A014963 -id:A014963 - cp's OEIS frontend

A110968 a(n) is the starting position of the first run of n ones in A014963.

1, 14, 20, 33, 54, 1025, 90, 513, 140, 536870913, 200, 144115188075855860, 294, 65522, 1832, 8193, 1070, 147573952589676412910, 888, 524289, 1130, 549755813889, 4178, 17179869185, 2478, 16385, 2972, 131073, 1332, 34359738338, 5592, 18014398509481952, 8468

Offset: 1

Franz Vrabec, Sep 27 2005

nonn

Probably a(n) exists for every n in N.

If n = 2*k, then a(n) - 1 or a(n) + n is of the form 2^e. - Jinyuan Wang, Mar 21 2020

			a(3) = 20 because the first run of 3 ones in A014963 begins at position 20.

Franz Vrabec, Table of n, a(n) for n = 1..250

Cf. A014963.

a(n) = if(n%2, my(c=0); for(k=1, oo, if(isprimepower(k), if(c==n, return(k-n), c=0), c++)), my(m=1); for(k=1, oo, m*=2; if(isprimepower(m-n-1) && sum(i=m-n, m-1, isprimepower(i))==0, return(m-n)); if(isprimepower(m+n+1) && sum(i=m+1, m+n, isprimepower(i))==0, return(m+1)))); \\ Jinyuan Wang, Mar 21 2020

A383293 Exponential of Mangoldt function applied to EKG-sequence: a(n) = A014963(A064413(n)).

Original entry on oeis.org

1, 2, 2, 1, 3, 3, 1, 2, 1, 5, 1, 1, 1, 7, 1, 1, 2, 1, 1, 11, 1, 3, 1, 5, 1, 1, 1, 13, 1, 1, 2, 1, 17, 1, 1, 1, 19, 1, 1, 1, 1, 1, 23, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 29, 1, 1, 1, 31, 1, 1, 2, 1, 1, 37, 1, 1, 1, 1, 1, 1, 41, 1, 3, 1, 1, 1, 1, 43, 1, 1, 1, 1, 1, 1, 1, 47, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 53

Offset: 1

Antti Karttunen, Apr 22 2025

nonn

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

Cf. A014963, A064413, A383294 (positions of terms > 1).

Cf. also A265576.

A139553 Triangle read by rows: T(n,k) = if n>=4k and n<4k*A014963(k) then k else 1; T(n,0)=1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Mats Granvik, Apr 27 2008

Row products give A139554.

			Row products of the triangle are:
1 = 1
1*1 = 1
1*1*1 = 1
1*1*1*1 = 1
1*1*1*1*1 = 1
1*1*1*1*1*1 = 1
1*1*1*1*1*1*1 = 1
1*1*1*1*1*1*1*1 = 1
1*1*2*1*1*1*1*1*1 = 2

Antti Karttunen, Table of n, a(n) for n = 0..23219; the first 215 rows of triangle

Cf. A139554, A139547, A003418.

=if(and(row()-1>=(column()-1)*4;row()-1 < A014963(k-1)*(column()-1)*4);column()-1;1)

up_to = 23220; \\ binomial(215+1,2)
A014963(n) = { ispower(n, , &n); if(isprime(n), n, 1); }; \\ From A014963 by Charles R Greathouse IV, Jun 10 2011
A139553tr(n, k) = if(0==k,1,if((n>=(4*k))&&(n<(4*k*A014963(k))),k,1));
A139553list(up_to) = { my(v = vector(up_to), i=0); for(n=1,oo, for(k=1,n, i++; if(i > up_to, return(v)); v[i] = A139553tr(n-1,k-1))); (v); };
v139553 = A139553list(up_to);
A139553(n) = v139553[1+n]; \\ Antti Karttunen, Jan 03 2019

Typo in the definition corrected by Antti Karttunen, Jan 03 2019

A140580 a(n) = (n^2)/A048671(n), = nA014963(n) = A140579 [1, 2, 3, ...].

Original entry on oeis.org

1, 4, 9, 8, 25, 6, 49, 16, 27, 10, 121, 12, 169, 14, 15, 32, 289, 18, 361, 20, 21, 22, 529, 24, 125, 26, 81, 28, 841, 30, 961, 64, 33, 34, 35, 36, 1369, 38, 39, 40, 1681, 42, 1849, 44, 45, 46, 2209, 48, 343, 50, 51, 52, 2809, 54, 55, 56, 57, 58, 3481, 60, 3721, 62, 63

Offset: 1

Gary W. Adamson, May 17 2008

nonn

a(n) gives the last row of columns in A133233. - Mats Granvik, Jun 07 2008

			a(9) = 27 = 81/3 where 9^2 = 81 and A048671(9) = 3.
a(9) = 27 = 9*A014963(n) = 9*3.

