A006005 -id:A006005 - cp's OEIS frontend

A325960 a(n) is k-n for the least k >= n+(A020639(n)-1) such that n-k and n-(sigma(n)-k) are relatively prime, or 0 if no such k <= sigma(n) exists.

0, 1, 0, 1, 0, 1, 0, 1, 2, 1, 0, 1, 0, 1, 5, 1, 0, 1, 0, 1, 3, 1, 0, 1, 4, 1, 3, 1, 0, 1, 0, 1, 5, 1, 5, 1, 0, 1, 3, 1, 0, 1, 0, 1, 5, 1, 0, 1, 6, 1, 7, 1, 0, 1, 5, 1, 3, 1, 0, 1, 0, 1, 3, 1, 5, 1, 0, 1, 5, 1, 0, 1, 0, 1, 3, 1, 7, 1, 0, 1, 2, 1, 0, 1, 5, 1, 5, 1, 0, 1, 9, 1, 3, 1, 9, 1, 0, 1, 5, 1, 0, 1, 0, 1, 5

Offset: 1

Antti Karttunen, May 29 2019

nonn

By definition, if n is neither an odd prime nor an odd perfect number, then a(n) >= (A020639(n)-1).

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to sigma(n)

Cf. A000203, A000396, A020639, A324213, A325817, A325818, A325826, A325961, A325962, A325965, A325966, A325970.

Cf. A006005 (positions of zeros, provided no odd perfect numbers exist).

A020639(n) = if(1==n, n, factor(n)[1, 1]);
A325960(n) = { my(s=sigma(n)); for(i=(-1)+n+A020639(n), s, if(1==gcd(n-i, n-(s-i)), return(i-n))); (0); };

a(n) = (A325961(n) - A325962(n)) / 2, assuming no odd perfect numbers exist.

a(2n) = 1.

A333570 Number of nonnegative values c such that c^n == -c (mod n).

Original entry on oeis.org

1, 2, 1, 2, 1, 4, 1, 2, 1, 4, 1, 4, 1, 4, 3, 2, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 8, 1, 8, 1, 2, 1, 4, 3, 4, 1, 4, 3, 4, 1, 8, 1, 4, 1, 4, 1, 4, 1, 4, 3, 8, 1, 4, 3, 4, 1, 4, 1, 8, 1, 4, 1, 2, 1, 24, 1, 4, 1, 16, 1, 4, 1, 4, 3, 8, 1, 8, 1, 4, 1, 4, 1, 8, 5, 4, 3, 4, 1, 8, 7, 4, 1, 4, 3

Offset: 1

Juri-Stepan Gerasimov, Mar 27 2020

nonn

a(n) is the number of nonnegative bases c < n such that c^n + c == 0 (mod n).
a(2^k) = 2 for k > 0.
a(p^m) = 1 for odd prime p with m >= 0.
Let fy(n) = (the number of values b in Z/nZ such that b^y = b)/(the number of values c in Z/nZ such that -c^y = c) for nonnegative y, then:
f0(n) = A000012(n),
f1(n) = A026741(n),
f2(n) = A000012(n),
1 <= f3(n) <= n,
f4(n) = A000012(n), ...,
1 <= fn(n) = A182816(n)/a(n) <= n, where fn(n) = n for odd noncomposite numbers A006005 and Carmichael numbers A002997.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000012, A000079, A000961, A002997, A006005, A026741, A182816.

[#[c: c in [0..n-1] | -c^n mod n eq c]: n in [1..95]];

a(n) = sum(c=1, n, Mod(c, n)^n == -c); \\ Michel Marcus, Mar 27 2020

a(n) = A182816(n)/r for some odd r.

A162800 a(n) = n-th grid point that is covered by the zig-zag function for prime numbers such that the grid point is a vertex in the graph of the function.

Original entry on oeis.org

0, 2, 4, 6, 9, 12, 15, 18, 21, 26, 30, 34, 39, 42, 45, 50, 56, 60, 64, 69, 72, 76, 81, 86, 93, 99, 102, 105, 108, 111, 120, 129, 134, 138, 144, 150, 154, 160, 165, 170, 176, 180, 186, 192, 195, 198, 205, 217, 225, 228, 231, 236, 240, 246, 254, 260, 266, 270, 274, 279

Offset: 0

Omar E. Pol, Jul 16 2009

Also {0, 2} together the numbers A024675.

See A162345 for the first differences.

Paolo Xausa, Table of n, a(n) for n = 0..10000

Cf. A000040, A006005, A024675, A058296, A079424, A162345, A162801, A162802.

Join[{0, 2}, Most[#] + Differences[#]/2] & [Prime[Range[2, 100]]] (* Paolo Xausa, Jun 17 2024 *)

Edited by Omar E. Pol, Jul 18 2009

A348608 a(n) = Sum_{d|n, d <= sqrt(n)} (-1)^(d + n/d) * d.

Original entry on oeis.org

1, -1, 1, 1, 1, -3, 1, 1, 4, -3, 1, -2, 1, -3, 4, 5, 1, -6, 1, -3, 4, -3, 1, 2, 6, -3, 4, -3, 1, -11, 1, 5, 4, -3, 6, 0, 1, -3, 4, 0, 1, -12, 1, -3, 9, -3, 1, 8, 8, -8, 4, -3, 1, -12, 6, -2, 4, -3, 1, -5, 1, -3, 11, 13, 6, -12, 1, -3, 4, -15, 1, 0, 1, -3, 9, -3, 8, -12, 1, 8

Offset: 1

Ilya Gutkovskiy, Oct 25 2021

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000593, A006005, A046897, A066839, A109506, A305152, A333782, A037213, A348953.

Table[DivisorSum[n, (-1)^(# + n/#) # &, # <= Sqrt[n] &], {n, 1, 80}]
nmax = 80; CoefficientList[Series[Sum[k x^(k^2)/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

a(n) = sumdiv(n, d, if (d<=sqrt(n), (-1)^(d + n/d)*d)); \\ Michel Marcus, Oct 25 2021

G.f.: Sum_{k>=1} k * x^(k^2) / (1 + x^k).
a(n) = 1 if n = 1 or n is an odd prime (A006005) or n = 4 or n = 8. - Bernard Schott, Dec 18 2021
a(n) = A037213(n) - A348953(n). - Ridouane Oudra, Aug 21 2025

A383300 Numbers k such that primorial base expansion of k has the primorial base expansion of k' as its suffix, where k' stands for the arithmetic derivative of k (A003415).

Original entry on oeis.org

0, 1, 3, 4, 5, 7, 11, 13, 17, 19, 23, 27, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331

Offset: 1

Antti Karttunen, May 15 2025

nonn

a(n) = A348283(n) for n=1 and n=3..334432. a(334433) = 4784261, which is not present in A348283 (see examples). - R. J. Mathar and Antti Karttunen, May 16 2025

			0 is a term as A003415(0) = 0.
1 is a term as A003415(1) = 0, whose primorial base expansion is here understood as an empty sequence of digits, thus it is a suffix of A049345(1) = 1.
3, like all odd primes, is a term as A003415(3) = 1, with A049345(3) = 11 and A049345(1) = 1.
4 and 27 are terms as they are in A051674 (the nonzero fixed points of A003415).
4784261 is a term as A003415(4784261) = 189671, with A049345(4784261) = 96411121 and A049345(189671) = 6411121. 4784261 is the first term > 1 of this sequence that is not in A348283. See more examples in A383301.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to primorial base.

Cf. A003415, A049345, A235224.
Disjoint union of {1}, A348283\{2} and A383301.
Cf. A006005, A051674 (other subsequences).
Subsequence of A383299.
Cf. also A383933.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
isA383300(n) = if(n<2, 1, my(p=2, k=A003415(n)); while(k, if((k%p)!=(n%p), return(0)); n = n\p; k = k\p; p = nextprime(1+p)); (1));

A002110(n) = prod(i=1,n,prime(i));
A235224(n) = { my(s=0, p=2); while(n, s++; n = n\p; p = nextprime(1+p)); (s); };
isA383300(n) = { my(ad=A003415(n)); (n%A002110(A235224(ad))==ad); };

{k such that A003415(k) is equal to k modulo A002110(A235224(A003415(k)))}.

A108501 Number of factorizations of 4*n into even numbers.

Original entry on oeis.org

2, 3, 2, 5, 2, 4, 2, 7, 3, 4, 2, 7, 2, 4, 3, 11, 2, 6, 2, 7, 3, 4, 2, 12, 3, 4, 3, 7, 2, 7, 2, 15, 3, 4, 3, 12, 2, 4, 3, 12, 2, 7, 2, 7, 4, 4, 2, 19, 3, 6, 3, 7, 2, 8, 3, 12, 3, 4, 2, 14, 2, 4, 4, 22, 3, 7, 2, 7, 3, 7, 2, 21, 2, 4, 4, 7, 3, 7, 2, 19, 4, 4, 2, 14, 3, 4, 3, 12, 2, 11, 3, 7, 3, 4, 3, 30, 2

Offset: 1

Christian G. Bower, Jun 06 2005

nonn

a(n) = 2 iff n is 1 or an odd prime (A006005); in this case, the two factorizations are 4n = 2 * 2n. - Bernard Schott, Nov 30 2020

			a(6)=4 because 6*4=24 can be factored as 24=12*2=6*4=6*2*2.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A000041, A001055, A006005, A108502, A108503.

with(numtheory):
b:= proc(n, i) option remember; `if`(n<=i, 1, 0)+
      add(`if`(d<=i and irem(d, 2)=0 and irem(n/d, 2)=0,
      b(n/d, min(d, i)), 0), d=divisors(n) minus {1, n})
    end:
a:= n-> b(4*n$2):
seq(a(n), n=1..100);  # Alois P. Heinz, Feb 17 2015

b[n_, i_] := b[n, i] = If[n <= i, 1, 0] + Sum[If[d <= i && Mod[d, 2]==0 && Mod[n/d, 2]==0, b[n/d, Min[d, i]], 0], {d, Divisors[n][[2 ;; -2]]}];
a[n_] := b[4n, 4n];
Array[a, 100] (* Jean-François Alcover, Nov 05 2020, after Alois P. Heinz *)

a(2^k) = A000041(k+2). - Bernard Schott, Dec 01 2020

A162350 The path of the primes: Pairs (x,y) such that the points P(x,y) represent the position of the nonnegative integers in the graph of the "mountain path" function for prime numbers.

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 2, 1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 4, 4, 5, 5, 5, 5, 4, 6, 4, 6, 5, 6, 6, 6, 7, 7, 7, 7, 6, 8, 6, 8, 7, 8, 8, 8, 9, 9, 9, 9, 8, 9, 7, 9, 6, 10, 6, 10, 7, 10, 8, 10, 9, 10, 10, 11, 10, 11, 9, 11, 8, 12, 8, 12, 9, 12, 10

Offset: 0

Omar E. Pol, Jul 03 2009

			n ..... Point
0 ..... P(0,0)
1 ..... P(1,0)
2 ..... P(2,0)
3 ..... P(2,1)
4 ..... P(2,2)
5 ..... P(3,2)
6 ..... P(4,2)
7 ..... P(4,3)
8 ..... P(4,4)
9 ..... P(4,5)
10 .... P(5,5)
11 .... P(5,4)

Omar E. Pol, Graph of the mountain path function for prime numbers
Omar E. Pol, Illustration: The path of the primes

Cf. A000040, A006005, A008578, A162200, A162201, A162202, A162203, A162340, A162341, A162342, A162343, A162344, A162351, A162352.

A383299 Numbers k such that A276086(k) is a multiple of A276086(A003415(k)), where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

Original entry on oeis.org

0, 1, 3, 4, 5, 7, 9, 11, 13, 15, 17, 19, 23, 27, 29, 31, 37, 41, 43, 45, 47, 51, 53, 59, 61, 67, 71, 73, 79, 83, 87, 89, 97, 101, 103, 107, 109, 113, 117, 119, 127, 131, 137, 139, 141, 147, 149, 151, 157, 161, 163, 165, 167, 171, 173, 177, 179, 181, 191, 193, 197, 199, 203, 207, 209, 211, 223, 227, 229, 233, 239

Offset: 1

Antti Karttunen, May 15 2025

nonn

The sequence contains the intersection of A048103, A369650, and A328387. That is, {1, 15, 5005}, at least.

			5 is a term as A003415(5) = 1, and A276086(5) = 18 is a multiple of A276086(1) = 2, and ditto for all odd primes.
9 is a term as A003415(9) = 6, and A276086(9) = 30 is a multiple of A276086(6) = 5.
15 is a term as A003415(15) = 8, and A276086(15) = 150 is a multiple of A276086(8) = 15.
5005 is a term as A003415(5005) = 2556, and A276086(5005) = 39055266250 = 7803250 * A276086(2556) = 7803250 * 5005. See also A369650.
See also examples in A383300.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to primorial base.

Cf. A003415, A048103, A276086, A327859, A328387, A383298 (characteristic function).
Cf. A006005, A051674, A383300, A383301 (subsequences).
Cf. also A369650.

PARI
```
isA383299 = A383298;
```

A061285 a(n) = 2^((prime(n) - 1)/2).

Original entry on oeis.org

2, 4, 8, 32, 64, 256, 512, 2048, 16384, 32768, 262144, 1048576, 2097152, 8388608, 67108864, 536870912, 1073741824, 8589934592, 34359738368, 68719476736, 549755813888, 2199023255552, 17592186044416, 281474976710656, 1125899906842624, 2251799813685248

Offset: 2

Labos Elemer, May 22 2001

nonn

Square root of 2^(prime(n) - 1), i.e., the smallest number that has prime(n) divisors.

Index to divisibility sequences

Cf. A000005, A000010, A005097, A005179, A006005, A006093, A034785, A217054.

Table[2^((Prime[n] - 1)/2), {n, 2, 25}] (* Amiram Eldar, Dec 23 2020 *)

a(n) = sqrt(min(x; A000005(x) = prime(n))) = sqrt(A034785(n)/2) = sqrt(2^(prime(n) - 1)) = sqrt(2^A006093(n)) = sqrt(2^phi(prime(n))) = sqrt(2^A000010(A000040(n))).

Sum_{n>=1} 1/a(n) = A217054. - Amiram Eldar, Dec 23 2020

A224502 Prime numbers (together with one) whose representation in balanced ternary are palindromes.

Original entry on oeis.org

1, 7, 13, 43, 61, 73, 103, 367, 421, 457, 547, 601, 613, 757, 859, 1039, 1093, 3823, 4021, 4561, 4723, 4759, 5743, 6211, 6373, 6481, 6949, 7219, 7489, 7933, 8563, 8941, 9103, 9679, 29527, 30013, 31147, 31741, 33037, 35251, 36061, 36097, 36583, 37717, 39607, 41011, 42667, 43963, 44773, 45691, 47581, 49201

Offset: 1

Malachi de Ælfweald, Apr 08 2013

Intersection of A006005 and A134027.

			For n=5, a(5)=61 and in balanced ternary notation is 1ī1ī1.

Malachi de Ælfweald and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 199 terms from Malachi de Ælfweald)
Wikipedia, Balanced ternary

bt(k,n)={
    sum(i=0,(n-1)\2,
        my(t=k%3-1);
        k\=3;
        n--;
        if(n==i,3^n,3^i+3^n)*t
    )
};
do(N)={
    my(v=List([1]),t);
    for(n=1,N,
        forstep(k=2,3^((n+1)\2)-1,3,
            t=bt(k,n);
            if(isprime(t),listput(v,t))
        )
    );
    vecsort(Vec(v))
}; \\ Charles R Greathouse IV, Apr 08 2013

cp's OEIS Frontend

A325960 a(n) is k-n for the least k >= n+(A020639(n)-1) such that n-k and n-(sigma(n)-k) are relatively prime, or 0 if no such k <= sigma(n) exists.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A333570 Number of nonnegative values c such that c^n == -c (mod n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

PARI

Formula

A162800 a(n) = n-th grid point that is covered by the zig-zag function for prime numbers such that the grid point is a vertex in the graph of the function.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions

A348608 a(n) = Sum_{d|n, d <= sqrt(n)} (-1)^(d + n/d) * d.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A383300 Numbers k such that primorial base expansion of k has the primorial base expansion of k' as its suffix, where k' stands for the arithmetic derivative of k (A003415).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI

Formula

A108501 Number of factorizations of 4*n into even numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A162350 The path of the primes: Pairs (x,y) such that the points P(x,y) represent the position of the nonnegative integers in the graph of the "mountain path" function for prime numbers.

Views

Author

Keywords

Examples

Links

Crossrefs

A383299 Numbers k such that A276086(k) is a multiple of A276086(A003415(k)), where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

Views

Author

Keywords

Comments

Examples

Links