A291750 -id:A291750 - cp's OEIS frontend

A323368 Lexicographically earliest sequence such that a(i) = a(j) => A000035(i) = A000035(j) and A003557(i) = A003557(j) and A048250(i) = A048250(j), for all i, j.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 15, 23, 24, 25, 26, 27, 28, 29, 21, 30, 31, 32, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 29, 31, 43, 44, 45, 46, 47, 48, 49, 46, 50, 51, 52, 53, 54, 55, 39, 56, 57, 58, 59, 60, 61, 62, 59, 46, 63, 64, 65, 66, 67, 62, 68, 51, 69, 70, 71, 58, 72, 73, 74, 75, 76, 77, 78, 79, 54, 80, 59, 75, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90

Offset: 1

Antti Karttunen, Jan 12 2019

nonn

For all i, j:
  a(i) = a(j) => A007814(i) = A007814(j),
  a(i) = a(j) => A291751(i) = A291751(j),
  a(i) = a(j) => A296089(i) = A296089(j),
  a(i) = a(j) => A323238(i) = A323238(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000035, A003557, A048250, A291750, A291751, A323238, A323369.
Differs from A296089 for the first time at n=103, where a(103)=88, while A296089(103)=56.
Cf. also A323366.

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); };
A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
v323368 = rgs_transform(vector(up_to, n, [(n%2), A003557(n), A048250(n)]));
A323368(n) = v323368[n];

A323369 Lexicographically earliest such sequence a that for all i, j, a(i) = a(j) => f(i) = f(j), where f(n) = 0 for odd primes, and f(n) = A323368(n) for any other number.

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, 24, 26, 3, 27, 28, 29, 3, 30, 3, 31, 32, 22, 3, 33, 34, 35, 36, 37, 3, 38, 36, 39, 40, 41, 3, 42, 3, 30, 43, 44, 45, 46, 3, 47, 48, 46, 3, 49, 3, 50, 51, 52, 48, 53, 3, 54, 55, 56, 3, 57, 58, 59, 60, 61, 3, 62, 63, 42, 64, 46, 60, 65, 3, 66, 67, 68, 3, 69, 3

Offset: 1

Antti Karttunen, Jan 12 2019

nonn

Restricted growth sequence transform of function f, where f(n) = 0 for odd primes, and for any other number, f(n) = [A000035(n), A003557(n), A048250(n)].
For all i, j:
  A305801(i) = A305801(j) => a(i) = a(j),
  a(i) = a(j) => A007814(i) = A007814(j),
  a(i) = a(j) => A322588(i) = A322588(j),
  a(i) = a(j) => A323238(i) = A323238(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000035, A003557, A048250, A291750, A291751, A305801, A322588, A323238, A323368.

Cf. also A323367.

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); };
A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
Aux323369(n) = if((n>2)&&isprime(n),0,[(n%2), A003557(n), A048250(n)]);
v323369 = rgs_transform(vector(up_to, n, Aux323369(n)));
A323369(n) = v323369[n];

A323372 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => A003557(i) = A003557(j) and A323363(i) = A323363(j).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 32, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 44, 49, 50, 51, 44, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 58, 62, 29, 65, 66, 67, 68, 69, 58, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 79

Offset: 1

Antti Karttunen, Jan 13 2019

nonn

Restricted growth sequence transform of ordered pair [A003557(n), A323363(n)].
For all i, j:
  a(i) = a(j) => A291751(i) = A291751(j),
  a(i) = a(j) => A323364(i) = A323364(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A001615, A003557, A291750, A291751, A323363, A323364.

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; };
DirInverse(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = -sumdiv(n, d, if(dA001615(n) = (n * sumdivmult(n, d, issquarefree(d)/d)); \\ From A001615
v323363 = DirInverse(vector(up_to,n,A001615(n)));
A323363(n) = v323363[n];
A003557(n) = { my(f=factor(n)); for(i=1, #f~, f[i, 2] = f[i, 2]-1); factorback(f); };
v323372 = rgs_transform(vector(up_to, n, [A003557(n), A323363(n)]));
A323372(n) = v323372[n];

A291752 Compound filter: a(n) = P(A101296(n), A291751(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 5, 8, 18, 17, 40, 30, 71, 58, 82, 23, 126, 80, 124, 124, 197, 57, 196, 138, 237, 214, 235, 93, 359, 256, 304, 356, 412, 327, 570, 173, 640, 469, 500, 469, 791, 498, 599, 634, 828, 255, 912, 668, 867, 909, 410, 408, 1237, 864, 1041, 410, 1087, 437, 1233, 410, 1283, 1132, 1180, 530, 1724, 1178, 707, 1437, 1967, 1435, 1779, 1433, 1717, 707, 1779, 353, 2361

Offset: 1

Antti Karttunen, Sep 06 2017

nonn

This filter combines information about A101296(n) (prime signature of n, A046523), A003557(n) and A048250(n). - Antti Karttunen, Oct 08 2017

Antti Karttunen, Table of n, a(n) for n = 1..16385

Cf. A000027, A003557, A046523, A048250, A101296, A291750, A291751.

