A346471 -id:A346471 - cp's OEIS frontend

A346470 a(n) = psi(A276086(n)), where psi is Dedekind psi function A001615, and A276086 is the prime product form of primorial base expansion of n.

1, 3, 4, 12, 12, 36, 6, 18, 24, 72, 72, 216, 30, 90, 120, 360, 360, 1080, 150, 450, 600, 1800, 1800, 5400, 750, 2250, 3000, 9000, 9000, 27000, 8, 24, 32, 96, 96, 288, 48, 144, 192, 576, 576, 1728, 240, 720, 960, 2880, 2880, 8640, 1200, 3600, 4800, 14400, 14400, 43200, 6000, 18000, 24000, 72000, 72000, 216000, 56, 168, 224, 672

Offset: 0

Antti Karttunen, Jul 21 2021

Antti Karttunen, Table of n, a(n) for n = 0..2310
Index entries for sequences related to primorial base

Cf. A001615.
Other number-theoretical functions similarly applied to A276086: A267263 (omega), A276150 (bigomega), A324650 (phi), A324653 (sigma), A324655 (tau), A327860 (arithmetic derivative).
Cf. also A346471, A346475.

A346470(n) = { my(m=1, p=2, e); while(n, e = (n%p); if(e, m *= (p+1)*(p^(e-1))); n = n\p; p = nextprime(1+p)); (m); };

a(n) = A001615(A276086(n)).

A347126 a(n) = A347129(A276086(n)).

Original entry on oeis.org

0, 1, 1, 10, 3, 21, 1, 14, 16, 124, 39, 246, 3, 27, 33, 222, 72, 423, 6, 44, 56, 344, 114, 636, 10, 65, 85, 490, 165, 885, 1, 18, 20, 164, 51, 330, 24, 236, 284, 1976, 636, 3804, 57, 438, 552, 3468, 1143, 6462, 104, 696, 904, 5296, 1776, 9624, 165, 1010, 1340, 7460, 2535, 13290, 3, 33, 39, 282, 90, 549, 51, 414, 516

Offset: 0

Antti Karttunen, Aug 25 2021

Antti Karttunen, Table of n, a(n) for n = 0..11550
Index entries for sequences related to primorial base

Cf. A003415, A003557, A276086, A347129, A347130.

Cf. also A342002, A346471, A347232.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003557(n) = (n/factorback(factorint(n)[, 1]));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A347130(n) = sumdiv(n,d,d*A003415(n/d));
A347129(n) = (A347130(n) / A003557(n));
A347126(n) = A347129(A276086(n));

a(n) = A347129(A276086(n)).

A347396 a(n) = A347395(A276086(n)), where A347395 is Dirichlet convolution of Liouville's lambda with A342001.

Original entry on oeis.org

0, 1, 1, 3, 1, 2, 1, 5, 6, 14, 5, 9, 1, 2, 3, 5, 2, 3, 2, 6, 8, 16, 6, 10, 2, 3, 5, 7, 3, 4, 1, 7, 8, 20, 7, 13, 10, 34, 44, 92, 34, 58, 7, 13, 20, 32, 13, 19, 16, 40, 56, 104, 40, 64, 13, 19, 32, 44, 19, 25, 1, 2, 3, 5, 2, 3, 5, 9, 14, 22, 9, 13, 2, 3, 5, 7, 3, 4, 6, 10, 16, 24, 10, 14, 3, 4, 7, 9, 4, 5, 2, 8, 10, 22

Offset: 0

Antti Karttunen, Sep 02 2021

The scatter plot looks quite peculiar. - Antti Karttunen, Sep 20 2021

Antti Karttunen, Table of n, a(n) for n = 0..11550
Index entries for sequences related to primorial base

Cf. A003415, A003557, A008836, A276086, A342001, A347395.

