A332221 -id:A332221 - cp's OEIS frontend

A347380 Length of the common prefix in the binary expansions of A156552(n) and A332221(n) = A156552(sigma(n)).

0, 1, 1, 1, 2, 3, 1, 1, 1, 1, 2, 2, 4, 2, 3, 1, 1, 1, 3, 4, 1, 1, 2, 3, 1, 3, 1, 6, 2, 1, 1, 1, 2, 1, 3, 1, 8, 2, 4, 2, 3, 2, 5, 3, 2, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 3, 2, 2, 2, 11, 2, 3, 1, 3, 1, 7, 3, 2, 1, 1, 1, 12, 7, 1, 2, 3, 3, 3, 3, 2, 3, 3, 3, 1, 4, 2, 2, 2, 2, 3, 3, 1, 1, 2, 2, 1, 1, 4, 1, 6, 1, 6, 2, 3

Offset: 1

Antti Karttunen, Aug 30 2021

nonn

Cf. A000203, A156552, A252464, A332221, A347381, A348040.

Abincompreflen(n, m) = { my(x=binary(n),y=binary(m),u=min(#x,#y)); for(i=1,u,if(x[i]!=y[i],return(i-1))); (u);};
A156552(n) = {my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res}; \\ From A156552
A347380(n) = Abincompreflen(A156552(n), A156552(sigma(n)));

a(n) = A252464(n) - A347381(n).

a(n) = A348040(n, A000203(n)). - Antti Karttunen, Jan 30 2022

A332216 Fixed points of A332221: Numbers k such that A156552(sigma(k)) is equal to k.

Original entry on oeis.org

2, 3, 5, 7, 11, 19, 23, 31, 47, 55, 79, 87, 127, 191, 383, 1279, 5119, 6143, 8191, 20479, 81919, 131071, 524287, 786431, 1310719, 2147483647

Offset: 1

Antti Karttunen, Feb 10 2020

Equally, numbers k such that sigma(k) is equal to A005940(1+k).
The primes in this sequence are obtained by subtracting 1 from those terms of A029747 that are one more than a prime.
Questions: Are there other composite terms than 55 and 87? Are there other even terms than 2? (All such even terms should also occur in A332218).

Cf. A000203, A005940, A029747, A156552, A332217, A332218, A332221.

Subsequences: A000668, A007505, A050522.

Select[Range[10^5], DivisorSigma[1, #] == Block[{p = Partition[Split[Join[IntegerDigits[#, 2], {2}]], 2], q}, Times @@ Flatten[Table[q = Take[p, -i]; Prime[Count[Flatten[q], 0] + 1]^q[[1, 1]], {i, Length[p]}]]] &] (* Michael De Vlieger, Feb 12 2020, after Robert G. Wilson v at A005940 *)

A332218 Numbers k such that A332221(k) = A156552(sigma(k)) is 2*{an odd square}.

Original entry on oeis.org

2, 162, 441, 2704, 4225, 275194921

Offset: 1

Antti Karttunen, Feb 11 2020

Any even term of A332216 must occur also in this sequence.

			  a(n)              -> sigma(a(n))              -> A156552(sigma(a(n)))
     2 = 2^1 * 1^2  ->    3 = 3^1               ->      2 = 2^1 * 1^1,
   162 = 2^1 * 3^4  ->  363 = 3^1 * 11^2        ->     98 = 2^1 * 7^2,
   441 = 3^2 * 7^2  ->  741 = 3^1 * 13^1 * 19^1 ->    578 = 2^1 * 17^2,
  2704 = 2^4 * 13^2 -> 5673 = 3^1 * 31^1 * 61^1 -> 526338 = 2^1 * 3^6 * 19^2,
  4225 = 5^2 * 13^2 -> 5673 = 3^1 * 31^1 * 61^1 -> 526338 = 2^1 * 3^6 * 19^2,
and
275194921 = 53^2 * 313^2 -> 281384229 = 3^1 * 7^1 * 181^2 * 409^1 -> 9671406556943421676716050 = 2^1 * 5^2 * 7^2 * 62829235873^2.

Index entries for sequences related to sigma(n)

Cf. A000203, A156552, A332216, A332217, A332221.

Subsequence of A332217 ⊂ A067051 ⊂ A028982.