(define (A291752 n) (* (/ 1 2) (+ (expt (+ (A101296 n) (A291751 n)) 2) (- (A101296 n)) (- (* 3 (A291751 n))) 2)))

a(n) = (1/2)*(2 + ((A101296(n) + A291751(n))^2) - A101296(n) - 3*A291751(n)).

Formula corrected by Antti Karttunen, Oct 08 2017

A319698 Filter sequence combining A003557(n) [n divided by largest squarefree divisor of n] with A319697(n) [sum of even squarefree divisors of n].

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Oct 31 2018

nonn

Restricted growth sequence transform of ordered pair [A003557(n), A319697(n)].

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A003557, A319697.

Cf also A291750, A291751.

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); }; \\ From A003557
A319697(n) = sumdiv(n, d, (!(d%2))*issquarefree(d)*d);
v319698 = rgs_transform(vector(up_to,n,[A003557(n),A319697(n)]));
A319698(n) = v319698[n];

A291758 Compound filter (prime signature of n & sum of squarefree divisors of n): a(n) = P(A046523(n), A048250(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 8, 12, 19, 23, 142, 38, 53, 25, 259, 80, 265, 107, 412, 412, 169, 173, 265, 212, 418, 672, 826, 302, 619, 40, 1087, 63, 607, 467, 5080, 530, 593, 1384, 1717, 1384, 1117, 743, 2086, 1836, 844, 905, 7780, 992, 1093, 607, 2932, 1178, 1759, 59, 418, 2932, 1390, 1487, 619, 2932, 1105, 3576, 4471, 1832, 8575, 1955, 5056, 915, 2209, 3922, 14908, 2348, 2092, 5056

Offset: 1

Antti Karttunen, Sep 10 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16385

Cf. A000027, A046523, A048250, A291750, A291752, A291757.

A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ This function from Charles R Greathouse IV, Aug 17 2011
A048250(n) = if(n<1, 0, sumdiv(n, d, if(core(d)==d, d)));
A291758(n) = (1/2)*(2 + ((A046523(n)+A048250(n))^2) - A046523(n) - 3*A048250(n));

a(n) = (1/2)*(2 + ((A046523(n)+A048250(n))^2) - A046523(n) - 3*A048250(n)).

A326199 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A003557(n), A046523(n), A048250(n)] for all other numbers, except f(n) = 0 for odd primes.

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

Offset: 1

Antti Karttunen, Jul 13 2019

nonn

For all i, j:
  A295300(i) = A295300(j) => a(i) = a(j),
  A305801(i) = A305801(j) => a(i) = a(j),
  a(i) = a(j) => A294877(i) = A294877(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A003557, A046523, A048250, A294877, A295300, A305801.

Differs from A323401 for the first time at n = 382 where a(382) = 253, while A323401(382) = 140.

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) = n/factorback(factor(n)[, 1]); \\ From A003557
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A048250(n) = if(n<1, 0, sumdiv(n, d, if(core(d)==d, d)));
A291750(n) = (1/2)*(2 + ((A003557(n)+A048250(n))^2) - A003557(n) - 3*A048250(n));
Aux326199(n) = if((n>2)&&isprime(n),0,(1/2)*(2 + ((A046523(n) + A291750(n))^2) - A046523(n) - 3*A291750(n)));
v326199 = rgs_transform(vector(up_to,n,Aux326199(n)));
A326199(n) = v326199[n];

A296087 Numbers n such that there is k < n for which A003557(k) = A003557(n), A048250(k) = A048250(n) and A173557(k) = A173557(n).

Original entry on oeis.org

15265, 27962, 30217, 30530, 45795, 50541, 54379, 54905, 57598, 60434, 61060, 64255, 66526, 72357, 72713, 89585, 90651, 91590, 101082, 101949, 108758, 109810, 120868, 122120, 128510, 136555, 137385, 137883, 138761, 144714, 145426, 149739, 151085, 152633, 161386, 163137, 164715, 166315, 179170, 181302, 181543, 182942

Offset: 1

Antti Karttunen, Dec 08 2017

nonn

Because Euler phi(n) = A000010(n) = A003557(n) * A173557(n), Dedekind psi(n) = A001615(n) = A003557(n) * A048250(n), and because also sigma(n) (A000203) can be computed from those three elements (see A291750), these numbers form also a subset of the positions of such duplicated occurrences of values computed for those functions. See for example A069822 and A296214.

a(11) = 61060 is the first term that is not squarefree.

