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.

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A352898 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A046523(n), A352892(n)], except f(n) = -n when <= 2.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Apr 08 2022

Keywords

Comments

For all i, j: A305801(i) = A305801(j) => a(i) = a(j) => A352897(i) = A352897(j).

Links

Crossrefs

Cf. A046523, A101296, A305801, A305897, A352892, A352897, A352899.

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); };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
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)};
A156552(n) = { my(f = factor(n), p, 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 };
A329603(n) = A005940(2+(3*A156552(n)));
A341515(n) = if(n%2, A064989(n), A329603(n));
A348717(n) = { my(f=factor(n)); if(#f~>0, my(pi1=primepi(f[1, 1])); for(k=1, #f~, f[k, 1] = prime(primepi(f[k, 1])-pi1+1))); factorback(f); }; \\ From A348717
A352892(n) = A348717(A341515(n));
Aux352898(n) = if(n<=2,-n,[A046523(n),A352892(n)]);
v352898 = rgs_transform(vector(up_to, n, Aux352898(n)));
A352898(n) = v352898[n];

A352899 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = A352892(n), except f(n) = -n when <= 2.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Apr 08 2022

Keywords

Comments

Restricted growth sequence transform of function f(n) = -n if n < 3, and otherwise f(n) = A352892(n).
For all i, j:
A305801(i) = A305801(j) => A352898(i) = A352898(j) => a(i) = a(j),
a(i) = a(j) => A352893(i) = A352893(j),
a(i) = a(j) => A352896(i) = A352896(j).

Links

Crossrefs

Cf. A305801, A329603, A341515, A348717, A352892, A352893, A352896, A352898.

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); };
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)};
A156552(n) = { my(f = factor(n), p, 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 };
A329603(n) = A005940(2+(3*A156552(n)));
A341515(n) = if(n%2, A064989(n), A329603(n));
A348717(n) = { my(f=factor(n)); if(#f~>0, my(pi1=primepi(f[1, 1])); for(k=1, #f~, f[k, 1] = prime(primepi(f[k, 1])-pi1+1))); factorback(f); }; \\ From A348717
A352892(n) = A348717(A341515(n));
Aux352899(n) = if(n<=2,-n,A352892(n));
v352899 = rgs_transform(vector(up_to, n, Aux352899(n)));
A352899(n) = v352899[n];

A355833 Lexicographically earliest infinite sequence such that a(i) = a(j) => A342671(i) = A342671(j) and A348717(i) = A348717(j) for all i, j >= 1.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jul 20 2022

Keywords

Comments

Restricted growth sequence transform of the ordered pair [A342671(n), A348717(n)].
Terms that occur in positions given by A349166 may occur only a finite number of times in this sequence. See also the array A355924.

Examples

			a(100) = a(3025) [= 66 as allotted by the rgs-transform] because 3025 = A003961(A003961(100)), therefore it is in the same column of the prime shift array A246278 as 100 is], and as A342671(100) = A342671(3025) = 7.
a(300) = a(21175) [= 200 as allotted by the rgs-transform], as 21175 = A003961(A003961(300)) and as A342671(300) = A342671(21175) = 7.
a(1215) = a(21875) [= 831 as allotted by the rgs-transform] because 21875 = A003961(1215), therefore it is in the same column of the prime shift array A246278 as 1215 is, and as A342671(1215) = A342671(21875) = 7.
a(2835) = a(48125) [= 1953 as allotted by the rgs-transform] because 48125 = A003961(2835) and as A342671(2835) = A342671(48125) = 11.

Links

Crossrefs

Cf. A000203, A003961, A246278, A342671, A348717, A349165, A349166, A355924.
Cf. also A305801, A305897, A355834, A355835.

