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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A353267 The least number with the same prime factorization pattern (A348717) as A332449(n) = A005940(1+(3*A156552(n))).

Original entry on oeis.org

1, 4, 4, 10, 4, 16, 4, 30, 10, 36, 4, 22, 4, 100, 16, 90, 4, 40, 4, 250, 36, 196, 4, 66, 10, 484, 30, 490, 4, 64, 4, 270, 100, 676, 16, 154, 4, 1156, 196, 750, 4, 144, 4, 1210, 22, 1444, 4, 198, 10, 84, 484, 1690, 4, 120, 36, 1470, 676, 2116, 4, 34, 4, 3364, 250, 810, 100, 400, 4, 2890, 1156, 324, 4, 462, 4, 3844
Offset: 1

Views

Author

Antti Karttunen, Apr 09 2022

Keywords

Links

Crossrefs

Cf. A005940, A156552, A332449, A348717.
Cf. also A305897 (rgs-transform), A352892, A353268.

Programs

  • PARI
    A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
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 };
A332449(n) = A005940(1+(3*A156552(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
A353267(n) = A348717(A332449(n));

Formula

a(n) = A348717(A332449(n)) = A332449(A348717(n)).

A355834 Lexicographically earliest infinite sequence such that a(i) = a(j) => A348717(i) = A348717(j) and A355931(i) = A355931(j) for all i, j >= 1.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jul 20 2022

Keywords

Comments

Restricted growth sequence transform of the ordered pair [A348717(n), A355931(n)], where A355931(n) = A000265(A009194(i)).

Examples

			a(450) = a(3675) [= 274 as allotted by rgs-transform] because A003961(450) = 3675, therefore 450 and 3675 are in the same column of the prime shift array A246278, and because A355931(450) = A355931(3675) = 3.
a(3185) = a(14399) [= 2020 as allotted by rgs-transform] because A003961(3185) = 14399 and A355931(3185) = A355931(14399) = 7.
a(5005) = a(17017) [= 3184 as allotted by rgs-transform] because A003961(5005) = 17017 and A355931(5005) = A355931(17017) = 7.

Links

Crossrefs

Cf. A000203, A000265, A009194, A246278, A348717, A355931.
Cf. also A300230, A305800, A305897, A355833.

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; };
A000265(n) = (n>>valuation(n,2));
A009194(n) = gcd(n, sigma(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));
Aux355834(n) = [A000265(A009194(n)), A348717(n)];
v355834 = rgs_transform(vector(up_to,n,Aux355834(n)));
A355834(n) = v355834[n];

A319339 Filter sequence combining A081373(n) with A246277(n).

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

Views

Author

Antti Karttunen, Sep 24 2018

Keywords

Comments

Restricted growth sequence transform of ordered pair [A081373(n), A246277(n)].

Links

Crossrefs

Cf. A081373, A246277, A300247, A305801, A305897, A318500.

Programs

  • PARI
    up_to = 65537;
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; };
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; };
v081373 = ordinal_transform(vector(up_to,n,eulerphi(n)));
A081373(n) = v081373[n];
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);
v319339 = rgs_transform(vector(up_to,n,[A081373(n),A246277(n)]));
A319339(n) = v319339[n];

A350068 Lexicographically earliest infinite sequence such that a(i) = a(j) => A046523(i) = A046523(j) and A350063(i) = A350063(j), for all i, j >= 1.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jan 29 2022

Keywords

Comments

Restricted growth sequence transform of the ordered pair [A046523(n), A350063(n)].
For all i, j >= 1: A305897(i) = A305897(j) => a(i) = a(j).

Links

Crossrefs

Cf. A000265, A046523, A156552, A322993, A350063.
Cf. A000040 (positions of 2's), A001248 (of 3's).
Cf. also A101296, A300226, A305897, A350065.

Programs

  • PARI
    up_to = 3003;
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>>valuation(n,2));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
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 };
A350063(n) = if(1==n,0,A046523(A000265(A156552(n))));
Aux350068(n) = [A046523(n),A350063(n)];
v350068 = rgs_transform(vector(up_to, n, Aux350068(n)));
A350068(n) = v350068[n];

A374478 Lexicographically earliest infinite sequence such that a(i) = a(j) => A348717(i) = A348717(j) and A364255(i) = A364255(j), for all i, j >= 1.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Aug 07 2024

Keywords

Comments

Restricted growth sequence transform of the ordered pair [A348717(n), A364255(n)].
For all i, j >= 1:
A305900(i) = A305900(j) => a(i) = a(j),
a(i) = a(j) => A305891(i) = A305891(j),
a(i) = a(j) => A374477(i) = A374477(j).

Links

Crossrefs

Cf. A163511, A348717, A364255.
Cf. also A305891, A305897, A364297, A364949, A365392, A365423, A374477.
Differs from A374040 first at n=77, where a(77) = 59, while A374040(77) = 50.
Differs from A305900 first at n=95, where a(95) = 39, while A305900(95) = 74.

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));
A163511(n) = if(!n, 1, my(p=2, t=1); while(n>1, if(!(n%2), (t*=p), p=nextprime(1+p)); n >>= 1); (t*p));
A364255(n) = gcd(n, A163511(n));
Aux374478(n) = [A348717(n), A364255(n)];
v374478 = rgs_transform(vector(up_to, n, Aux374478(n)));
A374478(n) = v374478[n];
Previous Showing 11-15 of 15 results.

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