cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 10 results.

A332459 Odd part of 1+sigma(n).

Original entry on oeis.org

1, 1, 5, 1, 7, 13, 9, 1, 7, 19, 13, 29, 15, 25, 25, 1, 19, 5, 21, 43, 33, 37, 25, 61, 1, 43, 41, 57, 31, 73, 33, 1, 49, 55, 49, 23, 39, 61, 57, 91, 43, 97, 45, 85, 79, 73, 49, 125, 29, 47, 73, 99, 55, 121, 73, 121, 81, 91, 61, 169, 63, 97, 105, 1, 85, 145, 69, 127, 97, 145, 73, 49, 75, 115, 125, 141, 97, 169, 81, 187, 61, 127
Offset: 1

Views

Author

Antti Karttunen, Feb 16 2020

Keywords

Links

Crossrefs

Cf. A000203, A000265, A002487, A088580, A161942, A202274, A324294, A332454, A332455.

Programs

  • PARI
    A332459(n) = { my(s=1+sigma(n)); (s>>valuation(s,2)); };

Formula

a(n) = A000265(A088580(n)) = A000265(1+sigma(n)).
A002487(a(n)) = A324294(n).
a(2^n) = 0 for all n >= 0. [Zero occurs at least also at a(25). See A202274]

A336694 a(n) = A329697(1+sigma(n)), where A329697 is totally additive with a(2) = 0 and a(p) = 1 + a(p-1) for odd primes.

Original entry on oeis.org

0, 0, 1, 0, 2, 2, 2, 0, 2, 3, 2, 3, 2, 2, 2, 0, 3, 1, 3, 4, 3, 3, 2, 3, 0, 4, 2, 4, 3, 3, 3, 0, 4, 3, 4, 3, 3, 3, 4, 4, 4, 2, 3, 2, 4, 3, 4, 3, 3, 4, 3, 4, 3, 4, 3, 4, 4, 4, 3, 4, 4, 2, 4, 0, 2, 4, 4, 5, 2, 4, 3, 4, 3, 4, 3, 5, 2, 4, 4, 3, 3, 5, 2, 4, 4, 5, 4, 4, 4, 5, 3, 4, 5, 4, 4, 5, 4, 4, 4, 4, 3, 5, 4, 5, 2
Offset: 1

Views

Author

Antti Karttunen, Jul 31 2020

Keywords

Links

Crossrefs

Cf. A000203, A088580, A324294, A329697, A332459, A336691, A336692, A336693, A336695, A336696.

Programs

Formula

a(n) = A329697(1+A000203(n)) = A329697(A088580(n)) = A329697(A332459(n)).

A336692 Binary weight of 1+sigma(n).

Original entry on oeis.org

1, 1, 2, 1, 3, 3, 2, 1, 3, 3, 3, 4, 4, 3, 3, 1, 3, 2, 3, 4, 2, 3, 3, 5, 1, 4, 3, 4, 5, 3, 2, 1, 3, 5, 3, 4, 4, 5, 4, 5, 4, 3, 4, 4, 5, 3, 3, 6, 4, 5, 3, 4, 5, 5, 3, 5, 3, 5, 5, 4, 6, 3, 4, 1, 4, 3, 3, 7, 3, 3, 3, 3, 4, 5, 6, 4, 3, 4, 3, 6, 5, 7, 4, 4, 5, 3, 5, 5, 5, 6, 4, 4, 2, 3, 5, 7, 4, 4, 5, 5, 5, 5, 4, 5, 3
Offset: 1

Views

Author

Antti Karttunen, Jul 31 2020

Keywords

Links

Crossrefs

Cf. A000120, A000203, A088580, A175548, A324294, A332459, A336691, A336693, A336694, A336695, A336696.

Programs

  • PARI
    A336692(n) = hammingweight(1+sigma(n));

Formula

a(n) = A000120(1+A000203(n)) = A000120(A088580(n)) = A000120(A332459(n)).

A336695 a(n) = A331410(1+sigma(n)), where A331410 is totally additive with a(2) = 0 and a(p) = 1 + a(p+1) for odd primes.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jul 31 2020

Keywords

Links

Crossrefs

Cf. A000203, A088580, A324294, A331410, A332459, A336691, A336692, A336693, A336694, A336696.

Programs

Formula

a(n) = A331410(1+A000203(n)) = A331410(A088580(n)) = A331410(A332459(n)).

