A297113 -id:A297113 - cp's OEIS frontend

A324202 a(n) = A046523(A332461(n)), where A332461(n) = Product_{d|n, d>1} prime(1+A297167(d)).

1, 2, 2, 6, 2, 12, 2, 30, 6, 12, 2, 120, 2, 12, 12, 210, 2, 180, 2, 420, 12, 12, 2, 2520, 6, 12, 30, 420, 2, 720, 2, 2310, 12, 12, 12, 7560, 2, 12, 12, 9240, 2, 720, 2, 420, 120, 12, 2, 138600, 6, 180, 12, 420, 2, 6300, 12, 60060, 12, 12, 2, 151200, 2, 12, 420, 30030, 12, 720, 2, 420, 12, 720, 2, 831600, 2, 12, 180, 420, 12, 720, 2, 360360

Offset: 1

Antti Karttunen, Feb 19 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8192
Index entries for sequences computed from indices in prime factorization

Cf. A032741, A061395, A297113, A297167, A324190, A324203 (rgs-transform), A324537, A332461.

A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A061395(n) = if(1==n, 0, primepi(vecmax(factor(n)[, 1])));
A297167(n) = if(1==n, 0, (A061395(n) + (bigomega(n)-omega(n)) - 1));
A324202(n) = A046523(factorback(apply(x -> prime(1+x),apply(A297167, select(d -> d>1,divisors(n))))));

a(n) = A046523(A332461(n)).
A001221(a(n)) = A324190(n).
A001222(a(n)) = A032741(n).

A297157 Restricted growth sequence transform of A297156, which is Möbius transform of A243354.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Dec 28 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8192

Cf. A006068, A156552, A243354, A297156.

Cf. also A297113, A297161, A297162.

up_to = 8192;
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])); }
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) = if(1==n, 0, if(!(n%2), 1+(2*A156552(n/2)), 2*A156552(A064989(n))));
A006068(n)= { my(s=1, ns); while(1, ns = n >> s; if(0==ns, break()); n = bitxor(n, ns); s <<= 1; ); return (n); } \\ Essentially Joerg Arndt's Jul 19 2012 code.
A243354(n) = A006068(A156552(n));
A297156(n) = sumdiv(n,d,moebius(n/d)*A243354(d));
write_to_bfile(1,rgs_transform(vector(up_to,n,A297156(n))),"b297157.txt");

A297161 Restricted growth sequence transform of A297171, which is Möbius transform of A243071.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Dec 27 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8192

Cf. A243071, A297171.

Cf. also A297113, A297157, A297162.

up_to = 8192;
rgs_transform(invec) = { my(occurrences = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(occurrences,invec[i]), my(pp = mapget(occurrences, invec[i])); outvec[i] = outvec[pp] , mapput(occurrences,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])); }
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)};
A243071(n) = if(n<=2, n-1, if(!(n%2), 2*A243071(n/2), 1+(2*A243071(A064989(n)))));
A297171(n) = sumdiv(n,d,moebius(n/d)*A243071(d));
write_to_bfile(1,rgs_transform(vector(up_to,n,A297171(n))),"b297161.txt");

A297162 Restricted growth sequence transform of A297172, which is Möbius transform of A253564.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Dec 27 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8192

Cf. A253564, A297172.

Cf. also A297113, A297157, A297161.

up_to = 8192;
rgs_transform(invec) = { my(occurrences = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(occurrences,invec[i]), my(pp = mapget(occurrences, invec[i])); outvec[i] = outvec[pp] , mapput(occurrences,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])); }
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)};
A122111(n) = if(1==n,n,prime(bigomega(n))*A122111(A064989(n)));
A156552(n) = if(1==n, 0, if(!(n%2), 1+(2*A156552(n/2)), 2*A156552(A064989(n))));
A253564(n) = A156552(A122111(n));
A297172(n) = sumdiv(n,d,moebius(n/d)*A253564(d));
write_to_bfile(1,rgs_transform(vector(up_to,n,A297172(n))),"b297162.txt");

A318891 Filter sequence combining the prime signature of n (A046523) with the largest prime factor of n (A006530).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Sep 16 2018

nonn

Restricted growth sequence transform of A286356.

For all i, j: a(i) = a(j) => A297112(i) = A297112(j). (Also, equivalently, A297113 or A297167 in place of A297112.)

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

Cf. A006530, A046523, A061395, A286356, A318890.

Cf. also A297112, A297113, A297167.

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; };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523
A061395(n) = if(1==n, 0, primepi(vecmax(factor(n)[, 1])));
A318891aux(n) = [A046523(n), A061395(n)];
v318891 = rgs_transform(vector(up_to,n,A318891aux(n)));
A318891(n) = v318891[n];

A307373 Heinz numbers of integer partitions with at least three parts, the third of which is 2.

Original entry on oeis.org

27, 45, 54, 63, 75, 81, 90, 99, 105, 108, 117, 126, 135, 147, 150, 153, 162, 165, 171, 180, 189, 195, 198, 207, 210, 216, 225, 231, 234, 243, 252, 255, 261, 270, 273, 279, 285, 294, 297, 300, 306, 315, 324, 330, 333, 342, 345, 351, 357, 360, 363, 369, 378, 387

Offset: 1

Gus Wiseman, Apr 05 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

The enumeration of these partitions by sum is given by A006918 (see Emeric Deutsch's comment there).

			The sequence of terms together with their prime indices begins:
   27: {2,2,2}
   45: {2,2,3}
   54: {1,2,2,2}
   63: {2,2,4}
   75: {2,3,3}
   81: {2,2,2,2}
   90: {1,2,2,3}
   99: {2,2,5}
  105: {2,3,4}
  108: {1,1,2,2,2}
  117: {2,2,6}
  126: {1,2,2,4}
  135: {2,2,2,3}
  147: {2,4,4}
  150: {1,2,3,3}
  153: {2,2,7}
  162: {1,2,2,2,2}
  165: {2,3,5}
  171: {2,2,8}
  180: {1,1,2,2,3}

Cf. A000726, A002620, A004250, A006918, A056239, A097701, A112798, A257990, A297113, A325164, A325169, A325170.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],PrimeOmega[#]>=3&&Reverse[primeMS[#]][[3]]==2&]

cp's OEIS Frontend

A324202 a(n) = A046523(A332461(n)), where A332461(n) = Product_{d|n, d>1} prime(1+A297167(d)).

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A297157 Restricted growth sequence transform of A297156, which is Möbius transform of A243354.

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A297161 Restricted growth sequence transform of A297171, which is Möbius transform of A243071.

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A297162 Restricted growth sequence transform of A297172, which is Möbius transform of A253564.

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A318891 Filter sequence combining the prime signature of n (A046523) with the largest prime factor of n (A006530).

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A307373 Heinz numbers of integer partitions with at least three parts, the third of which is 2.

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Mathematica