Select[Range@ 5000, And[IntegerQ[#], OddQ[#]] &@ Sqrt[#/2] &@ Floor@ Total@ Flatten@ MapIndexed[#1 2^(#2 - 1) &, Flatten[Table[2^(PrimePi@ #1 - 1), {#2}] & @@@ FactorInteger@ DivisorSigma[1, #]]] &] (* Michael De Vlieger, Feb 12 2020 *)

\\ Needs also code from A156552:
istosq(n) = ((1==valuation(n,2))&&issquare(n/2));
for(n=1,2^25,if(istosq(A156552(sigma(n*n))),print1(n*n,", ")); if(istosq(A156552(sigma(2*n*n))),print1(2*n*n,", ")));

A356320 Length of the common prefix in binary expansions of n and A332221(n) = A156552(sigma(A005940(1+n))).

Original entry on oeis.org

0, 1, 1, 1, 2, 3, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 2, 2, 1, 4, 3, 1, 2, 3, 1, 1, 1, 1, 1, 1, 2, 1, 4, 1, 2, 6, 1, 2, 3, 2, 3, 1, 3, 2, 2, 2, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 3, 4, 3, 3, 1, 4, 2, 3, 1, 1, 3, 3, 3, 6, 3, 2, 1, 3, 2, 1, 1, 2, 3, 2, 2, 2, 2, 2, 4, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Antti Karttunen, Aug 06 2022

nonn

Cf. A000203, A005940, A070939, A156552, A324054, A347380, A356321, A332222.

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
Abincompreflen(n, m) = { my(x=binary(n),y=binary(m),u=min(#x,#y)); for(i=1,u,if(x[i]!=y[i],return(i-1))); (u);};
A156552(n) = {my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res}; \\ From A156552
A347380(n) = Abincompreflen(A156552(n), A156552(sigma(n)));
A356320(n) = A347380(A005940(1+n));
\\ Alternatively as:
A356320(n) = Abincompreflen(n, A156552(sigma(A005940(1+n))));

A347381 Distance from n to the nearest common ancestor of n and sigma(n) in the Doudna-tree (A005940).

Original entry on oeis.org

0, 0, 1, 1, 1, 0, 3, 2, 2, 3, 3, 2, 2, 3, 1, 3, 6, 3, 5, 1, 4, 5, 7, 2, 3, 4, 3, 0, 8, 4, 10, 4, 4, 7, 2, 4, 4, 7, 3, 4, 10, 4, 9, 4, 3, 9, 13, 4, 4, 4, 7, 7, 15, 4, 5, 5, 6, 9, 15, 4, 7, 10, 3, 5, 4, 6, 12, 6, 8, 5, 19, 5, 9, 6, 4, 8, 3, 5, 19, 4, 3, 11, 20, 4, 7, 11, 9, 6, 22, 4, 4, 8, 11, 15, 7, 5, 24, 5, 3, 5, 20

Offset: 1

Antti Karttunen, Aug 30 2021

a(n) tells about the degree of relatedness between n and sigma(n) in Doudna tree (see the illustration in A005940). It is 0 for those n where sigma(n) is one of the descendants of n, 1 for those n where the nearest common ancestor of n and sigma(n) is the parent of n, 2 for those n where the nearest common ancestor of n and sigma(n) is the grandparent of n, and so on.

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 computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Indices of 0 .. 5 in this sequence are given by {2} U A336702, A347391, A347392, A347393, A347394, A374465.
Cf. A000203, A027687, A156552, A252463, A252464, A332221, A347380, A347383, A347384, A347390, A374481 [a(prime(n))], A374482 (indices of records), A374483 (record values).
Cf. A374200, A374204, A374214, A374218, A374219.
Cf. also A336834.

A000523(n) = logint(n,2);
Abincompreflen(x, y) = if(!x || !y, 0, my(xl=A000523(x), yl=A000523(y), s=min(xl,yl), k=0); x >>= (xl-s); y >>= (yl-s); while(s>=0 && !bitand(1,bitxor(x>>s,y>>s)), s--; k++); (k));
A156552(n) = {my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res}; \\ From A156552
A061395(n) = if(n>1, primepi(vecmax(factor(n)[, 1])), 0);
A252464(n) = if(1==n,0,(bigomega(n) + A061395(n) - 1));
A347381(n) = (A252464(n)-Abincompreflen(A156552(n), A156552(sigma(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)};
A252463(n) = if(!(n%2),n/2,A064989(n));
A347381(n) = if(1==n,0, my(lista=List([]), i, k=n, stemvec, stemlen, sbr=sigma(n)); while(k>1, listput(lista,k); k = A252463(k)); stemvec = Vecrev(Vec(lista)); stemlen = #stemvec; while(1, if((i=vecsearch(stemvec,sbr))>0, return(stemlen-i)); sbr = A252463(sbr)));

a(n) = A252464(n) - A347380(n), where A347380(n) is the length of the common prefix in binary expansions of A156552(n) and A332221(n) = A156552(sigma(n)).