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; };
A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A342671(n) = gcd(sigma(n), A003961(n));
A348717(n) = if(1==n, 1, my(f = factor(n), k = primepi(f[1, 1])-1); for (i=1, #f~, f[i, 1] = prime(primepi(f[i, 1])-k)); factorback(f));
Aux355833(n) = [A342671(n), A348717(n)];
v355833 = rgs_transform(vector(up_to,n,Aux355833(n)));
A355833(n) = v355833[n];

A355835 Lexicographically earliest infinite sequence such that a(i) = a(j) => A348717(i) = A348717(j) and A355442(i) = A355442(j) for all i, j >= 1.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jul 20 2022

Keywords

Comments

Restricted growth sequence transform of the ordered pair [A348717(n), A355442(n)].
For all i, j: a(i) = a(j) => A355836(i) = A355836(j).
Terms that occur in positions given by A355822 may occur only a finite number of times in this sequence. Most of these seem to be in the singular equivalence classes, i.e., have unique values, apart from exceptions like pairs {6, 15}, {273, 1729}, (see the examples and the array A355926). In a coarser variant A355836 multiple such finite equivalence classes may coalesce together into several infinite equivalence classes.

Examples

			a(6) = a(15) [= 5 as allotted by the rgs-transform] because 15 = A003961(6) [i.e., 15 is in the same column in prime shift array A246278 as 6 is], and because A355442(6) = A355442(15) = 5.
a(138) = a(435) [= 103 as allotted by the rgs-transform] because 435 = A003961(138), and A355442(138) = A355442(435) = 5.
a(273) = a(1729) [= 205 as allotted by the rgs-transform] because 1729 = A003961(A003961(273)) [i.e., 273 and 1729 are in the same column of A246278], and A355442(273) = A355442(1729) = 11.

Links

Crossrefs

Cf. A003961, A276086, A348717, A355442, A355821, A355822, A355926.
Cf. also A246278, A305801, A355833, A355834.

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; };
A348717(n) = if(1==n, 1, my(f = factor(n), k = primepi(f[1, 1])-1); for (i=1, #f~, f[i, 1] = prime(primepi(f[i, 1])-k)); factorback(f));
A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A355442(n) = gcd(A003961(n), A276086(n));
Aux355835(n) = [A348717(n), A355442(n)];
v355835 = rgs_transform(vector(up_to,n,Aux355835(n)));
A355835(n) = v355835[n];

A305890 Filter sequence for all such sequences b, for which b(A176997(k)) = constant for all k > 1, where A176997 is the union of odd primes and Fermat pseudoprimes.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jul 01 2018

Keywords

Links

Crossrefs

Cf. A001567, A062173, A176997, A257531.
Differs from A305801 for the first time at n=341, where a(341) = 3, while A305801(341) = 275.

Programs

  • PARI
    up_to = 100000;
A257531(n) = if(n==1, 0, if(Mod(2, n)^(n-1)==1, 1, 0));
partialsums(f,up_to) = { my(v = vector(up_to), s=0); for(i=1,up_to,s += f(i); v[i] = s); (v); }
vpartsums = partialsums(A257531, up_to);
Apartsums(n) = vpartsums[n];
A305890(n) = if(n<=2,n,if(A257531(n),3,1+n-Apartsums(n)));

Formula

For all i, j: A305801(i) = A305801(j) => a(i) = a(j) => A062173(i) = A062173(j).

A305896 Filter sequence combining prime signature of n (A046523) and the cardinality of invphi (A014197).

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jul 01 2018

Keywords

Comments

Restricted growth sequence transform of ordered pair [A046523(n), A014197(n)].
For all i, j: A305801(i) = A305801(j) => a(i) = a(j) => A097946(i) = A097946(j).

Links

Crossrefs

Cf. A014197, A046523, A305801.
Cf. also A097946.

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; };
A014197(n, m=1) = { n==1 && return(1+(m<2)); my(p, q); sumdiv(n, d, if( d>=m && isprime(d+1), sum( i=0, valuation(q=n\d, p=d+1), A014197(q\p^i, p))))} \\ This function from M. F. Hasler, Oct 05 2009
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
Aux305896(n) = [A046523(n), A014197(n)];
v305896 = rgs_transform(vector(up_to, n, Aux305896(n)));
A305896(n) = v305896[n];

A318500 Filter sequence combining A305897 and the parity of n.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Sep 24 2018

Keywords

Comments

Restricted growth sequence transform of ordered pair [A000035(n), A246277(n)], or equally, of ordered pair [A007814(n), A246277(n)].
For all i, j:
A305801(i) = A305801(j) => a(i) = a(j) => A305891(i) = A305891(j) => A291761(i) = A291761(j).

Links

Crossrefs

Cf. A246277, A291761, A305801, A305891, A305897.

