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.

Showing 1-10 of 14 results. Next

A331184 Number of values of k, 1 <= k <= n, with A318310(k) = A318310(n).

Original entry on oeis.org

1, 2, 1, 3, 1, 1, 1, 4, 1, 2, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 3, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 4, 1, 1, 1
Offset: 1

Views

Author

Antti Karttunen, Jan 12 2020

Keywords

Comments

Ordinal transform of A318310, or equally, of the ordered pair [A000120(n), A033879(n)], i.e., binary weight of n & deficiency of n.

Links

Crossrefs

Cf. A000120, A033879, A318310.
Cf. also A331181, A331183.

Programs

  • PARI
    up_to = 16384;
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; };
A318310aux(n) = [hammingweight(n), (2*n) - sigma(n)];
v331184 = ordinal_transform(vector(up_to,n,A318310aux(n)));
A331184(n) = v331184[n];

A331744 Lexicographically earliest infinite sequence such that a(i) = a(j) => A009194(i) = A009194(j) and A323901(i) = A323901(j) for all i, j.

Original entry on oeis.org

1, 1, 2, 1, 3, 4, 5, 1, 6, 7, 8, 9, 3, 10, 11, 1, 12, 13, 14, 7, 15, 16, 17, 18, 19, 7, 17, 20, 6, 21, 22, 1, 23, 24, 25, 6, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 9, 30, 19, 37, 7, 15, 38, 39, 40, 41, 42, 14, 43, 41, 44, 22, 1, 45, 46, 47, 24, 48, 49, 50, 13, 51, 52, 53, 54, 55, 56, 57, 7, 58, 59, 60, 61, 62, 63, 64, 33, 65, 66, 67, 68, 69, 70, 71, 18, 69, 30, 72, 19
Offset: 1

Views

Author

Antti Karttunen, Feb 04 2020

Keywords

Comments

Restricted growth sequence transform of the ordered pair [A009194(n), A323901(n)].

Links

Crossrefs

Cf. A002487, A009194, A163511, A323901.
Cf. also A324400, A318310, A324389, A331745, A331746, A331747.

Programs

  • PARI
    \\ Needs also code from A323901.
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; };
A009194(n) = gcd(n, sigma(n));
Aux331744(n) = [A009194(n),A323901(n)];
v331744 = rgs_transform(vector(up_to, n, Aux331744(n)));
A331744(n) = v331744[n];

Formula

a(2^n) = 1 for all n >= 0.

A334867 Lexicographically earliest infinite sequence such that a(i) = a(j) => A329697(i) = A329697(j) and A334204(i) = A334204(j) for all i, j >= 1.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jun 09 2020

Keywords

Comments

Restricted growth sequence transform of the ordered pair [A329697(n), A334204(n)].
For all i, j:
A365388(i) = A365388(j) => a(i) = a(j) => A334873(i) = A334873(j).

Links

Crossrefs

Cf. A000079 (positions of ones), A163511, A329697, A334204, A334873.
Cf. also A318310, A365386, A365388.

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; };
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
A054429(n) = ((3<<#binary(n\2))-n-1);
A163511(n) = if(!n,1,A005940(1+A054429(n)));
A329697(n) = if(!bitand(n,n-1),0,1+A329697(n-(n/vecmax(factor(n)[, 1]))));
A334204(n) = A329697(A163511(n));
A334867aux(n) = [A329697(n),A334204(n)];
v334867 = rgs_transform(vector(up_to,n,A334867aux(n)));
A334867(n) = v334867[n];

A318311 Filter sequence combining A278222(n) and A294898(n).

Original entry on oeis.org

1, 1, 2, 1, 3, 4, 5, 1, 6, 7, 8, 9, 10, 11, 12, 1, 13, 14, 15, 16, 17, 18, 19, 20, 21, 8, 22, 23, 24, 25, 26, 1, 27, 28, 29, 30, 31, 32, 19, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 1, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 10, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 62, 79
Offset: 1

Views

Author

Antti Karttunen, Aug 25 2018

Keywords

Comments

Restricted growth sequence transform of ordered pair [A278222(n), A294898(n)].
For all i, j: a(i) = a(j) => A318310(i) = A318310(j) => A033879(i) = A033879(j).

Links

Crossrefs

Cf. A033879, A278222, A286622, A294898, A318310.

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; };
A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A294898(n) = (A005187(n)-sigma(n));
A318311aux(n) = [A278222(n), A294898(n)]; \\ Needs also code from A286622.
v318311 = rgs_transform(vector(up_to,n,A318311aux(n)));
A318311(n) = v318311[n];

A323892 Lexicographically earliest sequence such that a(i) = a(j) => A002487(i) = A002487(j) and A033879(i) = A033879(j), for all i, j >= 1.