A336693 Period of binary representation of 1/(1+sigma(n)).

Original entry on oeis.org

1, 1, 4, 1, 3, 12, 6, 1, 3, 18, 12, 28, 4, 20, 20, 1, 18, 4, 6, 14, 10, 36, 20, 60, 1, 14, 20, 18, 5, 9, 10, 1, 21, 20, 21, 11, 12, 60, 18, 12, 14, 48, 12, 8, 39, 9, 21, 100, 28, 23, 9, 30, 20, 110, 9, 110, 54, 12, 60, 156, 6, 48, 12, 1, 8, 28, 22, 7, 48, 28, 9, 21, 20, 44, 100, 46, 48, 156, 54, 40, 60, 7, 8, 60
Offset: 1

Views

Author

Antti Karttunen, Jul 31 2020

Keywords

Links

Crossrefs

Cf. A000203, A007733, A088580, A324294, A332459, A336691, A336692, A336694, A336695.

Programs

Formula

a(n) = A007733(1+A000203(n)) = A007733(A088580(n)) = A007733(A332459(n)).

A336696 Sum of odd divisors of 1+sigma(n).

Original entry on oeis.org

1, 1, 6, 1, 8, 14, 13, 1, 8, 20, 14, 30, 24, 31, 31, 1, 20, 6, 32, 44, 48, 38, 31, 62, 1, 44, 42, 80, 32, 74, 48, 1, 57, 72, 57, 24, 56, 62, 80, 112, 44, 98, 78, 108, 80, 74, 57, 156, 30, 48, 74, 156, 72, 133, 74, 133, 121, 112, 62, 183, 104, 98, 192, 1, 108, 180, 96, 128, 98, 180, 74, 57, 124, 144, 156, 192, 98
Offset: 1

Views

Author

Antti Karttunen, Jul 31 2020

Keywords

Links

Crossrefs

Cf. A000203, A000593, A088580, A193337, A324294, A332459, A336691, A336692, A336693, A336694, A336696.

Programs

  • Mathematica
    Table[Total[Select[Divisors[1+DivisorSigma[1,n]],OddQ]],{n,80}] (* Harvey P. Dale, Jan 01 2022 *)
  • PARI
    A000593(n) = sigma(n>>valuation(n, 2));
A336696(n) = A000593(1+sigma(n));

Formula

a(n) = A000593(1+A000203(n)) = A000593(A088580(n)) = A000593(A332459(n)).

A332455 Starting from sigma(n)+1, number of tripling steps to reach 1 in '3x+1' problem, or -1 if 1 is never reached.

Original entry on oeis.org

0, 0, 1, 0, 5, 2, 6, 0, 5, 6, 2, 5, 5, 7, 7, 0, 6, 1, 1, 9, 8, 6, 7, 5, 0, 9, 40, 10, 39, 42, 8, 0, 7, 41, 7, 4, 11, 5, 10, 33, 9, 43, 4, 1, 11, 42, 7, 39, 5, 38, 42, 7, 41, 34, 42, 34, 6, 33, 5, 16, 39, 43, 12, 0, 1, 42, 3, 15, 43, 42, 42, 7, 3, 10, 39, 3, 43, 16, 6, 14, 5, 15, 1, 17, 41, 8, 34, 4, 33, 46, 2, 16, 44, 42, 34, 39
Offset: 1

Views

Author

Antti Karttunen, Feb 16 2020

Keywords

Links

Crossrefs

Cf. A000203, A006667, A088580, A202274, A332453, A332454.
Cf. also A324294, A332459.

Programs

  • PARI
    A006667(n) = { my(t=0); while(n>1, if(n%2, t++; n=3*n+1, n>>=1)); (t); };
A332455(n) = A006667(1+sigma(n));

Formula

a(n) = A006667(A088580(n)) = A006667(1+sigma(n)).
a(2^n) = 0 for all n >= 0. [Zero occurs at least also at a(25). See A202274]

A202274 Numbers k for which sigma(k) = 2^m - 1 for some m.