Programs

  • PARI
    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; };
A246277(n) = if(1==n, 0, my(f = factor(n), k = primepi(f[1,1])-1); for (i=1, #f~, f[i,1] = prime(primepi(f[i,1])-k)); factorback(f)/2);
v318500 = rgs_transform(vector(up_to,n,[(n%2),A246277(n)]));
A318500(n) = v318500[n];

A318888 Filter sequence combining the 2-adic valuation of n (A007814) with the differences between odd primes in the prime factorization of n.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Sep 24 2018

Keywords

Comments

Restricted growth sequence transform of an ordered pair [A007814(n), A318885(A000265(n))].
For all i, j: A305801(i) = A305801(j) => a(i) = a(j) => A305891(i) = A305891(j).

Links

Crossrefs

Cf. A305801, A305891, A318500, A318885, A318887.

Programs

  • PARI
    up_to = 100000;
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; };
A000265(n) = (n/2^valuation(n, 2));
A007814(n) = valuation(n,2);
A318885(n) = if(1==n,n,my(f=factor(n),m=2^f[1,2],i=1); for(k=2,#f~,i += (f[k,1]-f[k-1,1]); m *= prime(i)^f[k,2]); (m));
v318888 = rgs_transform(vector(up_to,n,[A007814(n), A318885(A000265(n))]));
A318888(n) = v318888[n];

A319338 Filter sequence combining the 2-adic valuation of n (A007814) with gcd(n,sigma(n)) (A009194).

Original entry on oeis.org

1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 6, 8, 9, 1, 10, 1, 11, 1, 6, 1, 12, 1, 6, 1, 13, 1, 4, 1, 14, 8, 6, 1, 3, 1, 6, 1, 15, 1, 4, 1, 7, 8, 6, 1, 16, 1, 2, 8, 11, 1, 4, 1, 17, 1, 6, 1, 18, 1, 6, 1, 19, 1, 4, 1, 11, 8, 6, 1, 20, 1, 6, 1, 7, 1, 4, 1, 21, 1, 6, 1, 13, 1, 6, 8, 22, 1, 23, 24, 7, 1, 6, 25, 26, 1, 2, 8, 3, 1, 4, 1, 27, 8
Offset: 1

Views

Author

Antti Karttunen, Sep 24 2018

Keywords

Comments

Restricted growth sequence transform of ordered pair [A007814(n), A009194(n)].

Links

Crossrefs

Cf. A007814, A009194, A305801, A319346.

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; };
A007814(n) = valuation(n,2);
A009194(n) = gcd(n, sigma(n));
v319338 = rgs_transform(vector(up_to,n,[A007814(n),A009194(n)]));
A319338(n) = v319338[n];

A319347 Filter sequence combining A000035(n) (parity of n), A003557(n), and A046523(n) (prime signature of n).

Original entry on oeis.org

1, 2, 3, 4, 3, 5, 3, 6, 7, 5, 3, 8, 3, 5, 9, 10, 3, 11, 3, 8, 9, 5, 3, 12, 13, 5, 14, 8, 3, 15, 3, 16, 9, 5, 9, 17, 3, 5, 9, 12, 3, 15, 3, 8, 18, 5, 3, 19, 20, 21, 9, 8, 3, 22, 9, 12, 9, 5, 3, 23, 3, 5, 18, 24, 9, 15, 3, 8, 9, 15, 3, 25, 3, 5, 26, 8, 9, 15, 3, 19, 27, 5, 3, 23, 9, 5, 9, 12, 3, 28, 9, 8, 9, 5, 9, 29, 3, 30, 18, 31, 3, 15, 3, 12, 32
Offset: 1

Views

Author

Antti Karttunen, Sep 24 2018

Keywords

Comments

Restricted growth sequence transform of triple [A000035(n), A003557(n), A046523(n)], or equally, of triple [A007814(n), A003557(n), A046523(n)], or equally, of ordered pair [A000035(n), A291757(n)].
For all i, j: A305801(i) = A305801(j) => a(i) = a(j) => A305891(i) = A305891(j).

Links

Crossrefs

Cf. A000035, A003557, A007814, A046523, A291757, A305801, A305891.

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; };
A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = f[i, 2]-1); factorback(f); };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p=0); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };
v319347 = rgs_transform(vector(up_to,n,[A003557(n),(n%2),A046523(n)]));
A319347(n) = v319347[n];
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