A324054 -id:A324054 - cp's OEIS frontend

A290077 a(n) = A000010(A005940(1+n)).

1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 8, 4, 20, 6, 18, 8, 10, 6, 12, 8, 24, 8, 24, 8, 42, 20, 40, 12, 100, 18, 54, 16, 12, 10, 20, 12, 40, 12, 36, 16, 60, 24, 48, 16, 120, 24, 72, 16, 110, 42, 84, 40, 168, 40, 120, 24, 294, 100, 200, 36, 500, 54, 162, 32, 16, 12, 24, 20, 48, 20, 60, 24, 72, 40, 80, 24, 200, 36, 108, 32, 120, 60, 120

Offset: 0

Antti Karttunen, Jul 19 2017

nonn

Each n occurs A014197(n) times in total in this sequence.

Antti Karttunen, Table of n, a(n) for n = 0..16384
Index entries for sequences related to binary expansion of n

Cf. A000010, A000040, A005940, A014197, A290076, A323915, A324052, A324054, A364568.

f[n_, i_, x_]:=f[n, i, x]=Which[n==0, x, EvenQ[n], f[n/2, i + 1, x], f[(n - 1)/2, i, x Prime[i]]]; a005940[n_]:=f[n - 1, 1, 1]; Table[EulerPhi[a005940[n + 1]], {n, 0, 100}] (* Indranil Ghosh, Jul 20 2017 *)

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
A290077(n) = eulerphi(A005940(1+n));

A290077(n) = { my(p=2,z=1); while(n, if(!(n%2), p=nextprime(1+p), z *= (p-(1==(n%4)))); n>>=1); (z); }; \\ Antti Karttunen, Aug 05 2023

def A290077(n):
    i = 1
    m = 1
    while n > 0:
      if 0==(n%2):
        n = n//2
        i += 1
      else:
        if(1==(n%4)):
          n = (n-1)//4
          m *= sloane.A000040(i)-1
          i += 1
        else:
          n = (n-1)//2
          m *= sloane.A000040(i)
    return m

(define (A290077 n) (A000010 (A005940 (+ 1 n))))

(define (A290077 n) (let loop ((n n) (m 1) (i 1)) (cond ((zero? n) m) ((even? n) (loop (/ n 2) m (+ 1 i))) ((= 1 (modulo n 4)) (loop (/ (- n 1) 4) (* m (- (A000040 i) 1)) (+ 1 i))) (else (loop (/ (- n 1) 2) (* m (A000040 i)) i))))) ;; Requires only an implementation of A000040, see for example under A083221.

a(n) = A000010(A005940(1+n)).

A324184 a(n) = sigma(A163511(n)).

Original entry on oeis.org

1, 3, 7, 4, 15, 13, 12, 6, 31, 40, 39, 31, 28, 24, 18, 8, 63, 121, 120, 156, 91, 124, 93, 57, 60, 78, 72, 48, 42, 32, 24, 12, 127, 364, 363, 781, 280, 624, 468, 400, 195, 403, 372, 342, 217, 228, 171, 133, 124, 240, 234, 248, 168, 192, 144, 96, 90, 104, 96, 72, 56, 48, 36, 14, 255, 1093, 1092, 3906, 847, 3124, 2343, 2801, 600

Offset: 0

Antti Karttunen, Feb 17 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16383
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65537

Cf. A000203, A054429, A163511, A324054, A324183, A324185, A324186, A324187, A324188, A324189.

A324184(n) = if(!n,1,my(p=2,mp=p*p,m=1); while(n>1, if(n%2, p=nextprime(1+p); mp = p*p, if((2==n)||!(n%4),mp *= p,m *= (mp-1)/(p-1))); n >>= 1); (m*(mp-1)/(p-1)));

A163511(n) = if(!n,1,my(p=2, t=1); while(n>1, if(!(n%2), (t*=p), p=nextprime(p+1)); n >>= 1); (t*p));
A324184(n) = sigma(A163511(n));

from sympy import nextprime
def A324184(n):
    if n:
        c, p = 1, 1
        while n:
            c *= ((p:=nextprime(p))**(s:=(~n&n-1).bit_length()+1)-1)//(p-1)
            n >>= s
        return c*(p**(s+1)-1)//(p**s-1)
    return 1 # Chai Wah Wu, Jul 25 2023

a(n) = A000203(A163511(n)).

For n >= 1, a(n) = A324054(A054429(n)).

A324057 a(n) = A106315(A005940(1+n)).