Name changed, old name is now in formula section. - Antti Karttunen, Jul 09 2024

A058063 Number of prime factors (when counted with multiplicity) of sigma(n), the sum of divisors of n.

Original entry on oeis.org

0, 1, 2, 1, 2, 3, 3, 2, 1, 3, 3, 3, 2, 4, 4, 1, 3, 2, 3, 3, 5, 4, 4, 4, 1, 3, 4, 4, 3, 5, 5, 3, 5, 4, 5, 2, 2, 4, 4, 4, 3, 6, 3, 4, 3, 5, 5, 3, 2, 2, 5, 3, 4, 5, 5, 5, 5, 4, 4, 5, 2, 6, 4, 1, 4, 6, 3, 4, 6, 6, 5, 3, 2, 3, 3, 4, 6, 5, 5, 3, 2, 4, 4, 6, 5, 4, 5, 5, 4, 4, 5, 5, 7, 6, 5, 5, 3, 3, 4, 2, 3, 6, 4, 4, 7

Offset: 1

Labos Elemer, Nov 23 2000

nonn

			n=35, sigma(35) = 35 + 5 + 7 + 1 = 48 = 2*2*2*2*3, so a(35)=5.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to sums of divisors

Cf. A000120, A000203, A001222, A331750, A332221.

with(numtheory):a:=proc(n) if n=0 then 0 else bigomega(sigma(n)) fi end: seq(a(n), n=1..105); # Zerinvary Lajos, Apr 11 2008

Array[PrimeOmega@ DivisorSigma[1, #] &, 105] (* Michael De Vlieger, Nov 08 2017 *)

a(n) = bigomega(sigma(n)); \\ Michel Marcus, Nov 07 2017

a(n) = A001222(A000203(n)).
From Antti Karttunen, Feb 12 2020: (Start)
Additive with a(p^e) = A001222(A000203(p^e)) = A001222(1 + p + p^2 + ... + p^e).
a(n) = A000120(A332221(n)).
(End)

Offset corrected by Antti Karttunen, Nov 07 2017

A347879 The nearest common ancestor of n and sigma(n) in the Doudna tree (A005940).

Original entry on oeis.org

1, 2, 2, 2, 3, 6, 2, 2, 2, 2, 3, 3, 7, 3, 6, 2, 2, 2, 5, 10, 2, 2, 3, 6, 2, 5, 2, 28, 3, 2, 2, 2, 3, 2, 6, 2, 19, 3, 7, 3, 5, 3, 11, 5, 3, 2, 3, 3, 2, 2, 2, 2, 2, 2, 2, 3, 5, 3, 3, 3, 31, 3, 5, 2, 5, 2, 17, 5, 3, 2, 2, 2, 37, 17, 2, 3, 6, 5, 5, 5, 4, 5, 5, 5, 2, 7, 3, 3, 3, 3, 5, 5, 2, 2, 3, 3, 2, 2, 7, 2, 13, 2

Offset: 1

Antti Karttunen, Oct 13 2021

nonn

The fixed points of this sequence is given by the union of {2} and A336702.
The positions x such that a(x) = A252463(x) is given by the union of {1} and A347391.
The positions x such that a(x) = A252463(A252463(x)) is given by the union of {1} and A347392.

Cf. A000203, A005940, A252463, A332221, A336702, A347381, A347391, A347392, A348040, A348041.

A347879(n) = A348041sq(n,sigma(n)); \\ Needs also code from A348041.

a(n) = A348041(n, A000203(n)).

A332222 a(n) = A156552(sigma(A005940(1+n))).

