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.

Previous Showing 11-19 of 19 results.

A351083 a(n) = gcd(n, A003415(A276086(n))), where A003415 is the arithmetic derivative and A276086 is the primorial base exp-function.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 1, 7, 8, 1, 1, 1, 2, 1, 1, 5, 16, 1, 3, 1, 10, 1, 1, 1, 4, 25, 1, 1, 2, 1, 1, 1, 2, 1, 17, 5, 12, 1, 1, 13, 2, 1, 1, 1, 4, 5, 1, 1, 2, 1, 25, 1, 4, 1, 3, 5, 2, 1, 1, 1, 2, 1, 1, 7, 4, 1, 1, 1, 2, 1, 7, 1, 24, 1, 1, 5, 2, 7, 1, 1, 80, 1, 1, 1, 14, 5, 1, 1, 8, 1, 3, 91, 4, 1, 1, 1, 2, 1, 49, 1, 4
Offset: 0

Views

Author

Antti Karttunen, Feb 03 2022

Keywords

Links

Crossrefs

Cf. A003415, A276086, A324198, A327860, A328572, A351080, A351084, A351087 (fixed points), A354823 (Dirichlet inverse), A373145, A373599 (indices of multiples of 3 in this sequence).
Cf. A166486 (a(n) mod 2, parity of terms, see comment in A327860).
Cf. also A345000.

Programs

  • Mathematica
    Array[Block[{i, m, n = #, p}, m = i = 1; While[n > 0, p = Prime[i]; m *= p^Mod[n, p]; n = Quotient[n, p]; i++]; GCD[#, If[m < 2, 0, m Total[#2/#1 & @@@ FactorInteger[m]]]]] &, 101, 0] (* Michael De Vlieger, Feb 04 2022 *)
  • PARI
    A327860(n) = { my(s=0, m=1, p=2, e); while(n, e = (n%p); m *= (p^e); s += (e/p); n = n\p; p = nextprime(1+p)); (s*m); };
A351083(n) = gcd(n, A327860(n));

Formula

a(n) = gcd(n, A327860(n)) = gcd(n, A003415(A276086(n))).
a(n) = A373145(A276086(n)). - Antti Karttunen, Jun 18 2024

A351236 Lexicographically earliest infinite sequence such that a(i) = a(j) => A344025(i) = A344025(j) and A351085(i) = A351085(j) for all i, j >= 1.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 6, 7, 2, 8, 2, 9, 10, 11, 2, 12, 2, 13, 14, 15, 2, 16, 17, 18, 19, 20, 2, 21, 2, 22, 23, 24, 25, 26, 2, 27, 28, 29, 2, 30, 2, 31, 32, 33, 2, 34, 35, 36, 37, 38, 2, 39, 28, 40, 41, 21, 2, 42, 2, 43, 44, 45, 46, 47, 2, 48, 49, 50, 2, 51, 2, 52, 53, 54, 46, 55, 2, 56, 57, 58, 2, 59, 41, 60, 61, 62, 2, 63, 64, 65
Offset: 1

Views

Author

Antti Karttunen, Feb 06 2022

Keywords

Comments

Restricted growth sequence transform of the 4-tuple [A003415(n), A003557(n), A327858(n), A345000(n)].
Question: If an image-analysis algorithm were to classify the scatter plot of this sequence, where it would cluster it? Nearer to A344025 than to A351085?

Links

Crossrefs

Cf. A003415, A003557, A327858, A345000, A351085.
Differs from A344025 for the first time at n=91, where a(91) = 64, while A344025(91) = 37.
Cf. also A305800, A351235, A351260.

Programs

  • PARI
    up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
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); };
A327858(n) = gcd(A003415(n),A276086(n));
A345000(n) = gcd(A003415(n),A003415(A276086(n)));
Aux351236(n) = [A003415(n), A003557(n), A327858(n), A345000(n)];
v351236 = rgs_transform(vector(up_to, n, Aux351236(n)));
A351236(n) = v351236[n];

A347389 Dirichlet convolution of A003415(n) and A003415(A276086(n)), where A003415(n) is the arithmetic derivative of n, and A276086(n) gives the prime product form of primorial base expansion of n.