Cf. also A342002, A346471, A347126, A347232, A347962.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003557(n) = (n/factorback(factorint(n)[, 1]));
A342001(n) = (A003415(n) / A003557(n));
A008836(n) = ((-1)^bigomega(n));
A347395(n) = sumdiv(n,d,A008836(n/d)*A342001(d));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A347396(n) = A347395(A276086(n));

A346475 a(n) = A342919(A276086(n)).

Original entry on oeis.org

0, 1, 1, 5, 1, 7, 1, 7, 1, 31, 13, 41, 1, 1, 11, 37, 2, 47, 1, 11, 7, 43, 19, 53, 2, 13, 17, 49, 11, 59, 1, 3, 5, 41, 17, 55, 1, 59, 71, 247, 53, 317, 19, 73, 23, 289, 127, 359, 13, 29, 113, 331, 37, 401, 11, 101, 67, 373, 169, 443, 1, 11, 13, 47, 5, 61, 17, 23, 43, 277, 121, 347, 1, 83, 107, 319, 71, 389, 31, 97, 2, 361, 163

Offset: 0

Antti Karttunen, Jul 21 2021

For n >= 1, each term a(n) is a divisor of A342002(n).

Antti Karttunen, Table of n, a(n) for n = 0..13860
Index entries for sequences related to primorial base

Cf. A001615, A003415, A276086, A327860, A342919, A346470, A346471.

Cf. also A342002, A345930, A346474 for sequences with similar scatter plots.

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))); \\ After code in A001615
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A342919(n) = { my(u=A003415(n)); (u/gcd(u, A001615(n))); };
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A346475(n) = A342919(A276086(n));

a(n) = A342919(A276086(n)).

a(n) = A327860(n) / gcd(A327860(n), A346470(n)).

A347232 a(n) = A346485(A276086(n)), where A346485 is Möbius transform of A342001.

Original entry on oeis.org

0, 1, 1, 3, 1, 1, 1, 5, 6, 14, 4, 4, 1, 1, 2, 2, 0, 0, 1, 1, 2, 2, 0, 0, 1, 1, 2, 2, 0, 0, 1, 7, 8, 20, 6, 6, 10, 34, 44, 92, 24, 24, 6, 6, 12, 12, 0, 0, 6, 6, 12, 12, 0, 0, 6, 6, 12, 12, 0, 0, 1, 1, 2, 2, 0, 0, 4, 4, 8, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 0, 0, 4, 4, 8, 8

Offset: 0

Antti Karttunen, Aug 26 2021

Antti Karttunen, Table of n, a(n) for n = 0..11550
Index entries for sequences related to primorial base

Cf. A003415, A003557, A276086, A342001, A346485.

Cf. also A342002, A346471, A347126, A347396.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003557(n) = (n/factorback(factorint(n)[, 1]));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A342001(n) = (A003415(n) / A003557(n));
A346485(n) = sumdiv(n,d,moebius(n/d)*A342001(d));
A347232(n) = A346485(A276086(n));

a(n) = A346485(A276086(n)).

A348997 a(n) = A348733(A276086(n)), where A348733(n) = gcd(A003959(n), A034448(n)), and A276086 gives the prime product form of primorial base expansion of n.

Original entry on oeis.org

1, 3, 4, 12, 2, 6, 6, 18, 24, 72, 12, 36, 2, 6, 8, 24, 4, 12, 18, 54, 72, 216, 36, 108, 2, 6, 8, 24, 4, 12, 8, 24, 32, 96, 16, 48, 48, 144, 192, 576, 96, 288, 16, 48, 64, 192, 32, 96, 144, 432, 576, 1728, 288, 864, 16, 48, 64, 192, 32, 96, 2, 6, 8, 24, 4, 12, 12, 36, 48, 144, 24, 72, 4, 12, 16, 48, 8, 24, 36, 108, 144

Offset: 0

Antti Karttunen, Nov 07 2021

Antti Karttunen, Table of n, a(n) for n = 0..11550
Index entries for sequences related to primorial base

Cf. A003959, A034448, A276086, A348733, A348949, A348996.

