michael De Vlieger, Dec 31 2018, after JungHwan Min at A122111 *)

A000265(n) = (n>>valuation(n, 2));
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)};
A122111(n) = if(1==n,n,prime(bigomega(n))*A122111(A064989(n)));
A322865(n) = A000265(A122111(n));

a(n) = A000265(A122111(n)).
a(n) = A122111(A322820(n)).
A000005(a(n)) = A322813(n).
A000203(a(n)) = A322819(n).
A122111(a(n)) = A322820(n).
A000120(a(n)) = A322867(n).

A322813 a(n) = A001227(A122111(n)).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 3, 2, 1, 2, 1, 2, 3, 2, 1, 4, 1, 2, 3, 2, 1, 2, 4, 2, 3, 2, 1, 4, 1, 2, 3, 2, 4, 4, 1, 2, 3, 2, 1, 4, 1, 2, 3, 2, 1, 2, 5, 6, 3, 2, 1, 4, 4, 2, 3, 2, 1, 4, 1, 2, 3, 2, 4, 4, 1, 2, 3, 6, 1, 4, 1, 2, 6, 2, 5, 4, 1, 2, 3, 2, 1, 4, 4, 2, 3, 2, 1, 4, 5, 2, 3, 2, 4, 2, 1, 8, 3, 6, 1, 4, 1, 2, 6

Offset: 1

Antti Karttunen, Dec 27 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences computed from indices in prime factorization

Cf. A001227, A122111, A322819.

A001227(n) = numdiv(n>>valuation(n, 2));
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)};
A122111(n) = if(1==n,n,prime(bigomega(n))*A122111(A064989(n)));
A322813(n) = A001227(A122111(n));

a(n) = A001227(A122111(n)).

A324118 Sum of odd divisors in A156552(n): a(1) = 0, for n > 1, a(n) = A000593(A156552(n)) = A000203(A322993(n)).

Original entry on oeis.org

0, 1, 1, 4, 1, 6, 1, 8, 4, 13, 1, 12, 1, 18, 6, 24, 1, 14, 1, 20, 13, 48, 1, 24, 4, 84, 8, 48, 1, 32, 1, 32, 18, 176, 6, 40, 1, 258, 48, 56, 1, 38, 1, 68, 12, 800, 1, 48, 4, 31, 84, 132, 1, 30, 13, 72, 176, 1302, 1, 44, 1, 2736, 20, 104, 18, 96, 1, 304, 258, 42, 1, 72, 1, 4356, 14, 624, 6, 160, 1, 80, 24, 10928, 1, 124, 48, 20520, 800, 240, 1, 78, 13

Offset: 1

Antti Karttunen, Feb 20 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..4473
Index entries for sequences computed from indices in prime factorization

Cf. A000203, A000593, A156552, A246277, A322819, A322993, A323243, A324117.

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)};
A156552(n) = if(1==n, 0, if(!(n%2), 1+(2*A156552(n/2)), 2*A156552(A064989(n))));
A246277(n) = { if(1==n, 0, while((n%2), n = A064989(n)); (n/2)); };
A322993(n) = A156552(2*A246277(n));
A324118(n) = if(1==n, 0, sigma(A322993(n)));

a(1) = 0; for n > 1, a(n) = A000593(A156552(n)) = A000203(A322993(n)) = A323243(2*A246277(n)).

A322826 Lexicographically earliest such sequence a that a(i) = a(j) => A052126(i) = A052126(j) for all i, j.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 4, 2, 1, 3, 1, 2, 4, 5, 1, 6, 1, 3, 4, 2, 1, 5, 7, 2, 8, 3, 1, 6, 1, 9, 4, 2, 7, 10, 1, 2, 4, 5, 1, 6, 1, 3, 8, 2, 1, 9, 11, 12, 4, 3, 1, 13, 7, 5, 4, 2, 1, 10, 1, 2, 8, 14, 7, 6, 1, 3, 4, 12, 1, 15, 1, 2, 16, 3, 11, 6, 1, 9, 17, 2, 1, 10, 7, 2, 4, 5, 1, 13, 11, 3, 4, 2, 7, 14, 1, 18, 8, 19, 1, 6, 1, 5, 16

Offset: 1

Antti Karttunen, Dec 27 2018

nonn

Restricted growth sequence transform of A052126, or equally, of A322820.
For all i, j:
  A300226(i) = A300226(j) => a(i) = a(j),
  a(i) = a(j) => A322813(i) = A322813(j),
  a(i) = a(j) => A322819(i) = A322819(j).
For all i, j > 1:
  a(i) = a(j) => A001222(i) = A001222(j).

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

Cf. A001222, A006530, A052126, A305800, A300226, A319994, A319996, A322813, A322819, A322820.

