A246277 -id:A246277 - cp's OEIS frontend

A252759 Manhattan distance of n in array A246278 from the top left corner: a(1) = 0; for n>1: a(n) = A055396(n) + A246277(n) - 1.

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

Offset: 1

Antti Karttunen, Jan 03 2015

nonn

			a(2) = 1, because 2 sits nearest to the top-left corner of the array A246278.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A055396, A246277, A246278, A252464.

(define (A252759 n) (if (= 1 n) 0  (+ -1 (A055396 n) (A246277 n))))

a(1) = 0; for n>1: a(n) = A055396(n) + A246277(n) - 1.

A304112 Restricted growth sequence transform of A246277(A064413(n)), related to indices in the prime factorization of EKG sequence.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, May 17 2018

nonn

For all i, j: a(i) = a(j) => A304113(i) = A304113(j).

Antti Karttunen, Table of n, a(n) for n = 1..65539
Index entries for sequences related to EKG sequence

Cf. A064413, A246277, A278263, A304113.

Cf. also A304114.

\\ Needs also code for A064413:
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)};
A246277(n) = { if(1==n, 0, while((n%2), n = A064989(n)); (n/2)); };
Aux304112(n) = A246277(A064413(n));
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
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(1,rgs_transform(vector(65539,n,Aux304112(n))),"b304112.txt");

A304114 Restricted growth sequence transform of A246277(A098550(n)), related to indices in the prime factorization of Yellowstone permutation.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, May 17 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65539

Cf. A098550, A246277, A304115.

Cf. also A304112.

\\ Needs also code for A098550:
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)};
A246277(n) = { if(1==n, 0, while((n%2), n = A064989(n)); (n/2)); };
Aux304114(n) = A246277(A098550(n));
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
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(1,rgs_transform(vector(65539,n,Aux304114(n))),"b304114.txt");

For all i, j: a(i) = a(j) => A304115(i) = A304115(j).

A304742 Restricted growth sequence transform of A246277(A280866(n)).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, May 18 2018

nonn

For all i, j: a(i) = a(j) => A304743(i) = A304743(j).

Antti Karttunen, Table of n, a(n) for n = 1..65539
Index entries for sequences computed from indices in prime factorization

Cf. A246277, A280866, A304743.

\\ Needs also code from A280866:
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)};
A246277(n) = { if(1==n, 0, while((n%2), n = A064989(n)); (n/2)); };
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; };
v304742 = rgs_transform(vector(65539,n,A246277(A280866(n))));
A304742(n) = v304742[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

Antti Karttunen, Sep 24 2018

nonn

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

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

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

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];

A329371 Dirichlet convolution of the identity function with A246277.

Original entry on oeis.org

0, 1, 1, 4, 1, 8, 1, 12, 5, 12, 1, 28, 1, 16, 11, 32, 1, 37, 1, 44, 15, 24, 1, 80, 7, 28, 19, 60, 1, 82, 1, 80, 21, 36, 15, 128, 1, 40, 27, 128, 1, 114, 1, 92, 49, 48, 1, 208, 9, 89, 33, 108, 1, 146, 21, 176, 39, 60, 1, 284, 1, 64, 69, 192, 25, 174, 1, 140, 45, 170, 1, 364, 1, 76, 70, 156, 21, 210, 1, 336, 65, 84, 1, 396, 33, 88, 55, 272, 1, 368, 25, 188, 63

Offset: 1

Antti Karttunen, Nov 12 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A246277.

Cf. also A305796, A323599, A329347, A329372, A329373.

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);
A329371(n) = sumdiv(n,d,(n/d)*A246277(d));

a(n) = Sum_{d|n} d * A246277(n/d).

A373983 Lexicographically earliest infinite sequence such that a(i) = a(j) = A246277(A324886(i)) = A246277(A324886(j)) and A278226(A328768(i)) = A278226(A328768(j)), for all i, j >= 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jun 25 2024

Restricted growth sequence transform of the ordered pair [A246277(A276086(A108951(n))), A046523(A276086(A328768(n)))].
For all i, j >= 1:
  A305800(i) = A305800(j) => a(i) = a(j),
  a(i) = a(j) => A329345(i) = A329345(j) => A329045(i) = A329045(j),
  a(i) = a(j) => A373982(i) = A373982(j) => A328771(i) = A328771(j).
It is hard to say for sure which graphical features in the scatter plot have their provenance in A373982, and which ones in A329345.