Original entry on oeis.org

0, 1, 2, 5, 4, 0, 1, 2, 6, 4, 12, 16, 13, 30, 28, 18, 10, 8, 20, 36, 44, 24, 36, 12, 33, 21, 78, 51, 32, 72, 42, 3, 12, 16, 36, 0, 4, 48, 66, 50, 20, 128, 72, 48, 58, 144, 120, 108, 97, 75, 198, 32, 102, 312, 10, 84, 172, 128, 504, 176, 1, 168, 2, 67, 16, 20, 44, 12, 8, 96, 126, 88, 28, 16, 168, 112, 162, 264, 232, 56, 68, 80, 312, 0, 200, 480, 36, 120

Offset: 0

Antti Karttunen, Feb 14 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 0..8191
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..70327

Cf. A005940, A106737, A106315, A324054, A324055, A324058.

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ From A005940
A106315(n) = (n*numdiv(n) % sigma(n));
A324057(n) = A106315(A005940(1+n));

a(n) = A106315(A005940(1+n)).

a(n) = A005940(1+n)*A106737(n) mod A324054(n).

A323915 a(n) = A023900(A005940(1+n)).

Original entry on oeis.org

1, -1, -2, -1, -4, 2, -2, -1, -6, 4, 8, 2, -4, 2, -2, -1, -10, 6, 12, 4, 24, -8, 8, 2, -6, 4, 8, 2, -4, 2, -2, -1, -12, 10, 20, 6, 40, -12, 12, 4, 60, -24, -48, -8, 24, -8, 8, 2, -10, 6, 12, 4, 24, -8, 8, 2, -6, 4, 8, 2, -4, 2, -2, -1, -16, 12, 24, 10, 48, -20, 20, 6, 72, -40, -80, -12, 40, -12, 12, 4, 120, -60, -120, -24, -240, 48, -48

Offset: 0

Antti Karttunen, Feb 22 2019

sign

Antti Karttunen, Table of n, a(n) for n = 0..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65537
Index entries for sequences related to binary expansion of n

Cf. A005940, A023900, A290077, A324052, A324054, A324055, A324057, A324058.

A323915(n) = { my(m1=1, p=2); while(n, if(!(n%2), p=nextprime(1+p), if(1==(n%4), m1 *= (1-p))); n>>=1); (m1); };

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ From A005940
A023900(n) = sumdivmult(n, d, d*moebius(d)); \\ From A023900
A323915(n) = A023900(A005940(1+n));

a(n) = A023900(A005940(1+n)).

A331733 a(n) = sigma(A225546(n)), where sigma is the sum of divisors.

Original entry on oeis.org

1, 3, 7, 4, 31, 15, 511, 12, 13, 63, 131071, 28, 8589934591, 1023, 127, 6, 36893488147419103231, 39, 680564733841876926926749214863536422911, 124, 2047, 262143, 231584178474632390847141970017375815706539969331281128078915168015826259279871, 60, 121, 17179869183, 91, 2044

Offset: 1

Antti Karttunen, Feb 02 2020

nonn

Cf. A000203, A048675, A225546, A331309, A331734, A331735, A331741.

Cf. A323243, A323173, A324054, A324184, A324545 for other permutations of sigma, and also A324573, A324653.

Array[If[# == 1, 1, DivisorSigma[1, #] &@ Apply[Times, Flatten@ Map[Function[{p, e}, Map[Prime[Log2@ # + 1]^(2^(PrimePi@ p - 1)) &, DeleteCases[NumberExpand[e, 2], 0]]] @@ # &, FactorInteger[#]]]] &, 28] (* Michael De Vlieger, Feb 08 2020 *)

A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
A331733(n) = if(1==n,1,my(f=factor(n),u=#binary(vecmax(f[, 2])),prods=vector(u,x,1),m=1,e); for(i=1,u,for(k=1,#f~, if(bitand(f[k,2],m),prods[i] *= f[k,1])); m<<=1); prod(i=1,u,(prime(i)^(1+A048675(prods[i]))-1)/(prime(i)-1)));

a(n) = A000203(A225546(n)).

For all n >= 1, A000035(a(A016754(n))) = 1. [Result is odd for all odd squares]

A324545 An analog of sigma (A000203) for nonstandard factorization based on the sieve of Eratosthenes (A083221).