			15265 is a term because A003557(15265) = 1 = A003557(15169), A048250(15265) = 19008 = A048250(15169), A173557(15265) = 11760 = A173557(15169).
27962 is a term because A003557(27962) = 1 = A003557(26355), A048250(27962) = 48384 = A048250(26355), A173557(27962) = 12000 = A173557(26355).

Antti Karttunen, Table of n, a(n) for n = 1..2876

Cf. A000010, A001615, A003557, A048250, A173557.

Subsequence of A069822 and of A296214.

search_up_to = (2^23);
A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = max(0,f[i, 2]-1)); factorback(f); };
A048250(n) = if(n<1, 0, sumdiv(n, d, if(core(d)==d, d)));
A173557(n) = my(f=factor(n)[, 1]); prod(k=1, #f, f[k]-1); \\ This function from Michel Marcus, Oct 31 2017
Anotsubmitted1(n) = (1/2)*(2 + ((A003557(n)+A173557(n))^2) - A003557(n) - 3*A173557(n));
Akaikki3(n) = (1/2)*(2 + ((A048250(n)+Anotsubmitted1(n))^2) - A048250(n) - 3*Anotsubmitted1(n));
om = Map(); m = 0; i=0; for(n = 1, search_up_to, k = Akaikki3(n); if(!mapisdefined(om,k), mapput(om,k,n), i++; write("b296087.txt", i, " ", n)));

A332230 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A003557(n), A046523(n), A048250(n)] for all other numbers, except f(2^k) = 0 for k >= 2.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 4, 8, 9, 10, 11, 12, 13, 13, 4, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 4, 29, 30, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 41, 46, 47, 48, 41, 49, 50, 51, 52, 53, 54, 55, 56, 4, 57, 58, 59, 60, 55, 58, 61, 62, 63, 64, 65, 66, 55, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 76

Offset: 1

Antti Karttunen, Feb 22 2020

nonn

For all i, j:
  A295300(i) = A295300(j) => a(i) = a(j),
  a(i) = a(j) => A048250(i) = A048250(j),
  a(i) = a(j) => A332455(i) = A332455(j),
  a(i) = a(j) => A332459(i) = A332459(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A003557, A046523, A048250, A295300, A332455, A332459.

Cf. also A326199.

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) = n/factorback(factor(n)[, 1]); \\ From A003557
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A048250(n) = if(n<1, 0, sumdiv(n, d, if(core(d)==d, d)));
A209229(n) = (n && !bitand(n,n-1));
A291750(n) = (1/2)*(2 + ((A003557(n)+A048250(n))^2) - A003557(n) - 3*A048250(n));
Aux332230(n) = if((n>2)&&A209229(n),0,(1/2)*(2 + ((A046523(n) + A291750(n))^2) - A046523(n) - 3*A291750(n)));
v332230 = rgs_transform(vector(up_to,n,Aux332230(n)));
A332230(n) = v332230[n];

cp's OEIS Frontend

A323368 Lexicographically earliest sequence such that a(i) = a(j) => A000035(i) = A000035(j) and A003557(i) = A003557(j) and A048250(i) = A048250(j), for all i, j.

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A323369 Lexicographically earliest such sequence a that for all i, j, a(i) = a(j) => f(i) = f(j), where f(n) = 0 for odd primes, and f(n) = A323368(n) for any other number.

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A323372 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => A003557(i) = A003557(j) and A323363(i) = A323363(j).

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A291752 Compound filter: a(n) = P(A101296(n), A291751(n)), where P(n,k) is sequence A000027 used as a pairing function.

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A319698 Filter sequence combining A003557(n) [n divided by largest squarefree divisor of n] with A319697(n) [sum of even squarefree divisors of n].

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A291758 Compound filter (prime signature of n & sum of squarefree divisors of n): a(n) = P(A046523(n), A048250(n)), where P(n,k) is sequence A000027 used as a pairing function.

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A326199 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A003557(n), A046523(n), A048250(n)] for all other numbers, except f(n) = 0 for odd primes.

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A296087 Numbers n such that there is k < n for which A003557(k) = A003557(n), A048250(k) = A048250(n) and A173557(k) = A173557(n).

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A332230 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A003557(n), A046523(n), A048250(n)] for all other numbers, except f(2^k) = 0 for k >= 2.

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