Original entry on oeis.org

1, 1, 2, 1, 3, 4, 5, 1, 6, 7, 8, 9, 10, 3, 11, 1, 12, 13, 14, 15, 16, 17, 18, 19, 20, 8, 21, 22, 23, 24, 25, 1, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 35, 47, 48, 49, 50, 51, 52, 53, 54, 55, 1, 56, 57, 58, 8, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 47, 72, 73, 74, 75, 76, 77, 78, 79, 80, 31, 81
Offset: 1

Views

Author

Antti Karttunen, Feb 09 2019

Keywords

Comments

Restricted growth sequence transform of the ordered pair [A002487(n), A033879(n)].

Links

Crossrefs

Cf. A002487, A033879.
Cf. also A318310, A323889.

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; };
A002487(n) = { my(a=1, b=0); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (b); }; \\ From A002487
A033879(n) = (2*n-sigma(n));
A323892aux(n) = [A002487(n), A033879(n)];
v323892 = rgs_transform(vector(up_to,n,A323892aux(n)));
A323892(n) = v323892[n];

Formula

a(2^n) = 1 for all n >= 0.

A324389 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A009194(n), A318458(n)] for all other numbers, except f(1) = -1.

Original entry on oeis.org

1, 2, 3, 2, 3, 4, 3, 2, 2, 5, 3, 6, 3, 7, 8, 2, 3, 9, 3, 10, 3, 11, 3, 12, 2, 13, 14, 15, 3, 16, 3, 2, 17, 18, 3, 19, 3, 11, 3, 20, 3, 21, 3, 22, 23, 7, 3, 6, 2, 24, 25, 26, 3, 27, 28, 29, 28, 30, 3, 31, 3, 32, 33, 2, 3, 34, 3, 18, 17, 35, 3, 36, 3, 5, 3, 37, 3, 38, 3, 39, 2, 18, 3, 40, 41, 11, 17, 42, 3, 43, 44, 45, 3, 46, 47, 12, 3, 48, 23, 49, 3, 50, 3
Offset: 1

Views

Author

Antti Karttunen, Mar 05 2019

Keywords

Comments

For all i, j:
A324401(i) = A324401(j) => a(i) = a(j).
Regarding the scatter plot of this sequence, see also comments in A318310. - Antti Karttunen, Feb 04 2020

Links

Crossrefs

Cf. A009194, A318310, A318458, A324390, A324401, A324530, A331744, A331746, A331747.

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; };
A009194(n) = gcd(n,sigma(n));
A318458(n) = bitand(n,sigma(n)-n);
Aux324389(n) = if(1==n,-1,[A009194(n), A318458(n)]);
v324389 = rgs_transform(vector(up_to,n,Aux324389(n)));
A324389(n) = v324389[n];

A323168 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j), where f(n) = [A322867(n), A323174(n)] for n > 1, and f(1) = 0.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 6, 7, 2, 3, 2, 8, 9, 10, 2, 11, 2, 12, 13, 14, 2, 15, 16, 17, 18, 19, 2, 20, 2, 21, 22, 23, 20, 21, 2, 24, 25, 26, 2, 27, 2, 28, 29, 30, 2, 31, 32, 33, 34, 35, 2, 36, 37, 38, 39, 40, 2, 41, 2, 42, 43, 44, 45, 46, 2, 47, 48, 49, 2, 50, 2, 51, 52, 53, 54, 55, 2, 15, 32, 56, 2, 57, 58, 59, 60, 61, 2, 62, 63, 64, 65, 66, 67, 21, 2, 68, 69, 70, 2, 71, 2, 57
Offset: 1

Views

Author

Antti Karttunen, Jan 10 2019

Keywords

Comments

Restricted growth sequence transform of function f, with f(1) = 0 and f(n) = [A322867(n), A323174(n)] for n > 1.
Equally, restricted growth sequence transform of function f, with f(1) = 0 and f(n) = A318310(A122111(n)) for n > 1.
For all i, j:
a(i) = a(j) => A322867(i) = A322867(j),
a(i) = a(j) => A323167(i) = A323167(j),
a(i) = a(j) => A323174(i) = A323174(j).

Links

Crossrefs

Cf. A000203, A005187, A122111, A294898, A318310, A322867, A323167, A323173, A323174.
Cf. also A323240.

Programs

  • PARI
    up_to = 4096;
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; };
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A122111(n) = if(1==n,n,prime(bigomega(n))*A122111(A064989(n)));
A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A294898(n) = (A005187(n)-sigma(n));
A318310aux(n) = [hammingweight(n), A294898(n)];
A323168aux(n) = if(1==n,0,A318310aux(A122111(n)));
v323168 = rgs_transform(vector(up_to, n, A323168aux(n)));
A323168(n) = v323168[n];

A323898 Lexicographically earliest sequence such that a(i) = a(j) => A000120(i) = A000120(j) and A083254(i) = A083254(j), for all i, j >= 1.

