Antti Karttunen, - cp's OEIS frontend

A387408 Partial sums of A387412, where A387412(n) is the length of the maximal common prefix of the binary expansions of n and A003961(n).

1, 2, 3, 6, 7, 9, 10, 11, 12, 16, 17, 18, 19, 20, 21, 23, 26, 31, 33, 34, 35, 37, 38, 39, 43, 46, 48, 50, 53, 55, 56, 57, 62, 63, 66, 67, 69, 72, 74, 77, 81, 85, 88, 89, 92, 95, 96, 100, 102, 103, 104, 105, 107, 108, 109, 110, 116, 117, 120, 121, 122, 124, 125, 127, 128, 129, 133, 135, 138, 139, 142, 144, 148, 149

Offset: 1

Antti Karttunen, Sep 03 2025

Cf. A000203, A387412, A387407, A387409.

Cf. also A387425.

up_to = 65537;
A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A387412(n) = { my(a=binary(n), b=binary(A003961(n)), i=1); while(i<=#a,if(a[i]!=b[i],return(i-1)); i++); (#a); };
A387408list(up_to) = { my(v=vector(up_to)); v[1] = A387412(1); for(n=2,up_to,v[n] = v[n-1]+A387412(n)); (v); };
v387408 = A387408list(up_to);
A387408(n) = v387408[n];

a(1) = 1; and for n > 1, a(n) = a(n-1) + A387412(n).

a(n) = A387407(n) - A387409(n).

A387407 Partial sums of A387422, where A387422 is the length of the maximal common prefix of the binary expansions of n and sigma(n).

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 9, 10, 11, 13, 14, 16, 18, 20, 22, 23, 26, 30, 32, 36, 38, 40, 41, 43, 45, 46, 47, 52, 55, 56, 57, 58, 59, 60, 61, 63, 67, 68, 69, 72, 76, 77, 80, 83, 85, 87, 88, 90, 92, 93, 94, 97, 101, 103, 104, 107, 108, 109, 112, 113, 117, 119, 121, 122, 124, 127, 131, 132, 133, 136, 139, 140, 145, 146, 147

Offset: 1

Antti Karttunen, Sep 03 2025

Cf. A003961, A387422, A387408, A387409.

Cf. also A387424.

up_to = 65537;
A387422(n) = { my(a=binary(n), b=binary(sigma(n)), i=1); while(i<=#a,if(a[i]!=b[i],return(i-1)); i++); (#a); };
A387407list(up_to) = { my(v=vector(up_to)); v[1] = A387422(1); for(n=2,up_to,v[n] = v[n-1]+A387422(n)); (v); };
v387407 = A387407list(up_to);
A387407(n) = v387407[n];

a(1) = 1; and for n > 1, a(n) = a(n-1) + A387422(n).

a(n) = A387408(n) + A387409(n).

A387409 Partial sums of A387422 minus partial sums of A387412.

Original entry on oeis.org

0, 0, 0, -2, -2, -1, -1, -1, -1, -3, -3, -2, -1, 0, 1, 0, 0, -1, -1, 2, 3, 3, 3, 4, 2, 0, -1, 2, 2, 1, 1, 1, -3, -3, -5, -4, -2, -4, -5, -5, -5, -8, -8, -6, -7, -8, -8, -10, -10, -10, -10, -8, -6, -5, -5, -3, -8, -8, -8, -8, -5, -5, -4, -5, -4, -2, -2, -3, -5, -3, -3, -4, -3, -3, -3, -1, -3, -2, -2, -1, -2, -3, -2

Offset: 1

Antti Karttunen, Sep 03 2025

Also partial sums of A387413 minus partial sums of A387423.
This sequence gives some measure of how much longer the common prefix of the binary expansions of n and sigma(n) is - on average - than the common prefix of the binary expansions of n and A003961(n).
Question: Does the sequence eventually grow without limit and is it just because A003961(n) >= A000203(n)? Does ratio a(n)/n converge to any limit?

Cf. A000203, A003961, A387407, A387408, A387412, A387413, A387422, A387423, A387424, A387425.

PARI
```
A387409(n) = (A387407(n) - A387408(n));
```

a(n) = A387407(n) - A387408(n).

a(n) = A387425(n) - A387424(n).

A387425 Partial sums of A387413.

Original entry on oeis.org

0, 1, 2, 2, 4, 5, 7, 10, 13, 13, 16, 19, 22, 25, 28, 31, 33, 33, 36, 40, 44, 47, 51, 55, 56, 58, 61, 64, 66, 69, 73, 78, 79, 84, 87, 92, 96, 99, 103, 106, 108, 110, 113, 118, 121, 124, 129, 131, 135, 140, 145, 150, 154, 159, 164, 169, 169, 174, 177, 182, 187, 191, 196, 201, 207, 213, 216, 221, 225, 231, 235, 240, 243

Offset: 1

Antti Karttunen, Sep 01 2025

Cf. A003961, A387413.

Cf. also A387424.

up_to = 65537;
A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A387413(n) = { my(a=binary(n), b=binary(A003961(n)), i=1); while(i<=#a,if(a[i]!=b[i],return(#a-(i-1))); i++); (0); };
A387425list(up_to) = { my(v=vector(up_to)); v[1] = 0; for(n=2,up_to,v[n] = v[n-1]+A387413(n)); (v); };
v387425 = A387425list(up_to);
A387425(n) = v387425[n];

A387424 Partial sums of A387423.

Original entry on oeis.org

0, 1, 2, 4, 6, 6, 8, 11, 14, 16, 19, 21, 23, 25, 27, 31, 33, 34, 37, 38, 41, 44, 48, 51, 54, 58, 62, 62, 64, 68, 72, 77, 82, 87, 92, 96, 98, 103, 108, 111, 113, 118, 121, 124, 128, 132, 137, 141, 145, 150, 155, 158, 160, 164, 169, 172, 177, 182, 185, 190, 192, 196, 200, 206, 211, 215, 218, 224, 230, 234, 238, 244

Offset: 1

Antti Karttunen, Sep 01 2025

Cf. A000203, A387423.

Cf. also A387425.

A387423[n_] := BitLength[n] - LengthWhile[Transpose[IntegerDigits[{n, DivisorSigma[1, n]}, 2][[All, ;; BitLength[n]]]], Equal @@ # &];
Accumulate[Array[A387423, 100]] (* Paolo Xausa, Sep 03 2025 *)

up_to = 65537;
A387423(n) = { my(a=binary(n), b=binary(sigma(n)), i=1); while(i<=#a,if(a[i]!=b[i],return(#a-(i-1))); i++); (0); };
A387424list(up_to) = { my(v=vector(up_to)); v[1] = 0; for(n=2,up_to,v[n] = v[n-1]+A387423(n)); (v); };
v387424 = A387424list(up_to);
A387424(n) = v387424[n];

a(1) = 0; for n > 1, a(n) = a(n-1) + A387423(n).

A387422 The length of the maximal common prefix of the binary expansions of n and sigma(n), where sigma is the sum of divisors function.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Sep 01 2025

Cf. A000203, A000523, A387423.

Cf. also A347380, A387412.

A387422[n_] := LengthWhile[Transpose[IntegerDigits[{n, DivisorSigma[1, n]}, 2][[All, ;; BitLength[n]]]], Equal @@ # &];
Array[A387422, 100] (* Paolo Xausa, Sep 03 2025 *)

A387422(n) = { my(a=binary(n), b=binary(sigma(n)), i=1); while(i<=#a,if(a[i]!=b[i],return(i-1)); i++); (#a); };

from os.path import commonprefix
from sympy import divisor_sigma
def A387422(n): return len(commonprefix([bin(n)[2:],bin(divisor_sigma(n))[2:]])) # Chai Wah Wu, Sep 03 2025

a(n) = (1+A000523(n)) - A387423(n).

A387413 The length of binary expansion of n minus the length of the maximal common prefix of the binary expansions of n and A003961(n), where A003961 is fully multiplicative with a(p) = nextprime(p).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Sep 01 2025

Cf. A000523, A003961, A387412, A387414 (positions of 0's).

Cf. also A387423.

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A387413(n) = { my(a=binary(n), b=binary(A003961(n)), i=1); while(i<=#a,if(a[i]!=b[i],return(#a-(i-1))); i++); (0); };

a(n) = (1+A000523(n)) - A387412(n).

A387412 The length of the maximal common prefix of the binary expansions of n and A003961(n), where A003961 is fully multiplicative with a(p) = nextprime(p).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Sep 01 2025

Cf. A000523, A003961, A387413.

Cf. also A387422.

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A387412(n) = { my(a=binary(n), b=binary(A003961(n)), i=1); while(i<=#a,if(a[i]!=b[i],return(i-1)); i++); (#a); };

from os.path import commonprefix
from math import prod
from sympy import factorint, nextprime
def A387412(n): return len(commonprefix([bin(n)[2:],bin(prod(nextprime(p)**e for p, e in factorint(n).items()))[2:]])) # Chai Wah Wu, Sep 03 2025

a(n) = (1+A000523(n)) - A387413(n).

A387423 The length of binary expansion of n minus the length of the maximal common prefix of the binary expansions of n and sigma(n), where sigma is the sum of divisors function.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Sep 01 2025

Positions of 0's in this sequence is given by such numbers n that sigma(n) = 2^k * n + r, for some n >= 1, k >= 0, 0 <= r < 2^k. These would include also quasi-perfect numbers and their generalizations, numbers n such that sigma(n) = 2^k * n + 2^k - 1, for some n > 1, k > 0 (see comments in A332223), if such numbers exist. However, it is conjectured that there are no other zeros than those given by A336702.

Cf. A000203, A000523, A332223, A336700, A336701, A336702 (conjectured positions of 0's), A387422.

Cf. also A347381, A387413.

A387423[n_] := BitLength[n] - LengthWhile[Transpose[IntegerDigits[{n, DivisorSigma[1, n]}, 2][[All, ;; BitLength[n]]]], Equal @@ # &];
Array[A387423, 100] (* Paolo Xausa, Sep 03 2025 *)

A387423(n) = { my(a=binary(n), b=binary(sigma(n)), i=1); while(i<=#a,if(a[i]!=b[i],return(#a-(i-1))); i++); (0); };

from os.path import commonprefix
from sympy import divisor_sigma
def A387423(n): return n.bit_length()-len(commonprefix([bin(n)[2:],bin(divisor_sigma(n))[2:]])) # Chai Wah Wu, Sep 03 2025

a(n) = (1+A000523(n)) - A387422(n).

A387414 Numbers k such that the binary expansion of k is a prefix of the binary expansion of A003961(k), where A003961 is fully multiplicative with a(p) = nextprime(p).

Original entry on oeis.org

1, 4, 10, 18, 57, 348, 1054, 2626, 60625, 68727, 129260, 192276, 675348, 960320, 5368464, 12371554, 30078308, 356311953, 1158654378, 1673018314

Offset: 1

Antti Karttunen, Sep 01 2025

Numbers k such that A003961(k) = 2^e * k + r, for some k >= 1, e >= 0, 0 <= r < 2^e.

			A007088(4) = 100, and A007088(A003961(4)) = A007088(9) = 1001 begins with the same binary string, therefore 4 is included.
A007088(18) = 10010, and A007088(A003961(18)) = A007088(75) = 1001011 begins with the same binary string, therefore 18 is included as a term. Also, 75 = 2^2 * 18 + 3.

Positions of 0's in A387413.
Cf. A000523, A003961, A007088, A387412.
Subsequences: A348514 (which is also a subsequence of A387411).

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
is_A387414(n) = { my(s=A003961(n)); while(s>n, s >>= 1); (s==n); };

cp's OEIS Frontend

User: Antti Karttunen,

A387408 Partial sums of A387412, where A387412(n) is the length of the maximal common prefix of the binary expansions of n and A003961(n).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A387407 Partial sums of A387422, where A387422 is the length of the maximal common prefix of the binary expansions of n and sigma(n).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A387409 Partial sums of A387422 minus partial sums of A387412.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A387425 Partial sums of A387413.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

A387424 Partial sums of A387423.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A387422 The length of the maximal common prefix of the binary expansions of n and sigma(n), where sigma is the sum of divisors function.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A387413 The length of binary expansion of n minus the length of the maximal common prefix of the binary expansions of n and A003961(n), where A003961 is fully multiplicative with a(p) = nextprime(p).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A387412 The length of the maximal common prefix of the binary expansions of n and A003961(n), where A003961 is fully multiplicative with a(p) = nextprime(p).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Python

Formula

A387423 The length of binary expansion of n minus the length of the maximal common prefix of the binary expansions of n and sigma(n), where sigma is the sum of divisors function.

Views