Antti Karttunen - cp's OEIS frontend

A387425 Partial sums of A387413.

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.

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];

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).

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); };

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

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); };

A387419 Numbers k such that the odd part of (1+k) divides (1 + odd part of A003959(k)), where A003959 is multiplicative with a(p^e) = (p+1)^e.

Original entry on oeis.org

1, 3, 4, 7, 15, 31, 40, 63, 127, 255, 511, 639, 1023, 2047, 2175, 2431, 4095, 5247, 8191, 14335, 16383, 32767, 40959, 44031, 57855, 65535, 90111, 131071, 204799, 262143, 376831, 524287, 923647, 1048575, 1632255, 2056191, 2097151, 2621439, 2744319, 4194303, 6815743, 8388607, 8781823, 16777215, 19922943, 24068095

Offset: 1

Antti Karttunen, Sep 01 2025

Like in many sequences of this type, the criterion seems to strongly select for numbers with a long tail of trailing 1-bits. Terms 1, 4 and 40 are probably the only terms that are not in A004767.

Cf. A000225 (subsequence), A000265, A003959, A004767.

For similar sequences, see A336700, A387410, A387411, A387415, A387418.

A000265(n) = (n>>valuation(n,2));
A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
isA387419(n) = !((1+A000265(A003959(n)))%A000265(1+n));

A387418 Numbers k such that the odd part of (1+k) divides (1 + odd part of A034448(k)), where A034448 is unitary sigma (usigma).

Original entry on oeis.org

1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, 1791, 2047, 2431, 4095, 8191, 14335, 14847, 16383, 27391, 32767, 44031, 57855, 65535, 114687, 131071, 204799, 262143, 376831, 524287, 923647, 1048575, 1632255, 2056191, 2097151, 2744319, 4194303, 6815743, 8388607, 8781823, 8978431, 12058623, 16777215, 19922943, 24068095

Offset: 1

Antti Karttunen, Sep 01 2025

Like in many sequences of this type, the criterion seems to strongly select for numbers with a long tail of trailing 1-bits. The initial 1 is probably the only term that is not in A004767.

Cf. A000225 (subsequence), A000265, A002827, A004767, A034448.

For similar sequences, see A336700, A387410, A387415, A387419.

A000265(n) = (n>>valuation(n,2));
A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); };
isA387418(n) = !((1+A000265(A034448(n)))%A000265(1+n));

A387415 Numbers k such that the odd part of (1+k) divides (1 + odd part of A001615(k)), where A001615 is Dedekind's psi-function.

Original entry on oeis.org

1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, 2047, 2431, 4095, 8191, 14335, 16383, 27135, 32767, 44031, 57855, 65535, 75775, 131071, 204799, 262143, 376831, 524287, 667135, 923647, 1048575, 1441791, 1632255, 2056191, 2097151, 2315775, 2744319, 4194303, 6768639, 6815743, 8388607, 8781823, 16777215, 19922943, 24068095

Offset: 1

Antti Karttunen, Sep 01 2025

Like in many sequences of this type, the criterion seems to strongly select for numbers with a long tail of trailing 1-bits. The initial 1 is probably the only term that is not in A004767.

Cf. A000225 (subsequence), A000265, A001615.

For similar sequences, see A336700, A387410, A387418, A387419.

A000265(n) = (n>>valuation(n,2));
A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1)));
isA387415(n) = !((1+A000265(A001615(n)))%A000265(1+n));

cp's OEIS Frontend

User: Antti Karttunen

A387425 Partial sums of A387413.

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A387424 Partial sums of A387423.

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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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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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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).

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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.

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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).

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A387419 Numbers k such that the odd part of (1+k) divides (1 + odd part of A003959(k)), where A003959 is multiplicative with a(p^e) = (p+1)^e.

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A387418 Numbers k such that the odd part of (1+k) divides (1 + odd part of A034448(k)), where A034448 is unitary sigma (usigma).

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