A324545 -id:A324545 - cp's OEIS frontend

A324544 a(n) = A009194(A250246(n)) = gcd(A250246(n), A324545(n)).

1, 1, 1, 1, 1, 6, 1, 1, 1, 2, 1, 4, 1, 2, 3, 1, 1, 3, 1, 2, 1, 2, 1, 12, 1, 2, 1, 28, 1, 6, 1, 1, 3, 2, 1, 1, 1, 2, 3, 10, 1, 6, 1, 4, 1, 2, 1, 4, 1, 1, 1, 2, 1, 6, 1, 8, 1, 2, 1, 12, 1, 2, 1, 1, 1, 18, 1, 2, 15, 2, 1, 3, 1, 2, 3, 4, 1, 6, 1, 2, 3, 2, 1, 4, 1, 2, 3, 4, 1, 3, 1, 4, 1, 2, 1, 12, 1, 1, 1, 1, 1, 6, 1, 2, 1

Offset: 1

Antti Karttunen, Mar 06 2019

nonn

Fixed points are: 1, 6, 28, 120, 496, 8128, etc,

Positions where a(n) == A250246(n) are: 1, 6, 28, 120, 496, 864, 8128, 11424, 15240, ..., which is sequence A250245(A007691(n)) sorted into ascending order.

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

Cf. A007691, A009194, A250246, A252754, A324394, A324545, A324546.

Differs from A009194 for the first time at n=39. Here a(39) = 3.

up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
A020639(n) = if(n>1, if(n>n=factor(n, 0)[1, 1], n, factor(n)[1, 1]), 1); \\ From A020639
A055396(n) = if(1==n,0,primepi(A020639(n)));
v078898 = ordinal_transform(vector(up_to,n,A020639(n)));
A078898(n) = v078898[n];
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A250246(n) = if(1==n,n,my(k = 2*A250246(A078898(n)), r = A055396(n)); if(1==r, k, while(r>1, k = A003961(k); r--); (k)));
A009194(n) = gcd(n, sigma(n));
A324544(n) = A009194(A250246(n));

a(n) = A009194(A250246(n)) = gcd(A250246(n), A324545(n)).

a(n) = A324394(A252754(n)).

A323243 a(1) = 0; for n > 1, a(n) = A000203(A156552(n)).

Original entry on oeis.org

0, 1, 3, 4, 7, 6, 15, 8, 12, 13, 31, 12, 63, 18, 18, 24, 127, 14, 255, 20, 39, 48, 511, 24, 28, 84, 24, 48, 1023, 32, 2047, 32, 54, 176, 42, 40, 4095, 258, 144, 56, 8191, 38, 16383, 68, 36, 800, 32767, 48, 60, 31, 252, 132, 65535, 30, 91, 72, 528, 1302, 131071, 44, 262143, 2736, 60, 104, 126, 96, 524287, 304, 774, 42, 1048575, 72, 2097151, 4356, 42

Offset: 1

Antti Karttunen, Jan 10 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000 (based on Hans Havermann's factorization of A156552)
Index entries for sequences related to binary expansion of n
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Cf. A000203, A156552, A323244, A323247, A323248, A324118, A324543 (Möbius transform), A324396, A324823.

Cf. A323173, A324054, A324184, A324545 for other permutations of sigma, and also A324573, A324653.

Array[If[# == 0, 0, DivisorSigma[1, #]] &@ Floor@ Total@ Flatten@ MapIndexed[#1 2^(#2 - 1) &, Flatten[Table[2^(PrimePi@ #1 - 1), {#2}] & @@@ FactorInteger@ #]] &, 75] (* Michael De Vlieger, Apr 21 2019 *)

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))));
A323243(n) = if(1==n, 0, sigma(A156552(n)));

\\ For computing terms a(n), with n > ~4000 use Hans Havermann's factorization file https://oeis.org/A156552/a156552.txt
v156552sigs = readvec("a156552.txt"); \\ First read it in as a PARI-vector.
A323243(n) = if(n<=2,n-1,my(prsig=v156552sigs[n],ps=prsig[1],es=prsig[2]); prod(i=1,#ps,((ps[i]^(1+es[i]))-1)/(ps[i]-1))); \\ Then play sigma
\\ Antti Karttunen, Mar 15 2019

from sympy import divisor_sigma, primepi, factorint
def A323243(n): return divisor_sigma(sum((1< 1 else 0 # Chai Wah Wu, Mar 10 2023

a(1) = 0; for n > 1, a(n) = A000203(A156552(n)).
a(n) = 2*A156552(n) - A323244(n).
a(n) = A323247(n) - A323248(n).
From Antti Karttunen, Mar 12 2019: (Start)
a(A000040(n)) = A000225(n).
a(n) = Sum_{d|n} A324543(d).
For n > 1, a(2*A246277(n)) = A324118(n).
gcd(a(n), A156552(n)) = A324396(n).
A000035(a(n)) = A324823(n).
(End)

A324546 An analog of deficiency (A033879) for nonstandard factorization based on the sieve of Eratosthenes (A083221).