Original entry on oeis.org

1, 2, 3, 2, 4, 5, 6, 2, 4, 5, 7, 8, 9, 10, 11, 2, 12, 13, 14, 8, 15, 10, 16, 17, 18, 10, 19, 20, 21, 22, 23, 2, 24, 5, 25, 26, 27, 10, 19, 17, 28, 29, 30, 20, 31, 32, 33, 34, 27, 35, 36, 20, 37, 38, 39, 40, 41, 32, 42, 43, 44, 45, 46, 2, 47, 48, 49, 8, 50, 51, 52, 53, 54, 10, 55, 20, 56, 57, 58, 34, 59, 10, 60, 61, 56, 32, 39, 40, 62, 63, 64, 65, 66, 45, 67, 68, 69, 70, 16
Offset: 1

Views

Author

Antti Karttunen, Feb 09 2019

Keywords

Comments

Restricted growth sequence transform of the ordered pair [A000120(n), A083254(n)].

Links

Crossrefs

Cf. A000120, A083254.
Cf. also A318310, A323892, A323898.

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; };
A083254(n) = (2*eulerphi(n)-n);
A323898aux(n) = [hammingweight(n), A083254(n)];
v323898 = rgs_transform(vector(up_to,n,A323898aux(n)));
A323898(n) = v323898[n];

Formula

a(2^n) = 2 for all n >= 1.

A324344 Lexicographically earliest positive sequence such that a(i) = a(j) => A000120(i) = A000120(j) and A324342(i) = A324342(j), for all i, j >= 0.

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 4, 5, 2, 3, 4, 6, 7, 8, 9, 10, 2, 3, 4, 11, 4, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 2, 3, 4, 11, 22, 11, 15, 19, 14, 23, 24, 25, 26, 13, 10, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 2, 3, 4, 11, 44, 5, 23, 45, 44, 12, 16, 46, 32, 33, 33, 47, 48, 18, 16, 46, 26, 37, 49, 50, 16, 51, 52, 53, 13, 54, 55, 56, 57, 36, 58, 38, 59, 49, 60
Offset: 0

Views

Author

Antti Karttunen, Feb 24 2019

Keywords

Comments

Restricted growth sequence transform of the ordered pair [A000120(n), A324342(n)].

Links

Crossrefs

Cf. A000120, A324342, A324343.
Cf. also A318310.

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; };
A002110(n) = prod(i=1,n,prime(i));
A030308(n,k) = bittest(n,k);
A283477(n) = prod(i=0,#binary(n),if(0==A030308(n,i),1,A030308(n,i)*A002110(1+i)));
A276150(n) = { my(s=0,m); forprime(p=2, , if(!n, return(s)); m = n%p; s += m; n = (n-m)/p); };
A324342(n) = A276150(A283477(n));
A324344aux(n) = [hammingweight(n), A324342(n)];
v324344 = rgs_transform(vector(1+up_to,n,A324344aux(n-1)));
A324344(n) = v324344[1+n];

A331745 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278222(i) = A278222(j) and A323901(i) = A323901(j) for all i, j.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 3, 5, 2, 6, 4, 7, 3, 8, 5, 9, 2, 10, 6, 11, 4, 12, 7, 13, 3, 14, 8, 15, 5, 16, 9, 17, 2, 18, 10, 19, 6, 20, 11, 21, 4, 22, 12, 23, 7, 24, 13, 25, 3, 26, 14, 27, 8, 28, 15, 29, 5, 30, 16, 31, 9, 32, 17, 33, 2, 34, 18, 35, 10, 36, 19, 37, 6, 38, 20, 39, 11, 24, 21, 40, 4, 41, 22, 42, 12, 43, 23, 44, 7, 45, 24, 46, 13, 47, 25, 48, 3, 49, 26, 50, 14, 51, 27, 52, 8, 45
Offset: 0

Views

Author

Antti Karttunen, Feb 04 2020

Keywords

Comments

Restricted growth sequence transform of the ordered pair [A278222(n), A323901(n)].

Links

Crossrefs

Cf. A278222, A323901.
Cf. also A324400, A286622, A318310, A324389, A331744, A331746.

Programs

  • PARI
    \\ Needs also code from A323901.
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; };
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1)));
t };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A278222(n) = A046523(A005940(1+n));
Aux331745(n) = [A278222(n),A323901(n)];
v331745 = rgs_transform(vector(1+up_to, n, Aux331745(n-1)));
A331745(n) = v331745[1+n];

Formula

a(2^n) = 2 for all n >= 0.
Showing 1-10 of 14 results. Next

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