A329345 -id:A329345 - cp's OEIS frontend

A329348 The least significant nonzero digit in the primorial base expansion of primorial inflation of n, A108951(n).

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

Offset: 1

Antti Karttunen, Nov 11 2019

nonn

Number of occurrences of the least primorial present in the greedy sum of primorials adding to A108951(n).

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 = 1*30 + 3*6, and as the factor of the least primorial in the sum is 3, we have a(24) = 3.

Cf. A002110, A034386, A067029, A108951, A117366, A276086, A276088, A324886, A324888, A329040, A329345, A329343, A329344, A329349, A331188, A331289, A331290, A331291.

Cf. also A331292 (the next digit to left of this one), A331293.

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
A276088(n) = { my(e=0, p=2); while(n && !(e=(n%p)), n = n/p; p = nextprime(1+p)); (e); };
A329348(n) = A276088(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));
A067029(n) = if(1==n, 0, factor(n)[1, 2]); \\ From A067029
A329348(n) = A067029(A324886(n));

A002110(n) = prod(i=1, n, prime(i));
A329348(n) = if(1==n, n, my(f=factor(n), p=nextprime(1+vecmax(f[, 1]))); prod(i=1, #f~, A002110(primepi(f[i, 1]))^(f[i, 2]-(#f~==i)))%p); \\ Antti Karttunen, Jan 15 2020

a(n) = A067029(A324886(n)) = A276088(A108951(n)).
a(n) <= A324888(n).
From Antti Karttunen, Jan 15-17 2020: (Start)
a(n) = A331188(n) mod A117366(n).
a(n) = A001511(A246277(A324886(n))).
(End)

Name changed by Antti Karttunen, Jan 17 2020

A329620 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A046523(n), A246277(A324886(n))].

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Nov 18 2019

nonn

Restricted growth sequence transform of the ordered pair [A046523(n), A246277(A324886(n))].
For all i, j:
  A305800(i) = A305800(j) => a(i) = a(j),
  a(i) = a(j) => A101296(i) = A101296(j),
  a(i) = a(j) => A329345(i) = A329345(j),
  a(i) = a(j) => A329618(i) = A329618(j),
  a(i) = a(j) => A329619(i) = A329619(j).

Cf. A046523, A034386, A108951, A246277, A276086, A305800, A324886, A329345, A329618, A329619.

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; };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523
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));
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);
Aux329620(n) = [A046523(n), A246277(A324886(n))];
v329620 = rgs_transform(vector(up_to, n, Aux329620(n)));
A329620(n) = v329620[n];

A329349 Number of occurrences of the largest primorial 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, 1, 4, 2, 1, 4, 1, 1, 1, 1, 6, 2, 2, 4, 1, 2, 6, 1, 1, 1, 1, 4, 5, 2, 1, 3, 1, 8, 6, 4, 1, 2, 2, 8, 6, 2, 1, 3, 1, 2, 3, 2, 1, 12, 1, 4, 6, 5, 1, 1, 1, 2, 2, 4, 16, 12, 1, 2, 6, 2, 1, 2, 1, 2, 6, 8, 1, 10, 12, 4, 6, 2, 1, 6, 1, 2, 2, 1, 1, 12, 1, 8, 1

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 = 21 = 3 * 7, A108951(21) = A034386(3) * A034386(7) = 6 * 210, so the factor of the largest primorial present (210) in the greedy sum is 6 (as 1260 = 210 + 210 + 210 + 210 + 210 + 210), thus a(21) = 6.
For n = 24 = 2^3 * 3, A108951(24) = A034386(2)^3 * A034386(3) = 2^3 * 6 = 48 = 1*30 + 3*6, and as the factor of the largest primorial in the sum is 1, we have a(24) = 1.

Cf. A002110, A034386, A071178, A108951, A276086, A276153, A324886, A324888, A329040, A329343, A329344, A329345, A329348.

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
A276153(n) = { my(e=0, p=2); while(n, e=n%p; n = n\p; p = nextprime(1+p)); (e); };
A329349(n) = A276153(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));
A071178(n) = if(1==n,0,my(es=factor(n)[,2]); es[#es]);
A329349(n) = A071178(A324886(n));

a(n) = A276153(A108951(n)) = A071178(A324886(n)).

a(n) <= A324888(n).

A329038 a(n) = A246277(A276086(n)).

Original entry on oeis.org

0, 1, 1, 3, 2, 9, 1, 5, 3, 15, 6, 45, 2, 25, 9, 75, 18, 225, 4, 125, 27, 375, 54, 1125, 8, 625, 81, 1875, 162, 5625, 1, 7, 5, 21, 10, 63, 3, 35, 15, 105, 30, 315, 6, 175, 45, 525, 90, 1575, 12, 875, 135, 2625, 270, 7875, 24, 4375, 405, 13125, 810, 39375, 2, 49, 25, 147, 50, 441, 9, 245, 75, 735, 150, 2205, 18, 1225, 225, 3675, 450, 11025, 36

Offset: 0

Antti Karttunen, Nov 08 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 0..2310
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..32768
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to primorial base

Cf. A046523, A246277, A276086, A278226, A329048 (rgs-transform).

Cf. also A329345.

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); };
A329038(n) = A246277(A276086(n));

a(n) = A246277(A276086(n)).

For n >= 1, A046523(2*a(n)) = A278226(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];

A329048 Lexicographically earliest infinite sequence such that a(i) = a(j) => A329038(i) = A329038(j) for all i, j.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Nov 11 2019

nonn

Restricted growth sequence transform of A329038, i.e., of function f(n) = A246277(A276086(n)).
For all i, j:
  a(i) = a(j) => A286626(i) = A286626(j),
  a(i) = a(j) => A276088(i) = A276088(j),
  a(i) = a(j) => A276153(i) = A276153(j),

Antti Karttunen, Table of n, a(n) for n = 0..32768

Cf. A046523, A246277, A276086, A276088, A276153, A278226, A286626, A329038.

Cf. also A329345.

up_to = 32768;
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; };
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); };
A329038(n) = A246277(A276086(n));
v329048 = rgs_transform(vector(1+up_to, n, A329038(n-1)));
A329048(n) = v329048[1+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];

A351949 Lexicographically earliest infinite sequence such that a(i) = a(j) => A246277(A329044(i)) = A246277(A329044(j)) and A003557(i) = A003557(j), for all i, j >= 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Apr 05 2022

nonn

Restricted growth sequence transform of the ordered pair [A003557(n), A329345(n)].

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

Cf. A003557, A305800, A329044, A329345, A329620.

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(factorint(n)[, 1]));
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]) };
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);
v351949 = rgs_transform(vector(up_to, n, [A003557(n), A246277(A329044(n))]));
A351949(n) = v351949[n];

cp's OEIS Frontend

A329348 The least significant nonzero digit in the primorial base expansion of primorial inflation of n, A108951(n).

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A329620 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A046523(n), A246277(A324886(n))].

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A329349 Number of occurrences of the largest primorial present in the greedy sum of primorials adding to A108951(n).

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A329038 a(n) = A246277(A276086(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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A329048 Lexicographically earliest infinite sequence such that a(i) = a(j) => A329038(i) = A329038(j) for all i, j.

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

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