Original entry on oeis.org

1, 1, 2, 1, 4, 0, 6, 1, 5, 2, 10, -4, 12, 4, 6, 1, 16, -3, 18, -2, 14, 8, 22, -12, 19, 10, 10, 0, 28, -12, 30, 1, 12, 14, 22, -19, 36, 16, 18, -10, 40, -12, 42, 4, 41, 20, 46, -28, 41, 7, 26, 6, 52, -12, 94, -8, 22, 26, 58, -48, 60, 28, 22, 1, 38, -54, 66, 10, 30, -4, 70, -51, 72, 34, 30, 12, 58, -12, 78, -26, 42, 38, 82, -64, 102, 40, 18, -4, 88

Offset: 1

Antti Karttunen, Mar 06 2019

sign

Even positions for zeros is given by the even terms of A000396, because they are among the fixed points of permutation A250246. Whether there are any zeros in odd positions depends on whether there are any odd perfect numbers. If such zeros exist, they would not necessarily be in the same positions as in A033879.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65539
Index entries for sequences generated by sieves
Index entries for sequences related to sigma(n)

Cf. A000396, A033879, A083221, A250246, A324535, A324545.

Cf. also A323244, A323174, A324574, A324575.

up_to = 65539;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
A020639(n) = if(n>1, if(n>n=factor(n, 0)[1, 1], n, factor(n)[1, 1]), 1); \\ From A020639
A055396(n) = if(1==n,0,primepi(A020639(n)));
v078898 = ordinal_transform(vector(up_to,n,A020639(n)));
A078898(n) = v078898[n];
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A250246(n) = if(1==n,n,my(k = 2*A250246(A078898(n)), r = A055396(n)); if(1==r, k, while(r>1, k = A003961(k); r--); (k)));
A324546(n) = { my(k=A250246(n)); (k+k - sigma(k)); };

a(n) = A033879(A250246(n)) = 2*A250246(n) - A324545(n).

a(n) = A250246(n) - A324535(n).

A331733 a(n) = sigma(A225546(n)), where sigma is the sum of divisors.

Original entry on oeis.org

1, 3, 7, 4, 31, 15, 511, 12, 13, 63, 131071, 28, 8589934591, 1023, 127, 6, 36893488147419103231, 39, 680564733841876926926749214863536422911, 124, 2047, 262143, 231584178474632390847141970017375815706539969331281128078915168015826259279871, 60, 121, 17179869183, 91, 2044

Offset: 1

Antti Karttunen, Feb 02 2020

nonn

Cf. A000203, A048675, A225546, A331309, A331734, A331735, A331741.

Cf. A323243, A323173, A324054, A324184, A324545 for other permutations of sigma, and also A324573, A324653.

Array[If[# == 1, 1, DivisorSigma[1, #] &@ Apply[Times, Flatten@ Map[Function[{p, e}, Map[Prime[Log2@ # + 1]^(2^(PrimePi@ p - 1)) &, DeleteCases[NumberExpand[e, 2], 0]]] @@ # &, FactorInteger[#]]]] &, 28] (* Michael De Vlieger, Feb 08 2020 *)

A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
A331733(n) = if(1==n,1,my(f=factor(n),u=#binary(vecmax(f[, 2])),prods=vector(u,x,1),m=1,e); for(i=1,u,for(k=1,#f~, if(bitand(f[k,2],m),prods[i] *= f[k,1])); m<<=1); prod(i=1,u,(prime(i)^(1+A048675(prods[i]))-1)/(prime(i)-1)));

a(n) = A000203(A225546(n)).

For all n >= 1, A000035(a(A016754(n))) = 1. [Result is odd for all odd squares]

A324535 An analog of sigma(n)-n (A001065) for nonstandard factorization based on the sieve of Eratosthenes (A083221).

Original entry on oeis.org

0, 1, 1, 3, 1, 6, 1, 7, 4, 8, 1, 16, 1, 10, 9, 15, 1, 21, 1, 22, 13, 14, 1, 36, 6, 16, 11, 28, 1, 42, 1, 31, 33, 20, 13, 55, 1, 22, 15, 50, 1, 66, 1, 40, 40, 26, 1, 76, 8, 43, 49, 46, 1, 54, 31, 64, 41, 32, 1, 108, 1, 34, 17, 63, 17, 144, 1, 58, 105, 74, 1, 123, 1, 40, 21, 64, 19, 78, 1, 106, 57, 44, 1, 172, 73, 46, 87, 92, 1, 201, 57, 76, 121

Offset: 1

Antti Karttunen, Mar 08 2019

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 generated by sieves
Index entries for sequences related to sigma(n)

Cf. A001065, A250246, A324545, A324546.

up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
A020639(n) = if(n>1, if(n>n=factor(n, 0)[1, 1], n, factor(n)[1, 1]), 1); \\ From A020639
A055396(n) = if(1==n,0,primepi(A020639(n)));
v078898 = ordinal_transform(vector(up_to,n,A020639(n)));
A078898(n) = v078898[n];
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A250246(n) = if(1==n,n,my(k = 2*A250246(A078898(n)), r = A055396(n)); if(1==r, k, while(r>1, k = A003961(k); r--); (k)));
A001065(n) = (sigma(n)-n);
A324535(n) = A001065(A250246(n));

a(n) = A001065(A250246(n)) = A324545(n) - A250246(n).

a(n) = A250246(n) - A324546(n).

cp's OEIS Frontend

A324544 a(n) = A009194(A250246(n)) = gcd(A250246(n), A324545(n)).

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A323243 a(1) = 0; for n > 1, a(n) = A000203(A156552(n)).

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A324546 An analog of deficiency (A033879) for nonstandard factorization based on the sieve of Eratosthenes (A083221).

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PARI

Formula

A331733 a(n) = sigma(A225546(n)), where sigma is the sum of divisors.

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Mathematica

PARI

Formula

A324535 An analog of sigma(n)-n (A001065) for nonstandard factorization based on the sieve of Eratosthenes (A083221).

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Author

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Links

Crossrefs

Programs

PARI

Formula