Original entry on oeis.org

1, 2, 4, 8, 16, 25, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648, 4294967296, 8589934592, 17179869184, 34359738368, 68719476736, 137438953472
Offset: 1

Views

Author

Jaroslav Krizek, Dec 25 2011

Keywords

Comments

Original definition, now conjectural, is "Positive integers m in increasing order determined by these rules: a(1) = 1, for n>=1, if m is in the sequence then also are numbers h such that sigma(h) = 4m-1". This is certainly equal to the new definition if 25 is the only term that is not a power of 2, as there is no x such that sigma(x) = 99 = 4*25-1. - Antti Karttunen, Dec 13 2024
If 31 is only number h of form 2^k-1 for any k>=1 such that sigma(x) = h has solution for more than one value of x then a(n) is union number 25 with A000079 (powers of 2).
Numbers k such that A000203(k) is in A000225. If Goormaghtigh conjecture is valid, then it is certain that 25 is the only odd prime power (after 1) in this sequence. - Antti Karttunen, Dec 13 2024

Examples

			These examples relate to the original definition:
m=1, 4m-1=3, sigma(h)=3 for h=2; number 2 is in sequence.
m=2, 4m-1=7, sigma(h)=7 for h=4; number 4 is in sequence.
m=4, 4m-1=15, sigma(h)=15 for h=8; number 8 is in sequence.
m=8, 4m-1=31, sigma(h)=31 for h=16 and 25; numbers 16 and 25 are in sequence.

Links

Crossrefs

Cf. A000079 (subsequence), A000203, A000225, A002191, A292369 (conjectured subsequence).
Subsequence of A028982. Conjectured intersection of A028982 and A378983.
Positions of 0's in A336694, A336695.
Positions of 1's in A324294, A332459, A336692, A336693, A336696.
Cf. also A202273.

Programs

  • PARI
    is_A202274(n) = ((x->!bitand(x,x+1))(sigma(n)));
for(n=1,2^20,if(is_A202274(n^2), print1(n^2,", ")); if(n>1 && is_A202274(2*((n-1)^2)), print1(2*((n-1)^2),", "))); \\ Remember to sort! - Antti Karttunen, Dec 13 2024

Extensions

Data section corrected (terms 1024, 2048 were duplicated), more terms added, and the name replaced with a new definition, with the original definition moved to the comments - Antti Karttunen, Dec 13 2024

A324293 a(n) = A002487(sigma(n)).

Original entry on oeis.org

1, 2, 1, 3, 2, 2, 1, 4, 5, 4, 2, 3, 3, 2, 2, 5, 4, 10, 3, 8, 1, 4, 2, 4, 5, 8, 3, 3, 4, 4, 1, 6, 2, 8, 2, 19, 7, 4, 3, 12, 8, 2, 5, 8, 10, 4, 2, 5, 10, 16, 4, 9, 8, 4, 4, 4, 3, 12, 4, 8, 5, 2, 5, 7, 8, 4, 5, 6, 2, 4, 4, 20, 11, 10, 5, 9, 2, 8, 3, 16, 13, 6, 8, 3, 8, 6, 4, 12, 12, 18, 3, 8, 1, 4, 4, 6, 9, 34, 10, 27, 12, 8, 5, 18, 2
Offset: 1

Views

Author

Antti Karttunen, Feb 22 2019

Keywords

Links

Crossrefs

Cf. A000203, A002487, A324287, A324294.

Programs

  • PARI
    A002487(n) = { my(s=sign(n), a=1, b=0); n = abs(n); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (s*b); }; \\ Modified from the one given in A002487, sign not actually needed here.
A324293(n) = A002487(sigma(n));

Formula

a(n) = A002487(A000203(n)).

A332454 Starting from sigma(n)+1, number of halving steps to reach 1 in '3x+1' problem, or -1 if this never happens.

Original entry on oeis.org

1, 2, 4, 3, 11, 7, 13, 4, 12, 14, 7, 13, 12, 16, 16, 5, 14, 7, 6, 20, 18, 15, 16, 14, 5, 20, 69, 22, 67, 73, 18, 6, 17, 71, 17, 13, 23, 14, 22, 59, 20, 75, 12, 8, 24, 73, 17, 69, 14, 67, 73, 18, 71, 61, 73, 61, 16, 59, 14, 33, 68, 75, 26, 7, 8, 74, 11, 31, 75, 74, 73, 19, 11, 23, 69, 12, 75, 33, 16, 30, 15, 31, 8, 35, 72, 20
Offset: 1

Views

Author

Antti Karttunen, Feb 16 2020

Keywords

Links

Crossrefs

Cf. A000203, A006666, A088580, A332452, A332455.
Cf. also A324294.

Programs

Formula

a(n) = A006666(A088580(n)) = A006666(1+sigma(n)).
Showing 1-10 of 10 results.

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