A329045 -id:A329045 - cp's OEIS frontend

A329040 Number of distinct primorials in the greedy sum of primorials adding to A108951(n).

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 3, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 2, 2, 2, 1, 1, 1, 1, 2, 1, 3, 1, 1, 3, 1, 2, 1, 1, 2, 3, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 3, 2, 3, 1, 1, 1, 1, 3

Offset: 1

Antti Karttunen, Nov 11 2019

nonn

The greedy sum is also the sum with the minimal number of primorials used in the primorial base representation.

			For n = 18 = 2 * 3^2, A108951(18) = A034386(2) * A034386(3)^2 = 2 * 6^2 = 72 = 2*A002110(3) + 2*A002110(2) = 2*30 + 2*6, and because there occurs only two distinct primorials (30 and 6) in the sum, we have a(18) = 2.

Cf. A001221, A002110, A034386, A108951, A267263, A276086, A324886, A324888, A329051 (positions of records), A329343.

Cf. also A329045, A329046.

A034386(n) = prod(i=1, primepi(n), prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) };  \\ From A108951
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A324886(n) = A276086(A108951(n));
A329040(n) = omega(A324886(n));

a(n) = A001221(A324886(n)).
a(n) = A267263(A108951(n)).
a(n) <= A324888(n).

A329044 a(n) = A064989(A324886(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 9, 7, 6, 15, 25, 11, 81, 13, 49, 15625, 36, 17, 225, 19, 625, 117649, 121, 23, 135, 60025, 169, 21, 2401, 29, 21875, 31, 10, 1771561, 289, 697540921, 50625, 37, 361, 4826809, 35, 41, 77, 43, 14641, 84035, 529, 47, 375, 161212051, 3603000625, 24137569, 28561, 53, 441, 2474329, 5764801, 47045881, 841, 59, 42875, 61, 961

Offset: 1

Antti Karttunen, Nov 08 2019

nonn

Cf. A034386, A064989, A108951, A276086, A324886, A329045.

A034386(n) = prod(i=1, primepi(n), prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) };  \\ From A108951
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A324886(n) = A276086(A108951(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)};
A329044(n) = A064989(A324886(n));

a(n) = A064989(A324886(n)).

a(A000040(n)) = A000040(n).

A329345 Lexicographically earliest infinite sequence such that a(i) = a(j) => A246277(A329044(i)) = A246277(A329044(j)) for all i, j.

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 2, 4, 4, 3, 2, 5, 2, 3, 6, 7, 2, 7, 2, 5, 6, 3, 2, 8, 9, 3, 10, 5, 2, 11, 2, 10, 6, 3, 12, 13, 2, 3, 6, 4, 2, 4, 2, 5, 14, 3, 2, 15, 16, 17, 6, 5, 2, 18, 19, 20, 6, 3, 2, 21, 2, 3, 21, 18, 22, 23, 2, 5, 6, 24, 2, 25, 2, 3, 26, 5, 27, 23, 2, 7, 28, 3, 2, 7, 29, 3, 6, 20, 2, 30, 31, 5, 6, 3, 32, 33, 2, 34, 35, 16, 2, 23, 2, 20, 36

Offset: 1

Antti Karttunen, Nov 11 2019

nonn

Restricted growth sequence transform of function f(n) = A246277(A329044(n)).
For all i, j:
  A305800(i) = A305800(j) => a(i) = a(j),
  a(i) = a(j) => A329045(i) = A329045(j),
  a(i) = a(j) => A329343(i) = A329343(j),
  a(i) = a(j) => A329348(i) = A329348(j),
  a(i) = a(j) => A329349(i) = A329349(j).

Cf. A034386, A064989, A108951, A246277, A276086, A305800, A324886, A329044, A329045, A329343, A329348, A329349.