Cf. A140579, A014963, A048671.

Cf. A133233.

Corrected and extended by Mats Granvik, Jun 07 2008

A253141 If n is a prime power, then a(n) = lambda(tau(n)) = A014963(A000005(n)); otherwise, a(n) = 1.

Original entry on oeis.org

1, 2, 2, 3, 2, 1, 2, 2, 3, 1, 2, 1, 2, 1, 1, 5, 2, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 7, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 5, 1, 2, 1, 1, 1, 1

Offset: 1

Matthew Vandermast, Dec 27 2014

For any integer sequence a, the sequence b such that b(n) = Product_{d|n} a(d) is a divisibility sequence. Since A253139(n) = Product_{d|n} a(d), A253139 is a divisibility sequence.

a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24 = 2^3*3 and 375 = 3*5^3 both have prime signature (3,1).

			2 is a prime number, i.e., a prime power with 2 divisors; a(2) = A014963(2) = 2.
6 = 2*3 is not a prime power; a(6) = 1.
8 = 2^3 is a prime power with 4 divisors; a(8) = A014963(4) = 2.
32 = 2^5 is a prime power with 6 divisors; a(32) = A014963(6) = 1.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Morgan Ward, A note on divisibility sequences, Bull. Amer. Math. Soc., 45 (1939), 334-336.
Index entries for sequences computed from exponents in factorization of n

Cf. A000005, A014963, A253139.

Table[If[PrimePowerQ[n], Exp[MangoldtLambda[DivisorSigma[0, n]]], 1], {n, 1, 100}] (* Indranil Ghosh, Jul 19 2017 *)

A014963(n) = ispower(n, , &n); if(isprime(n), n, 1); \\ This function from Charles R Greathouse IV, Jun 10 2011
A253141(n) = if(1==omega(n), A014963(numdiv(n)), 1); \\ Antti Karttunen, Jul 19 2017

A306694 a(n) is the denominator of log(A014963(n))/log(n) if n > 1 and a(1) = 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2

Offset: 1

Mats Granvik, Mar 05 2019

nonn

Log(A112624(n)) is the inverse Möbius transform of log(a(n)).

Antti Karttunen, Table of n, a(n) for n = 1..100000
Index entries for sequences computed from exponents in factorization of n

Cf. A014963, A112624, A246655.

with(numtheory): pexp := n -> ifactors(n)[2][1][2]:
a := n -> if nops(factorset(n)) = 1 then pexp(n) else 1 fi:
seq(a(n), n=1..101); # Peter Luschny, Mar 19 2019

Table[Denominator[FullSimplify[MangoldtLambda[n]/Log[n]]], {n, 1, 101}]

A306694(n) = if((n=isprimepower(n))>0,n,1); \\ Antti Karttunen, Nov 17 2019

def a(n):
    F = n.factor()
    return 1 if len(F) != 1 else F[0][1]
print([a(n) for n in (1..101)]) # Peter Luschny, Mar 18 2019

If n is a prime power (in the sense of A246655) then a(n) is the exponent of this prime and otherwise a(n) is 1. - Peter Luschny, Mar 18 2019

Dirichlet generating function: zeta(s) + Sum_{n>=1} n*primezeta((n + 1)*s). - Mats Granvik, Mar 24 2019

Data section extended up to term a(121) by Antti Karttunen, Nov 17 2019

A369010 Exponential of Mangoldt function M(n) applied to primorial base exp-function: a(n) = A014963(A276086(n)).

Original entry on oeis.org

1, 2, 3, 1, 3, 1, 5, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Antti Karttunen, Jan 14 2024

nonn

Also LCM-transform of A276086, because A276086 has the S-property explained in the comments of A368900.

Antti Karttunen, Table of n, a(n) for n = 0..30030
Index entries for sequences related to primorial base

Cf. A014963, A060735 (positions of terms > 1), A276086, A368900.

A014963(n) = { ispower(n, , &n); if(isprime(n), n, 1); };
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A369010(n) = A014963(A276086(n));

up_to = 510511; \\ = 1+A002110(7);
LCMtransform(v) = { my(len = length(v), b = vector(len), g = vector(len)); b[1] = g[1] = 1; for(n=2,len, g[n] = lcm(g[n-1],v[n]); b[n] = g[n]/g[n-1]); (b); };
v369010 = LCMtransform(vector(up_to,n,A276086(n-1)));
A369010(n) = v369010[1+n];

a(n) = A014963(A276086(n)).

For n > 0, a(n) = lcm {1..A276086(n)} / lcm {1..A276086(n-1)}.