Antti Karttunen, Table of n, a(n) for n = 1..100000
Index entries for sequences related to primorial numbers

Cf. A002110, A046523, A108951, A246277, A276086, A278226, A324886, A328768.
Cf. A305800, A329045, A329345, A328771, A373982.
Cf. also A329620.

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; };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };
A108951(n) = { my(f=factor(n)); prod(i=1, #f~,  prod(i=1, primepi(f[i, 1]), prime(i))^f[i, 2]); };
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);
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A002110(n) = prod(i=1,n,prime(i));
A328768(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]*A002110(primepi(f[i,1])-1)/f[i, 1]));
Aux373983(n) = [A246277(A276086(A108951(n))), A046523(A276086(A328768(n)))];
v373983 = rgs_transform(vector(up_to, n, Aux373983(n)));
A373983(n) = v373983[n];

A286457 Compound filter: a(n) = P(A078898(n), A246277(n)), with a(1) = 0, where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

0, 1, 1, 5, 1, 13, 1, 25, 5, 41, 1, 61, 1, 85, 13, 113, 1, 145, 1, 181, 32, 221, 1, 265, 5, 313, 33, 365, 1, 421, 1, 481, 72, 545, 13, 613, 1, 685, 143, 761, 1, 841, 1, 925, 86, 1013, 1, 1105, 5, 1201, 219, 1301, 1, 1405, 32, 1513, 335, 1625, 1, 1741, 1, 1861, 201, 1985, 60, 2113, 1, 2245, 447, 2381, 1, 2521, 1, 2665, 223, 2813, 13, 2965, 1, 3121, 224, 3281

Offset: 1

Antti Karttunen, May 14 2017

nonn

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

Cf. A000027, A078898, A246277, A280492.

(define (A286457 n) (if (= 1 n) 0 (* (/ 1 2) (+ (expt (+ (A078898 n) (A246277 n)) 2) (- (A078898 n)) (- (* 3 (A246277 n))) 2))))

a(1) = 0 and for n > 1, a(n) = (1/2)*(2 + ((A078898(n)+A246277(n))^2) - A078898(n) - 3*A246277(n)).

A304731 Restricted growth sequence transform of A246277(A304531(n)).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, May 19 2018

nonn

For all i, j: a(i) = a(j) => A304732(i) = A304732(j).

Antti Karttunen, Table of n, a(n) for n = 1..74431

Cf. A246277, A304531.
Differs from related A304728 for the first time at n=66, where a(66) = 18, while A304728(66) = 50.
Cf. also A304535, A304732.

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)};
A246277(n) = { if(1==n, 0, while((n%2), n = A064989(n)); (n/2)); };
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; };
v304731 = rgs_transform(vector(76503,n,A246277(A304531(n)))); \\ Needs also code from A304531
A304731(n) = v304731[n];

A337202 a(n) = 2*A246277(A047802(n)).

Original entry on oeis.org

12, 120, 19399380, 195534950863140268380, 1678409980907129617069656971232406858049983380, 1193774258350145889842491509271710921616406416330926349273223856572483463433620

Offset: 0

Antti Karttunen, Aug 19 2020

nonn

Question: Are there any duplicate terms, not necessarily consecutive? That is, are there two or more terms of A047802 that occur in the same column of array A246278?

Antti Karttunen, Table of n, a(n) for n = 0..13 (Prepared from the b-file of A047802 provided by Jeppe Stig Nielsen)
Index entries for sequences computed from indices in prime factorization

Cf. A047802, A246277, A246278, A336389.

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);
A337202(n) = 2*A246277(A047802(n));

For all n >= 0, a(n) >= A336389(1+n).

cp's OEIS Frontend

A252759 Manhattan distance of n in array A246278 from the top left corner: a(1) = 0; for n>1: a(n) = A055396(n) + A246277(n) - 1.

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A304112 Restricted growth sequence transform of A246277(A064413(n)), related to indices in the prime factorization of EKG sequence.

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A304114 Restricted growth sequence transform of A246277(A098550(n)), related to indices in the prime factorization of Yellowstone permutation.

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A304742 Restricted growth sequence transform of A246277(A280866(n)).

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

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A329371 Dirichlet convolution of the identity function with A246277.

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A373983 Lexicographically earliest infinite sequence such that a(i) = a(j) = A246277(A324886(i)) = A246277(A324886(j)) and A278226(A328768(i)) = A278226(A328768(j)), for all i, j >= 1.

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A286457 Compound filter: a(n) = P(A078898(n), A246277(n)), with a(1) = 0, where P(n,k) is sequence A000027 used as a pairing function.

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A304731 Restricted growth sequence transform of A246277(A304531(n)).

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