Cf. also A329048.

up_to = 1024;
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; };
A034386(n) = prod(i=1, primepi(n), prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) };  \\ From A108951
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A324886(n) = A276086(A108951(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)};
A329044(n) = A064989(A324886(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);
v329345 = rgs_transform(vector(up_to, n, A246277(A329044(n))));
A329345(n) = v329345[n];

A329344 Number of times most frequent primorial is present in the greedy sum of primorials adding to A108951(n).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Nov 11 2019

nonn

The greedy sum is also the sum with the minimal number of primorials, used for example in the primorial base representation.

			For n = 24 = 2^3 * 3, A108951(24) = A034386(2)^3 * A034386(3) = 2^3 * 6 = 48 = 30 + 6 + 6 + 6, and as the most frequent primorial in the sum is 6 = A002110(2), we have a(24) = 3.

Cf. A002110, A034386, A051903, A108951, A276086, A324886, A324888, A328114, A329040, A329045, A329343, A329348, A329349.

With[{b = Reverse@ Prime@ Range@ 120}, Array[Max@ IntegerDigits[#, MixedRadix[b]] &@ Apply[Times, Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Times @@ Prime@ Range@ PrimePi@ p, e}]] &, 105] ] (* Michael De Vlieger, Nov 18 2019 *)

A034386(n) = prod(i=1, primepi(n), prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) };  \\ From A108951
A328114(n) = { my(s=0, p=2); while(n, s = max(s,(n%p)); n = n\p; p = nextprime(1+p)); (s); };
A329344(n) = A328114(A108951(n));

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A324886(n) = A276086(A108951(n));
A051903(n) = if((1==n),0,vecmax(factor(n)[, 2]));
A329344(n) = A051903(A324886(n));

a(n) = A328114(A108951(n)) = A051903(A324886(n)).

A373982 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278226(A328768(i)) = A278226(A328768(j)), for all i, j >= 0.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Jun 25 2024

nonn

Restricted growth sequence transform of A278226(A328768(n)).
For all i, j >= 1:
  A305800(i) = A305800(j) => A373983(i) = A373983(j) => a(i) = a(j).
For all i, j >= 0: a(i) = a(j) => A328771(i) = A328771(j).

Antti Karttunen, Table of n, a(n) for n = 0..100000
Index entries for sequences related to primorial numbers

Cf. A002110, A046523, A276085, A276086, A278226, A328768, A373983.

Cf. also A329045.

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]); };  \\ From A046523
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A278226(n) = A046523(A276086(n));
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]));
v373982 = rgs_transform(vector(1+up_to, n, A278226(A328768(n-1))));
A373982(n) = v373982[1+n];

A329046 a(n) = A000005(A324886(n)).

Original entry on oeis.org

2, 2, 2, 3, 2, 3, 2, 4, 4, 3, 2, 5, 2, 3, 7, 9, 2, 9, 2, 5, 7, 3, 2, 8, 15, 3, 4, 5, 2, 12, 2, 4, 7, 3, 27, 25, 2, 3, 7, 4, 2, 4, 2, 5, 12, 3, 2, 8, 28, 45, 7, 5, 2, 9, 15, 9, 7, 3, 2, 16, 2, 3, 16, 9, 28, 13, 2, 5, 7, 36, 2, 24, 2, 3, 72, 5, 51, 13, 2, 9, 28, 3, 2, 9, 24, 3, 7, 9, 2, 33, 91, 5, 7, 3, 16, 21, 2, 117, 33, 28, 2, 13, 2, 9, 40

Offset: 1

Antti Karttunen, Nov 08 2019

nonn

Cf. A000005, A034386, A064989, A108951, A276086, A324655, A324886, A329045.

A034386(n) = prod(i=1, primepi(n), prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) };  \\ From A108951
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A324886(n) = A276086(A108951(n));
A329046(n) = numdiv(A324886(n));

a(n) = A000005(A324886(n)) = A000005(A276086(A108951(n))).

a(n) = A324655(A108951(n)).