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 14, 24, 24, 31, 18, 39, 20, 42, 40, 36, 24, 60, 31, 42, 32, 56, 30, 72, 32, 63, 78, 54, 48, 91, 38, 60, 48, 90, 42, 120, 44, 84, 121, 72, 48, 124, 57, 93, 124, 98, 54, 96, 156, 120, 104, 90, 60, 168, 62, 96, 56, 127, 72, 234, 68, 126, 240, 144, 72, 195, 74, 114, 72, 140, 96, 144, 80

Offset: 1

Antti Karttunen, Mar 06 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences generated by sieves
Index entries for sequences related to sigma(n)

Cf. A000203, A020639, A078898, A250246, A252754, A302044, A302045, A324054, A324535, A324544, A324546.

Cf. also A302051, A302055, A323243.

up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
A020639(n) = if(n>1, if(n>n=factor(n, 0)[1, 1], n, factor(n)[1, 1]), 1); \\ From A020639
A055396(n) = if(1==n,0,primepi(A020639(n)));
v078898 = ordinal_transform(vector(up_to,n,A020639(n)));
A078898(n) = v078898[n];
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A250246(n) = if(1==n,n,my(k = 2*A250246(A078898(n)), r = A055396(n)); if(1==r, k, while(r>1, k = A003961(k); r--); (k)));
A324545(n) = sigma(A250246(n));

\\ Or alternatively, using also A078898 defined above:
A000265(n) = (n/2^valuation(n, 2));
A001511(n) = 1+valuation(n,2);
A302045(n) = A001511(A078898(n));
A302044(n) = { my(c = A000265(A078898(n))); if(1==c,1,my(p = prime(-1+primepi(A020639(n))+primepi(A020639(c))), d = A078898(c), k=0); while(d, k++; if((1==k)||(A020639(k)>=p),d -= 1)); (k*p)); };
A324545(n) = if(1==n,n,my(p=A020639(n)); (((p^(A302045(n)+1))-1)/(p-1))*A324545(A302044(n)));

a(n) = A000203(A250246(n)) = A324535(n) + A250246(n).
a(1) = 1; for n > 1, let p = A020639(n) [the smallest prime factor of n], then a(n) = (((p^(1+A302045(n)))-1) / (p-1)) * a(A302044(n)).
a(n) = A324054(A252754(n)).

A324056 a(n) = A000593(A005940(1+n)).

Original entry on oeis.org

1, 1, 4, 1, 6, 4, 13, 1, 8, 6, 24, 4, 31, 13, 40, 1, 12, 8, 32, 6, 48, 24, 78, 4, 57, 31, 124, 13, 156, 40, 121, 1, 14, 12, 48, 8, 72, 32, 104, 6, 96, 48, 192, 24, 248, 78, 240, 4, 133, 57, 228, 31, 342, 124, 403, 13, 400, 156, 624, 40, 781, 121, 364, 1, 18, 14, 56, 12, 84, 48, 156, 8, 112, 72, 288, 32, 372, 104, 320, 6, 168, 96, 384, 48

Offset: 0

Antti Karttunen, Feb 14 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 0..8191
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..70327

Cf. A000593, A005940, A038712, A324054.

A000593(n) = sigma(n>>valuation(n, 2)); \\ From A000593
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ From A005940
A324056(n) = A000593(A005940(1+n));

a(n) = A000593(A005940(1+n)).

a(n) = A324054(n) / A038712(1+n).

A324349 a(n) = A324122(A005940(1+n)), where A005940 is the Doudna sequence and A324122(n) = sigma(n) - gcd(n*d(n), sigma(n)).

Original entry on oeis.org

0, 2, 2, 6, 4, 0, 12, 14, 6, 16, 12, 24, 30, 36, 36, 30, 10, 16, 28, 36, 44, 48, 72, 48, 54, 90, 122, 90, 152, 96, 120, 60, 12, 32, 36, 0, 68, 48, 102, 80, 92, 128, 168, 144, 246, 216, 120, 120, 132, 168, 222, 216, 336, 360, 402, 192, 396, 464, 600, 272, 780, 360, 362, 126, 16, 40, 52, 72, 80, 96, 150, 112, 84, 208, 264, 112, 366, 288, 312, 184, 164, 272, 360, 0, 568

Offset: 0

Antti Karttunen, Feb 24 2019

nonn

Zeros occur in the same positions as in A324057, and can be obtained by sorting into ascending order the terms obtained with A156552(A001599(n)), n >= 1.

Antti Karttunen, Table of n, a(n) for n = 0..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65537

Cf. A005940, A054429, A324054, A324057, A324058, A324189

Cf. also A001599.