Original entry on oeis.org

0, 1, 1, 5, 1, 11, 1, 22, 11, 29, 1, 48, 1, 17, 34, 76, 1, 84, 1, 160, 22, 137, 1, 172, 31, 61, 88, 130, 1, 404, 1, 456, 142, 725, 40, 411, 1, 297, 66, 900, 1, 1262, 1, 1984, 421, 4001, 1, 1244, 21, 1866, 730, 2382, 1, 6574, 160, 8740, 302, 22157, 1, 1930, 1, 43, 1249, 1530, 84, 2222, 1, 2968, 4006, 568, 1, 1860
Offset: 1

Views

Author

Antti Karttunen, Sep 02 2021

Keywords

Links

Crossrefs

Cf. A003415, A276086, A327860.
Cf. also A345000.

Programs

  • PARI
    A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A347389(n) = sumdiv(n,d,A003415(n/d) * A003415(A276086(d)));

Formula

a(n) = Sum_{d|n} A003415(n/d) * A327860(d).

A351086 a(n) = gcd(A003415(n), A328572(n)), where A003415 is the arithmetic derivative and A328572 converts the primorial base expansion of n into its prime product form, but with 1 subtracted from all nonzero digits.

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, 5, 1, 1, 1, 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 5, 3, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 35, 1, 1, 1, 1, 1, 49, 3, 1, 1, 7, 1, 7, 1, 7
Offset: 0

Views

Author

Antti Karttunen, Feb 03 2022

Keywords

Links

Crossrefs

Cf. A003415, A276086, A327858, A345000, A351084, A351085.

Programs

  • PARI
    A003415(n) = {my(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1]))}; \\ From A003415
A328572(n) = { my(m=1, p=2); while(n, if(n%p, m *= p^((n%p)-1)); n = n\p; p = nextprime(1+p)); (m); };
A351086(n) = gcd(A003415(n), A328572(n));
  • PARI
    A351086(n) = { my(m=1, p=2, orgn=A003415(n)); while(n, if(n%p, m *= (p^min((n%p)-1, valuation(orgn, p)))); n = n\p; p = nextprime(1+p)); (m); };

Formula

a(n) = gcd(A003415(n), A328572(n)).
a(n) = gcd(A327858(n), A345000(n)).

A355831 Lexicographically earliest infinite sequence such that a(i) = a(j) => A046523(i) = A046523(j) and A354347(i) = A354347(j) for all i, j >= 1.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 6, 4, 2, 7, 2, 4, 4, 8, 2, 9, 2, 7, 10, 4, 2, 11, 12, 10, 13, 7, 2, 14, 2, 15, 4, 4, 16, 17, 2, 4, 4, 18, 2, 19, 2, 20, 21, 10, 2, 22, 6, 23, 10, 24, 2, 25, 4, 18, 4, 4, 2, 26, 2, 4, 27, 28, 16, 14, 2, 7, 4, 29, 2, 30, 2, 4, 31, 32, 16, 19, 2, 33, 34, 4, 2, 35, 4, 10, 4, 36, 2, 37, 4, 38, 4, 39, 16, 40, 2, 41, 21, 42, 2, 43, 2, 44, 45
Offset: 1

Views

Author

Antti Karttunen, Jul 20 2022

Keywords

Comments

Restricted growth sequence transform of the ordered pair [A046523(n), A354347(n)].

Links

Crossrefs

Cf. A305800, A355830, A355832.

Programs

  • PARI
    up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A345000(n) = gcd(A003415(n),A003415(A276086(n)));
v354347 = DirInverseCorrect(vector(up_to,n,A345000(n)));
A354347(n) = v354347[n];
Aux355831(n) = [A046523(n), A354347(n)];
v355831 = rgs_transform(vector(up_to,n,Aux355831(n)));
A355831(n) = v355831[n];

A355832 Lexicographically earliest infinite sequence such that a(i) = a(j) => A354347(i) = A354347(j) for all i, j >= 1.

Original entry on oeis.org

1, 2, 2, 2, 2, 1, 2, 2, 3, 1, 2, 1, 2, 1, 1, 4, 2, 5, 2, 1, 6, 1, 2, 1, 7, 6, 3, 1, 2, 2, 2, 8, 1, 1, 2, 9, 2, 1, 1, 10, 2, 11, 2, 2, 3, 6, 2, 12, 3, 13, 6, 11, 2, 14, 1, 10, 1, 1, 2, 2, 2, 1, 15, 16, 2, 2, 2, 1, 1, 10, 2, 17, 2, 1, 18, 19, 2, 11, 2, 20, 3, 1, 2, 10, 1, 6, 1, 21, 2, 14, 1, 22, 1, 23, 2, 24, 2, 9, 3, 25, 2, 1, 2, 26, 26
Offset: 1

Views

Author

Antti Karttunen, Jul 19 2022

Keywords

Comments

Restricted growth sequence transform of A354347, which is the Dirichlet inverse of A345000(n) = gcd(A003415(n), A003415(A276086(n))).

