A387423 -id:A387423 - cp's OEIS frontend

A387424 Partial sums of A387423.

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.

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).

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).

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

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

cp's OEIS Frontend

A387424 Partial sums of A387423.

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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).

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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.

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