A344593 Lexicographically earliest infinite sequence such that a(i) = a(j) => A344592(i) = A344592(j), for all i, j >= 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, May 26 2021

nonn

Restricted growth sequence transform of A344592, where A344592(n) = A003557(A276086(A108951(n))).

For all i, j: a(i) = a(j) => A329344(i) = A329344(j).

			Both a(14) = 6 and a(32768) = 6, because A344592(14) = 11 is the sixth distinct value occurring in A344592, and A344592(32768) = A003557(A276086(A108951(32768))) = A003557(A276086(32768)) = A003557(401115) = A003557(3 * 5 * 11^2 * 13 * 17) = 11 also, which is the second time 11 occurs in A344592.

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

Cf. A003557, A108951, A276086, A324886, A329344, A344591 (positions of ones), A344592.

Cf. also A329045, A329345, A344594.

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; };
A034386(n) = prod(i=1, primepi(n), prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) }; \\ From A108951
A328572(n) = { my(m=1, p=2); while(n, if(n%p, m *= p^((n%p)-1)); n = n\p; p = nextprime(1+p)); (m); };
A344592(n) = A328572(A108951(n));
v344593 = rgs_transform(vector(up_to, n, A344592(n)));
A344593(n) = v344593[n];

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];

A351955 Lexicographically earliest infinite sequence such that a(i) = a(j) => A328571(A108951(i)) = A328571(A108951(j)) for all i, j >= 1.

Original entry on oeis.org

1, 2, 3, 2, 4, 3, 5, 6, 7, 4, 8, 3, 9, 5, 4, 6, 10, 7, 11, 4, 5, 8, 12, 7, 13, 9, 14, 5, 15, 13, 16, 17, 8, 10, 18, 7, 19, 11, 9, 13, 20, 18, 21, 8, 13, 12, 22, 7, 23, 13, 10, 9, 24, 14, 25, 5, 11, 15, 26, 13, 27, 16, 18, 17, 28, 8, 29, 10, 12, 18, 30, 31, 32, 19, 33, 11, 25, 9, 34, 13, 31, 20, 35, 18, 36, 21, 15, 8

Offset: 1

Antti Karttunen, Apr 03 2022

Restricted growth sequence transform of A346091, or equally, of A346093.
For all i, j:
  a(i) = a(j) => A006530(i) = A006530(j) [equally, A061395(i) = A061395(j)],
  a(i) = a(j) => A329040(i) = A329040(j) => A351956(i) = A351956(j),
  a(i) = a(j) => A329343(i) = A329343(j).
Interestingly, some of the rays in the scatter plot appear to be cut to discontinuous segments.

Cf. A108951, A328571, A346091, A346093.

Cf. also A006530, A061395, A329040, A329343, A351956, and A286621, A329045, A329345, A344594.

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; };
A002110(n) = prod(i=1,n,prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A002110(primepi(f[i, 1]))^f[i, 2]) };
A328571(n) = { my(m=1, p=2); while(n, m *= (p^!!(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A346091(n) = A328571(A108951(n));
v351955 = rgs_transform(vector(up_to, n, A346091(n)));
A351955(n) = v351955[n];

cp's OEIS Frontend

A329040 Number of distinct primorials in the greedy sum of primorials adding to A108951(n).

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A329044 a(n) = A064989(A324886(n)).

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A329345 Lexicographically earliest infinite sequence such that a(i) = a(j) => A246277(A329044(i)) = A246277(A329044(j)) for all i, j.

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A329344 Number of times most frequent primorial is present in the greedy sum of primorials adding to A108951(n).

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A373982 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278226(A328768(i)) = A278226(A328768(j)), for all i, j >= 0.

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A329046 a(n) = A000005(A324886(n)).

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A344593 Lexicographically earliest infinite sequence such that a(i) = a(j) => A344592(i) = A344592(j), for all i, j >= 1.

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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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