Links

Crossrefs

Cf. A003415, A276086, A345000, A354347, A355831.

Programs

  • PARI
    up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A345000(n) = gcd(A003415(n),A003415(A276086(n)));
v354347 = DirInverseCorrect(vector(up_to,n,A345000(n)));
v355832 = rgs_transform(v354347);
A355832(n) = v355832[n];

A369038 Numerator of ratio A003415(n) / A003415(A276086(n)), where A003415 is the arithmetic derivative and A276086 is the primorial base exp-function.

Original entry on oeis.org

0, 1, 1, 2, 1, 5, 1, 3, 6, 7, 1, 8, 1, 9, 8, 2, 1, 7, 1, 12, 2, 13, 1, 11, 2, 3, 27, 16, 1, 31, 1, 8, 14, 19, 4, 5, 1, 21, 16, 34, 1, 41, 1, 12, 39, 5, 1, 56, 14, 9, 4, 14, 1, 27, 16, 46, 22, 31, 1, 46, 1, 33, 51, 16, 6, 61, 1, 36, 26, 59, 1, 13, 1, 39, 1, 8, 6, 71, 1, 11, 108, 43, 1, 62, 22, 9, 32, 1, 1, 41, 20
Offset: 1

Views

Author

Antti Karttunen, Jan 20 2024

Keywords

Links

Crossrefs

Cf. A003415, A276086, A327860, A345000, A369039 (denominators).
Cf. also A351230, A351250.

Programs

  • PARI
    A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A327860(n) = { my(s=0, m=1, p=2, e); while(n, e = (n%p); m *= (p^e); s += (e/p); n = n\p; p = nextprime(1+p)); (s*m); };
A369038(n) = { my(u=A003415(n)); (u/gcd(u,A327860(n))); };

Formula

a(n) = A003415(n) / A345000(n) = A003415(n) / gcd(A003415(n), A327860(n)).

A369035 Numbers k for which A327860(k) is a multiple of 4, where A327860 is the arithmetic derivative of the primorial base exp-function.

Original entry on oeis.org

0, 8, 16, 24, 36, 44, 52, 64, 72, 80, 88, 92, 100, 108, 116, 120, 128, 136, 144, 156, 164, 172, 184, 192, 200, 208, 216, 224, 232, 244, 252, 260, 268, 272, 280, 288, 296, 300, 308, 316, 324, 336, 344, 352, 364, 372, 380, 388, 392, 400, 408, 416, 424, 432, 440, 448, 452, 460, 468, 476, 480, 488, 496, 504, 516, 524
Offset: 1

Views

Author

Antti Karttunen, Jan 20 2024

Keywords

Links

Crossrefs

Cf. A327860, A369034 (characteristic function).
Setwise difference A008586 \ A369037.
Positions of 0's in A353630. Positions of multiples of 4 in A345000.

Programs

A369039 Denominator of ratio A003415(n) / A003415(A276086(n)), where A003415 is the arithmetic derivative and A276086 is the primorial base exp-function.

Original entry on oeis.org

1, 1, 5, 3, 21, 1, 7, 2, 31, 39, 123, 5, 45, 55, 185, 15, 705, 25, 275, 175, 215, 1425, 3975, 125, 325, 425, 6125, 4125, 22125, 1, 9, 1, 41, 51, 55, 1, 59, 71, 247, 159, 951, 95, 365, 115, 1445, 381, 5385, 325, 2175, 565, 1655, 2775, 30075, 1375, 12625, 8375, 46625, 63375, 166125, 7, 77, 91, 329, 35, 427, 119, 483
Offset: 1

Views

Author

Antti Karttunen, Jan 20 2024

Keywords

Links

Crossrefs

Cf. A003415, A276086, A327860, A345000, A369038 (numerators).
Cf. also A351231, A351251.

Programs

  • PARI
    A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A327860(n) = { my(s=0, m=1, p=2, e); while(n, e = (n%p); m *= (p^e); s += (e/p); n = n\p; p = nextprime(1+p)); (s*m); };
A369039(n) = { my(u=A327860(n)); u/gcd(A003415(n),u); };

Formula

a(n) = A327860(n) / A345000(n) = A327860(n) / gcd(A003415(n), A327860(n)).
Previous Showing 11-19 of 19 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.