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ From A005940
A324122(n) = (sigma(n) - gcd(sigma(n),n*numdiv(n)));
A324349(n)  = A324122(A005940(1+n));

a(n) = A324122(A005940(1+n)).
a(n) = A324054(n) - A324058(n).
For n > 0, a(n) = A324189(A054429(n)).

A324394 a(n) = A009194(A005940(1+n)), where A005940 is the Doudna sequence and A009194(n) = gcd(n,sigma(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 1, 1, 1, 2, 3, 4, 1, 3, 1, 1, 1, 2, 1, 2, 1, 6, 3, 12, 1, 1, 1, 1, 1, 6, 1, 1, 1, 2, 3, 28, 1, 6, 1, 10, 1, 2, 3, 12, 1, 18, 15, 4, 1, 1, 3, 1, 1, 6, 1, 3, 1, 2, 3, 4, 1, 3, 1, 1, 1, 2, 1, 4, 1, 6, 3, 8, 7, 2, 3, 28, 1, 6, 1, 2, 1, 2, 3, 28, 1, 6, 3, 120, 1, 2, 1, 6, 1, 90, 3, 12, 1, 1, 1, 7, 1, 6, 3, 5, 7, 2

Offset: 0

Antti Karttunen, Mar 05 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65537
Index entries for sequences related to binary expansion of n
Index entries for sequences related to sigma(n)

Cf. A009194, A324054, A324055, A324058, A324349, A324395.

A324394(n) = { my(m1=1,m2=1,p=2,mp=p*p); while(n, if(!(n%2), p=nextprime(1+p); mp = p*p, m1 *= p; if(3==(n%4),mp *= p,m2 *= (mp-1)/(p-1))); n>>=1); gcd(m1,m2); };

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ From A005940
A009194(n) = gcd(n, sigma(n));
A324394(n) = A009194(A005940(1+n));

a(n) = A009194(A005940(1+n)) = gcd(A005940(1+n), A324054(n)).

A332222 a(n) = A156552(sigma(A005940(1+n))).

Original entry on oeis.org

0, 2, 3, 8, 5, 11, 32, 10, 7, 13, 23, 35, 1024, 66, 39, 1024, 11, 23, 31, 37, 47, 55, 133, 43, 258, 2050, 4099, 72, 267, 87, 48, 38, 17, 27, 47, 71, 55, 95, 263, 45, 95, 111, 191, 151, 8199, 269, 175, 4099, 264, 518, 1035, 2056, 1037, 8203, 2080, 138, 207, 539, 1071, 167, 1048592, 98, 291, 1073741824, 13, 37, 71, 75, 75, 111

Offset: 0

Antti Karttunen, Feb 12 2020

nonn

Cf. A000203, A005940, A156552, A324054, A332221.

Cf. also A332223.

Array[Floor@ Total@ Flatten@ MapIndexed[#1 2^(#2 - 1) &, Flatten[Table[2^(PrimePi@ #1 - 1), {#2}] & @@@ FactorInteger@ DivisorSigma[1, #]]] &@ Block[{p = Partition[Split[Join[IntegerDigits[#, 2], {2}]], 2], q}, Times @@ Flatten[Table[q = Take[p, -i]; Prime[Count[Flatten[q], 0] + 1]^q[[1, 1]], {i, Length[p]}]]] &, 70, 0] (* Michael De Vlieger, Feb 12 2020, after Robert G. Wilson v at A005940 *)

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ From A005940
A156552(n) = {my(f = factor(n), p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res}; \\ From A156552
A332222(n) = A156552(sigma(A005940(1+n)));

a(n) = A156552(A000203(A005940(1+n))).

a(n) = A332221(A005940(1+n)) = A156552(A324054(n)).

cp's OEIS Frontend

A290077 a(n) = A000010(A005940(1+n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Sage

Scheme

Scheme

Formula

A324184 a(n) = sigma(A163511(n)).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

PARI

Python

Formula

A324057 a(n) = A106315(A005940(1+n)).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A323915 a(n) = A023900(A005940(1+n)).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

PARI

Formula

A331733 a(n) = sigma(A225546(n)), where sigma is the sum of divisors.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A324545 An analog of sigma (A000203) for nonstandard factorization based on the sieve of Eratosthenes (A083221).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

PARI

Formula

A324056 a(n) = A000593(A005940(1+n)).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A324349 a(n) = A324122(A005940(1+n)), where A005940 is the Doudna sequence and A324122(n) = sigma(n) - gcd(n*d(n), sigma(n)).

Views

Author

Keywords

Comments