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; };
A006530(n) = if(n>1, vecmax(factor(n)[, 1]), 1);
A052126(n) = (n/A006530(n));
v322826 = rgs_transform(vector(up_to,n,A052126(n)));
A322826(n) = v322826[n];

A322820 a(n) = A052126(n) * A006530(A052126(n)).

Original entry on oeis.org

1, 1, 1, 4, 1, 4, 1, 8, 9, 4, 1, 8, 1, 4, 9, 16, 1, 18, 1, 8, 9, 4, 1, 16, 25, 4, 27, 8, 1, 18, 1, 32, 9, 4, 25, 36, 1, 4, 9, 16, 1, 18, 1, 8, 27, 4, 1, 32, 49, 50, 9, 8, 1, 54, 25, 16, 9, 4, 1, 36, 1, 4, 27, 64, 25, 18, 1, 8, 9, 50, 1, 72, 1, 4, 75, 8, 49, 18, 1, 32, 81, 4, 1, 36, 25, 4, 9, 16, 1, 54, 49, 8, 9, 4, 25, 64, 1, 98, 27, 100, 1, 18, 1, 16, 75

Offset: 1

Antti Karttunen, Dec 27 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A000265, A006530, A052126, A070003 (positions where a(n) = n for n > 1), A122111, A319988, A322813, A322819, A322826 (restricted growth sequence transform).

A322820(n) = if(isprime(n),1,if(1>=omega(n),n,my(f=factor(n), y=#f~); if(1==f[y,2], f[y,2] = 0; f[y-1,2]++); factorback(f)));

A006530(n) = if(n>1, vecmax(factor(n)[, 1]), 1);
A052126(n) = (n/A006530(n));
A322820(n) = A052126(n)*A006530(A052126(n));

A000265(n) = (n>>valuation(n, 2));
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)};
A122111(n) = if(1==n,n,prime(bigomega(n))*A122111(A064989(n)));
A322820(n) = A122111(A000265(A122111(n)));

a(n) = A052126(n) * A006530(A052126(n)).
a(n) = A122111(A000265(A122111(n))).
A052126(a(n)) = A052126(n).
a(n) <= n.
For all n > 1, A010051(n) + A319988(a(n)) = 1.

A336315 The number of divisors in the conjugated prime factorization of n: a(n) = A000005(A122111(n)).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jul 18 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..12000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65539
Index entries for sequences computed from indices in prime factorization

Cf. A000005, A034444, A122111, A286621, A323173, A322813, A322819, A336316.

A336315(n) = if(1==n,n,my(p=apply(primepi,factor(n)[,1]~),m=1+p[1]); for(i=2, #p, m *= (1+p[i]-p[i-1])); (m));

a(n) = A000005(A122111(n)).

a(n) = A336316(n) + A034444(n).

A337204 Sum of the odd divisors of the primorial inflation of n.

Original entry on oeis.org

1, 1, 4, 1, 24, 4, 192, 1, 13, 24, 2304, 4, 32256, 192, 78, 1, 580608, 13, 11612160, 24, 624, 2304, 278691840, 4, 403, 32256, 40, 192, 8360755200, 78, 267544166400, 1, 7488, 580608, 3224, 13, 10166678323200, 11612160, 104832, 24, 427000489574400, 624, 18788021541273600, 2304, 240, 278691840, 901825033981132800, 4, 22971, 403

Offset: 1

Antti Karttunen, Aug 22 2020

nonn

Row 1 of A337205.
Cf. A000593, A034386, A108951, A337203.
Cf. also A322819.

Array[DivisorSum[Apply[Times, FactorInteger[#] /. {p_, e_} /; e > 0 :> Apply[Times, Prime@ Range@ PrimePi@ p]^e], # &, OddQ] &, 50] (* Michael De Vlieger, Aug 27 2020 *)

A000593(n) = sigma(n>>valuation(n, 2));
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
A337204(n) = A000593(A108951(n));

a(n) = A000593(A108951(n)).

cp's OEIS Frontend

A323173 Sum of divisors computed for conjugated prime factorization: a(n) = A000203(A122111(n)).

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A322865 a(n) = A000265(A122111(n)).

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A322813 a(n) = A001227(A122111(n)).

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A324118 Sum of odd divisors in A156552(n): a(1) = 0, for n > 1, a(n) = A000593(A156552(n)) = A000203(A322993(n)).

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A322826 Lexicographically earliest such sequence a that a(i) = a(j) => A052126(i) = A052126(j) for all i, j.

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A322820 a(n) = A052126(n) * A006530(A052126(n)).

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A336315 The number of divisors in the conjugated prime factorization of n: a(n) = A000005(A122111(n)).

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A337204 Sum of the odd divisors of the primorial inflation of n.

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