A380118 a(n) = Sum_{k=1..n} (A014963(k) - A061397(k)).

Original entry on oeis.org

1, 1, 1, 3, 3, 4, 4, 6, 9, 10, 10, 11, 11, 12, 13, 15, 15, 16, 16, 17, 18, 19, 19, 20, 25, 26, 29, 30, 30, 31, 31, 33, 34, 35, 36, 37, 37, 38, 39, 40, 40, 41, 41, 42, 43, 44, 44, 45, 52, 53, 54, 55, 55, 56, 57, 58, 59, 60, 60, 61, 61, 62, 63, 65, 66, 67, 67, 68, 69, 70

Offset: 1

Peter Luschny, Jan 30 2025

nonn

Cf. A014963, A061397, A048671, A010051, A182936, A034387, A072107, A380117.

pSum := L -> ListTools:-PartialSums(L): h := n -> n/A048671(n) - n*A010051(n):
aList := upto -> pSum([seq(h(k), k = 1..upto)]): aList(70);

Accumulate[Table[Exp[MangoldtLambda[n]] - If[PrimeQ[n], n, 0] , {n, 1, 70}]]

a(n) = A072107(n) - A034387(n). - Amiram Eldar, Jan 30 2025

A139549 Triangle read by rows: T(n,k) = if n>=2k and n<2k*A014963(k-1) then k else 1 T(n,0)=1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 1, 3, 4, 1, 1, 1, 1, 1, 1, 1, 3, 4, 1, 1, 1, 1, 1, 1, 1, 1, 3, 4, 5, 1, 1, 1, 1, 1, 1, 1, 1, 3, 4, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 4, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 4, 5, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Mats Granvik, Apr 27 2008

Row products give A139550.

			Row products of the triangle are:
1 = 1
1*1 = 1
1*1*1 = 1
1*1*1*1 = 1
1*1*2*1*1 = 2
1*1*2*1*1*1 = 2
1*1*2*3*1*1*1 = 6
1*1*2*3*1*1*1*1 = 6
1*1*1*3*4*1*1*1*1 = 12

Cf. A139550, A139547, A003418.

=if(and(row()-1>=(column()-1)*2;row()-1 < A014963(k-1)*(column()-1)*2);column()-1;1)

A139551 Triangle read by rows: T(n,k) = if n>=3k and n<3k*A014963(k-1) then k else 1 T(n,0)=1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Mats Granvik, Apr 27 2008

Row products give A139552.

			Row products of the triangle are:
1 = 1
1*1 = 1
1*1*1 = 1
1*1*1*1 = 1
1*1*1*1*1 = 1
1*1*1*1*1*1 = 1
1*1*2*1*1*1*1 = 2
1*1*2*1*1*1*1*1 = 2
1*1*2*1*1*1*1*1*1 = 2

Cf. A139552, A139547, A003418.

=if(and(row()-1>=(column()-1)*3;row()-1 < A014963(k-1)*(column()-1)*3);column()-1;1)

cp's OEIS Frontend

A110968 a(n) is the starting position of the first run of n ones in A014963.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A383293 Exponential of Mangoldt function applied to EKG-sequence: a(n) = A014963(A064413(n)).

Views

Author

Keywords

Links

Crossrefs

A139553 Triangle read by rows: T(n,k) = if n>=4*k and n<4*k*A014963(k) then k else 1; T(n,0)=1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Excel

PARI

Extensions

A140580 a(n) = (n^2)/A048671(n), = n*A014963(n) = A140579 * [1, 2, 3, ...].

Views

Author

Keywords

Comments

Examples

Crossrefs

Extensions

A253141 If n is a prime power, then a(n) = lambda(tau(n)) = A014963(A000005(n)); otherwise, a(n) = 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A306694 a(n) is the denominator of log(A014963(n))/log(n) if n > 1 and a(1) = 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Sage

Formula

Extensions

A369010 Exponential of Mangoldt function M(n) applied to primorial base exp-function: a(n) = A014963(A276086(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

PARI

Formula

A380118 a(n) = Sum_{k=1..n} (A014963(k) - A061397(k)).

Views

Author

Keywords

Crossrefs

Programs

Maple

A139553 Triangle read by rows: T(n,k) = if n>=4k and n<4k*A014963(k) then k else 1; T(n,0)=1.

A140580 a(n) = (n^2)/A048671(n), = nA014963(n) = A140579 [1, 2, 3, ...].

A139549 Triangle read by rows: T(n,k) = if n>=2k and n<2k*A014963(k-1) then k else 1 T(n,0)=1.

A139551 Triangle read by rows: T(n,k) = if n>=3k and n<3k*A014963(k-1) then k else 1 T(n,0)=1.