Original entry on oeis.org

0, 2, 3, 8, 5, 11, 32, 10, 7, 13, 23, 35, 1024, 66, 39, 1024, 11, 23, 31, 37, 47, 55, 133, 43, 258, 2050, 4099, 72, 267, 87, 48, 38, 17, 27, 47, 71, 55, 95, 263, 45, 95, 111, 191, 151, 8199, 269, 175, 4099, 264, 518, 1035, 2056, 1037, 8203, 2080, 138, 207, 539, 1071, 167, 1048592, 98, 291, 1073741824, 13, 37, 71, 75, 75, 111

Offset: 0

Antti Karttunen, Feb 12 2020

nonn

Cf. A000203, A005940, A156552, A324054, A332221.

Cf. also A332223.

Array[Floor@ Total@ Flatten@ MapIndexed[#1 2^(#2 - 1) &, Flatten[Table[2^(PrimePi@ #1 - 1), {#2}] & @@@ FactorInteger@ DivisorSigma[1, #]]] &@ Block[{p = Partition[Split[Join[IntegerDigits[#, 2], {2}]], 2], q}, Times @@ Flatten[Table[q = Take[p, -i]; Prime[Count[Flatten[q], 0] + 1]^q[[1, 1]], {i, Length[p]}]]] &, 70, 0] (* Michael De Vlieger, Feb 12 2020, after Robert G. Wilson v at A005940 *)

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ From A005940
A156552(n) = {my(f = factor(n), p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res}; \\ From A156552
A332222(n) = A156552(sigma(A005940(1+n)));

a(n) = A156552(A000203(A005940(1+n))).

a(n) = A332221(A005940(1+n)) = A156552(A324054(n)).

A347876 a(n) = A323905(sigma(n)), where A323905(x) = A156552(x) - A048675(x).

Original entry on oeis.org

0, 0, 1, 0, 2, 7, 4, 4, 0, 8, 7, 25, 8, 18, 18, 0, 8, 32, 13, 26, 26, 21, 18, 35, 0, 26, 32, 60, 14, 48, 26, 26, 41, 22, 41, 32, 128, 35, 60, 36, 26, 88, 49, 63, 98, 48, 41, 3073, 128, 1024, 48, 32, 22, 78, 48, 78, 71, 36, 35, 138, 1024, 88, 228, 0, 63, 103, 193, 64, 88, 103, 48, 100, 2048, 386, 3073, 69, 88, 138, 71

Offset: 1

Antti Karttunen, Sep 19 2021

nonn

Cf. A000203, A048675, A156552, A323905, A331750, A332221, A347875.

A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; }; \\ From A048675
A156552(n) = {my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res}; \\ From A156552
A323905(n) = (A156552(n) - A048675(n));
A347876(n) = A323905(sigma(n));

a(n) = A323905(A000203(n)).

a(n) = A332221(n) - A331750(n).

A347875 Numbers k such that A323905(sigma(k)) is equal to A323905(2*k).

Original entry on oeis.org

1, 6, 21, 28, 496, 8128

Offset: 1

Antti Karttunen, Sep 19 2021

Numbers k such that A323905(sigma(k)) = A332221(k) - A331750(k) is equal to 2*A156552(k) - A048675(k) = A156552(k) + A323905(k).

Index entries for sequences where odd perfect numbers must occur, if they exist at all

Cf. A000203, A048675, A156552, A323905, A347876.

Cf. also A000396 (subsequence), A331751, A347392.

A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; }; \\ From A048675
A156552(n) = {my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res}; \\ From A156552
A323905(n) = (A156552(n) - A048675(n));
isA347875(n) = (A323905(sigma(n))==A323905(2*n));

cp's OEIS Frontend

A347380 Length of the common prefix in the binary expansions of A156552(n) and A332221(n) = A156552(sigma(n)).

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A332216 Fixed points of A332221: Numbers k such that A156552(sigma(k)) is equal to k.

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A332218 Numbers k such that A332221(k) = A156552(sigma(k)) is 2*{an odd square}.

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A356320 Length of the common prefix in binary expansions of n and A332221(n) = A156552(sigma(A005940(1+n))).

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A347381 Distance from n to the nearest common ancestor of n and sigma(n) in the Doudna-tree (A005940).

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A058063 Number of prime factors (when counted with multiplicity) of sigma(n), the sum of divisors of n.

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A347879 The nearest common ancestor of n and sigma(n) in the Doudna tree (A005940).

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A332222 a(n) = A156552(sigma(A005940(1+n))).

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