Cf. also A346471 for similar construction. (Compare the scatter plots).

A348997(n) = { my(m1=1, m2=1, p=2); while(n, if(n%p, m1 *= ((1+p)^(n%p)); m2 *= (1+(p^(n%p)))); n = n\p; p = nextprime(1+p)); gcd(m1, m2); };

a(n) = A348733(A276086(n)) = gcd(A348949(n), A348996(n)).

A346469 a(n) = A340070(A276086(n)).

Original entry on oeis.org

0, 1, 1, 5, 3, 3, 1, 7, 8, 31, 3, 3, 5, 5, 5, 5, 120, 15, 25, 25, 50, 25, 75, 75, 125, 125, 125, 125, 750, 375, 1, 9, 10, 41, 3, 3, 12, 59, 71, 247, 3, 3, 5, 5, 5, 5, 15, 15, 50, 25, 25, 25, 75, 75, 375, 125, 125, 125, 375, 375, 7, 7, 7, 7, 210, 21, 7, 7, 7, 7, 21, 21, 420, 35, 35, 35, 7455, 105, 175, 175, 175, 3325

Offset: 0

Antti Karttunen, Jul 21 2021

Antti Karttunen, Table of n, a(n) for n = 0..11550
Index entries for sequences related to primorial base

Cf. A003415, A069359, A276086, A327860, A329029, A340070, A345930.

Cf. also A346471.

A346469(n) = { my(s=0, t=0, m=1, p=2, e); while(n, e = (n%p); if(e, m *= (p^e); s += (1/p); t += (e/p)); n = n\p; p = nextprime(1+p)); (gcd(s,t)*m); };

a(n) = A340070(A276086(n)) = gcd(A327860(n), A329029(n)).

For n >= 1, a(n) = A327860(n) / A345930(n).

A366795 a(n) = A344695(A005940(1+n)), where A344695(n) = gcd(psi(n), sigma(n)), and A005940 is the Doudna sequence.

Original entry on oeis.org

1, 3, 4, 1, 6, 12, 1, 3, 8, 18, 24, 4, 1, 3, 4, 1, 12, 24, 32, 6, 48, 72, 6, 12, 1, 3, 4, 1, 6, 12, 1, 3, 14, 36, 48, 8, 72, 96, 8, 18, 96, 144, 192, 24, 8, 18, 24, 4, 1, 3, 4, 1, 6, 12, 1, 3, 8, 18, 24, 8, 1, 3, 4, 1, 18, 42, 56, 12, 84, 144, 12, 24, 112, 216, 288, 32, 12, 24, 32, 6, 168, 288, 384, 48, 576, 576

Offset: 0

Antti Karttunen, Oct 26 2023

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16384

Cf. A005940, A344695, A366796 (rgs-transform).

Cf. also A346471, A366801, A366803.

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))); \\ After code in A001615
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A344695(n) = gcd(sigma(n), A001615(n));
A366795(n) = A344695(A005940(1+n));

cp's OEIS Frontend

A346470 a(n) = psi(A276086(n)), where psi is Dedekind psi function A001615, and A276086 is the prime product form of primorial base expansion of n.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A347126 a(n) = A347129(A276086(n)).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A347396 a(n) = A347395(A276086(n)), where A347395 is Dirichlet convolution of Liouville's lambda with A342001.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A346475 a(n) = A342919(A276086(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A347232 a(n) = A346485(A276086(n)), where A346485 is Möbius transform of A342001.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A348997 a(n) = A348733(A276086(n)), where A348733(n) = gcd(A003959(n), A034448(n)), and A276086 gives the prime product form of primorial base expansion of n.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A346469 a(n) = A340070(A276086(n)).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A366795 a(n) = A344695(A005940(1+n)), where A344695(n) = gcd(psi(n), sigma(n)), and A005940 is the Doudna sequence.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI