A101296 -id:A101296 - cp's OEIS frontend

A292258 a(1) = 1; for n > 1, a(n) = prime(A101296(n)-1) * a(floor(n/2)).

1, 2, 2, 6, 4, 10, 4, 42, 18, 20, 8, 110, 20, 20, 20, 546, 84, 198, 36, 220, 100, 40, 16, 1870, 330, 100, 140, 220, 40, 380, 40, 12558, 2730, 420, 420, 5742, 396, 180, 180, 3740, 440, 1900, 200, 440, 440, 80, 32, 57970, 5610, 3630, 1650, 1100, 200, 2380, 700, 3740, 1100, 200, 80, 14060, 760, 200, 440, 514878

Offset: 1

Antti Karttunen, Sep 22 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8191

Cf. A000040, A000523, A004526, A007814, A078349, A101296, A292259 (rgs-version of this filter).

With[{nn = 64}, Block[{s = Function[s, Table[Position[Keys@ s, k_ /; MemberQ[k, n]][[1, 1]], {n, nn}]]@ Map[#1 -> #2 & @@ # &, Transpose@ {Values@ #, Keys@ #}] &@ PositionIndex@ Table[Times @@ MapIndexed[Prime[First@ #2]^#1 &, Sort[FactorInteger[n][[All, -1]], Greater]] - Boole[n == 1], {n, nn}], a}, a[n_] := a[n] = If[n == 1, 1, Prime[s[[n]] - 1]*a[Floor[n/2]]]; Array[a, nn]]] (* Michael De Vlieger, Sep 22 2017 *)

up_to = 8191
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; };
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
v101296 = rgs_transform(vector(up_to, n, A046523(n)));
A101296(n) = v101296[n];
A292258(n) = if(1==n,n,prime(A101296(n)-1) * A292258(n\2));

a(1) = 1; for n > 1, a(n) = A000040(A101296(n)-1) * a(A004526(n)).
Other identities. For all n >= 1:
A001222(a(n)) = A000523(n).
A007814(a(n)) = A078349(n).

A294897 a(n) = Product_{d|n, gcd(d,n/d)>1} prime(A101296(gcd(d,n/d))-1).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 4, 2, 1, 1, 4, 1, 1, 1, 12, 1, 4, 1, 4, 1, 1, 1, 16, 2, 1, 4, 4, 1, 1, 1, 36, 1, 1, 1, 80, 1, 1, 1, 16, 1, 1, 1, 4, 4, 1, 1, 144, 2, 4, 1, 4, 1, 16, 1, 16, 1, 1, 1, 16, 1, 1, 4, 252, 1, 1, 1, 4, 1, 1, 1, 1600, 1, 1, 4, 4, 1, 1, 1, 144, 12, 1, 1, 16, 1, 1, 1, 16, 1, 16, 1, 4, 1, 1, 1, 1296, 1, 4, 4, 80, 1, 1, 1, 16, 1

Offset: 1

Antti Karttunen, Nov 21 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from exponents in factorization of n

Cf. A005117 (the positions of ones).
Cf. A101296, A294895.
Cf. also A292258 (A292259), A293515, A294875 for similar filter sequences.

up_to = 16384
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; };
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
v101296 = rgs_transform(vector(up_to, n, A046523(n)));
A101296(n) = v101296[n];
A294897(n) = { my(m=1); fordiv(n,d,if(gcd(d,n/d)>1, m *= prime(A101296(gcd(d,n/d))-1))); m; };

a(n) = Product_{d|n} A008578(A101296(gcd(d,n/d))).

For n >= 1, A001222(a(n)) = A048105(n).

A295888 Filter combining prime signature of n (A101296) with Dedekind's psi (A001615).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Dec 03 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537
Eric Weisstein's World of Mathematics, Pairing Function
Wikipedia, Pairing Function

Cf. A001615, A046523, A101296, A291750, A295300, A295880, A295887, A296090.

allocatemem(2^30);
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; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A001615(n) = (n * sumdivmult(n, d, issquarefree(d)/d)); \\ This function from Charles R Greathouse IV, Sep 09 2014
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
Anotsubmitted8(n) = (1/2)*(2 + ((A046523(n)+A001615(n))^2) - A046523(n) - 3*A001615(n));
write_to_bfile(1,rgs_transform(vector(up_to,n,Anotsubmitted8(n))),"b295888.txt");

Restricted growth sequence transform of function f(n) = (1/2)*(2 + ((A046523(n) + A001615(n))^2) - A046523(n) - 3*A001615(n)), where values A046523(n) and A001615(n) are packed together to a(n) with the 2-argument form of A000027, also known as Cantor pairing-function.

A286454 Compound filter (prime signature & prime signature of conjugated prime factorization): a(n) = P(A101296(n), A286621(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 5, 8, 9, 12, 32, 23, 20, 13, 49, 38, 51, 47, 82, 49, 35, 68, 51, 80, 72, 124, 140, 122, 74, 18, 175, 26, 111, 155, 334, 192, 65, 257, 280, 82, 116, 255, 329, 355, 99, 327, 570, 380, 177, 72, 469, 437, 132, 31, 72, 532, 216, 498, 74, 257, 144, 599, 634, 597, 448, 632, 745, 159, 119, 784, 1044, 782, 331, 907, 570, 863, 186, 905, 1039, 72, 384, 140, 1335, 1037

Offset: 1

Antti Karttunen, May 14 2017

nonn

Here, instead of A046523 and A278221 we use as the components of a(n) their rgs-versions A101296 and A286621 because of the latter sequence's moderate growth rates.

For all i, j: a(i) = a(j) => A286356(i) = A286356(j).

Antti Karttunen, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pairing Function

Cf. A000027, A046523, A101296, A122111, A278221, A285334, A286621, A286356, A286455, A286456.

(define (A286454 n) (* (/ 1 2) (+ (expt (+ (A101296 n) (A286621 n)) 2) (- (A101296 n)) (- (* 3 (A286621 n))) 2)))

a(n) = (1/2)*(2 + ((A101296(n)+A286621(n))^2) - A101296(n) - 3*A286621(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

A296090 Filter combining the sum of divisors (A000203) and prime-signature (A101296) of n; restricted growth sequence transform of A286360.

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, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 57, 61, 64, 65, 66, 67, 68, 69, 57, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 79

Offset: 1

Antti Karttunen, Dec 07 2017

nonn

For all i, j:
  a(i) = a(j) => A286034(i) = A286034(j).
  a(i) = a(j) => A295880(i) = A295880(j).

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

Cf. A000203, A101296, A046523, A286034, A286360, A295300, A295888, A296089.

Differs from related A295880 for the first time at n=135, where a(135) = 123, while A295880(135) = 104.

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; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A000203(n) = sigma(n);
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
A286360(n) = (1/2)*(2 + ((A046523(n)+A000203(n))^2) - A046523(n) - 3*A000203(n));
write_to_bfile(1,rgs_transform(vector(up_to,n,A286360(n))),"b296090.txt");

A286467 Compound filter (prime signature of n & prime signature of the n-th Fibonacci number): a(n) = P(A101296(n), A286545(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 3, 5, 9, 5, 19, 5, 33, 18, 25, 5, 51, 5, 25, 40, 73, 5, 72, 12, 84, 40, 25, 5, 128, 69, 25, 71, 84, 5, 180, 12, 146, 40, 25, 40, 242, 23, 40, 40, 198, 12, 180, 5, 177, 177, 40, 5, 337, 31, 216, 40, 84, 12, 284, 59, 308, 140, 40, 12, 478, 12, 40, 177, 339, 40, 180, 23, 177, 140, 387, 12, 610, 12, 59, 216, 177, 59, 309, 12, 540, 332, 40, 5, 608, 59, 40, 59

Offset: 1

Antti Karttunen, May 17 2017

nonn

Nonsquare semiprimes pq for which F(pq) is also a semiprime is given by the positions where 25's occur in this sequence: 10, 14, 22, 26, 34, 94, (any more terms?). This is a subsequence of A072381.

Antti Karttunen, Table of n, a(n) for n = 1..1300 (based also on the b-file of A278245 provided by Hans Havermann)

Cf. A000027, A046523, A072381, A101296, A278245, A286545.

Cf. A083668 (positions of 5's).

(define (A286467 n) (* (/ 1 2) (+ (expt (+ (A101296 n) (A286545 n)) 2) (- (A101296 n)) (- (* 3 (A286545 n))) 2)))

a(n) = (1/2)*(2 + ((A101296(n) + A286545(n))^2) - A101296(n) - 3*A286545(n)).

A302046 A filter sequence analogous to A101296 for nonstandard factorization based on the sieve of Eratosthenes (A083221).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 3, 4, 2, 6, 2, 4, 4, 7, 2, 6, 2, 6, 5, 4, 2, 8, 3, 4, 4, 6, 2, 9, 2, 10, 6, 4, 4, 11, 2, 4, 4, 8, 2, 8, 2, 6, 7, 4, 2, 12, 3, 6, 6, 6, 2, 9, 5, 8, 6, 4, 2, 13, 2, 4, 4, 14, 4, 13, 2, 6, 8, 9, 2, 15, 2, 4, 4, 6, 4, 9, 2, 12, 6, 4, 2, 15, 6, 4, 9, 8, 2, 12, 5, 6, 10, 4, 4, 16, 2, 6, 4, 11, 2, 13, 2, 8, 11

Offset: 1

Antti Karttunen, Mar 31 2018

nonn

Restricted growth sequence transform of A278524.
See A302042 for the description of the nonstandard factorization employed here.
For all i, j:
  a(i) = a(j) => A253557(i) = A253557(j).
  a(i) = a(j) => A302041(i) = A302041(j).
  a(i) = a(j) => A302050(i) = A302050(j).
  a(i) = a(j) => A302051(i) = A302051(j) => A302052(i) = A302052(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences generated by sieves

Cf. A101296, A250246, A278524, A302042, A302044, A302045.

up_to = 32769;
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; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A020639(n) = { if(1==n,n,vecmin(factor(n)[, 1])); };
A078898(n) = { if(n<=1,n, my(spf=A020639(n),k=1,m=n/spf); while(m>1,if(A020639(m)>=spf,k++); m--); (k)); };
A001511(n) = 1+valuation(n,2);
A302045(n) = A001511(A078898(n));
A302044(n) = if(1==n,n,my(k=0); while((n%2), n = A268674(n); k++); n = (n/2^valuation(n, 2)); while(k>0, n = A250469(n); k--); (n));
A302041(n) = if(1==n, 0,1+A302041(A302044(n)));
Aux302046(n) = if(1==n,n, my(k=A302041(n), v = vector(k),i=1); while(n>1,v[i] = A302045(n); n = A302044(n); i++); vecsort(v));
write_to_bfile(1,rgs_transform(vector(up_to,n,Aux302046(n))),"b302046.txt");

A286566 Compound filter (prime signature of n & prime signature of the n-th Jacobsthal number): a(n) = P(A101296(n), A286566(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 3, 5, 9, 5, 19, 5, 26, 18, 19, 5, 51, 5, 19, 40, 73, 5, 72, 5, 72, 40, 40, 5, 113, 31, 19, 83, 111, 8, 129, 5, 101, 32, 19, 32, 221, 8, 19, 40, 179, 8, 199, 5, 84, 159, 40, 8, 312, 13, 84, 82, 84, 8, 239, 49, 261, 32, 82, 23, 419, 5, 19, 159, 224, 82, 334, 8, 84, 32, 334, 8, 543, 8, 32, 84, 84, 82, 285, 5, 243, 332, 32, 57, 478, 40, 32, 32, 218, 23, 419, 82

Offset: 1

Antti Karttunen, May 21 2017

nonn

Here, instead of A046523 and A278165 we use as the components of a(n) their rgs-versions A101296 and A286565 because of the latter sequences' more moderate growth rates.

Antti Karttunen, Table of n, a(n) for n = 1..1032 (computed using the b-file of A278165 provided by Hans Havermann)

Cf. A001045, A046523, A101296, A278165, A286565.
Cf. A000978 (positions of 5's).
Cf. A286467 (similar filter).

(define (A286566 n) (* (/ 1 2) (+ (expt (+ (A101296 n) (A286565 n)) 2) (- (A101296 n)) (- (* 3 (A286565 n))) 2)))

a(n) = (1/2)*(2 + ((A101296(n)+A286565(n))^2) - A101296(n) - 3*A286565(n)).

A289628 Compound filter (for the structure of the multiplicative group of integers modulo n & prime signature of n): a(n) = P(A289626(n), A101296(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 2, 5, 8, 9, 12, 14, 41, 19, 18, 27, 50, 35, 25, 63, 99, 54, 40, 65, 86, 102, 42, 90, 203, 134, 52, 101, 131, 135, 128, 152, 342, 228, 75, 250, 221, 230, 88, 250, 399, 275, 182, 299, 271, 295, 117, 324, 517, 323, 185, 403, 295, 377, 146, 462, 623, 525, 168, 495, 549, 527, 187, 698, 728, 663, 343, 629, 460, 738, 370, 702, 889, 740, 273, 523, 590, 858, 370

Offset: 1

Antti Karttunen, Jul 19 2017

nonn

Here, instead of A046523 and A289625 we use as the components of a(n) their rgs-versions A101296 and A289626 because of the latter sequence's more moderate growth rate.
For all i, j: a(i) = a(j) => A286160(i) = A286160(j).
For all i, j: a(i) = a(j) => A289622(i) = A289622(j).

Antti Karttunen, Table of n, a(n) for n = 1..65536

Cf. A000027, A046523, A101296, A286160, A289622, A289626, A286454.

(define (A289628 n) (* (/ 1 2) (+ (expt (+ (A289626 n) (A101296 n)) 2) (- (A289626 n)) (- (* 3 (A101296 n))) 2)))

a(n) = (1/2)*(2 + ((A289626(n)+A101296(n))^2) - A289626(n) - 3*A101296(n)).

cp's OEIS Frontend

A292258 a(1) = 1; for n > 1, a(n) = prime(A101296(n)-1) * a(floor(n/2)).

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A294897 a(n) = Product_{d|n, gcd(d,n/d)>1} prime(A101296(gcd(d,n/d))-1).

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A295888 Filter combining prime signature of n (A101296) with Dedekind's psi (A001615).

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A286454 Compound filter (prime signature & prime signature of conjugated prime factorization): a(n) = P(A101296(n), A286621(n)), where P(n,k) is sequence A000027 used as a pairing function.

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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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A296090 Filter combining the sum of divisors (A000203) and prime-signature (A101296) of n; restricted growth sequence transform of A286360.

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A286467 Compound filter (prime signature of n & prime signature of the n-th Fibonacci number): a(n) = P(A101296(n), A286545(n)), where P(n,k) is sequence A000027 used as a pairing function.

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A302046 A filter sequence analogous to A101296 for nonstandard factorization based on the sieve of Eratosthenes (A083221).

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A286566 Compound filter (prime signature of n & prime signature of the n-th Jacobsthal number): a(n) = P(A101296(n), A286566(n)), where P(n,k) is sequence A000027 used as a pairing function.

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