A331293 -id:A331293 - cp's OEIS frontend

A331294 Positions of records in A331293.

1, 8, 16, 25, 27, 49, 75, 81, 121, 169, 289, 361, 529, 841, 961, 1369, 1681, 1849, 2209, 2809, 3481, 3721, 4489, 5041, 5329, 6241, 6889, 7921, 9409, 10201, 10609, 11449, 11881, 12769, 16129, 17161, 18769, 19321, 22201, 22801, 24649, 26569, 27889, 29929, 32041, 32761, 36481, 37249, 38809, 39601, 44521, 49729, 51529, 52441, 54289

Offset: 1

Antti Karttunen, Jan 17 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..76

Cf. A001248, A331293.

m=-1; k=0; for(n=1,2^17,t=A331293(n); if(t > m, m=t; k++; print1(n,", "); write("b331294.txt", k, " ", n)));

Conjectured: for n >= 9, a(n) = A000040(n-4)^2 = A001248(n-4).

A329348 The least significant nonzero digit in the primorial base expansion of primorial inflation of n, 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, 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

A331188 Primorial inflation of A052126(n), where A052126(n) = n/(largest prime dividing n).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 4, 6, 2, 1, 4, 1, 2, 6, 8, 1, 12, 1, 4, 6, 2, 1, 8, 30, 2, 36, 4, 1, 12, 1, 16, 6, 2, 30, 24, 1, 2, 6, 8, 1, 12, 1, 4, 36, 2, 1, 16, 210, 60, 6, 4, 1, 72, 30, 8, 6, 2, 1, 24, 1, 2, 36, 32, 30, 12, 1, 4, 6, 60, 1, 48, 1, 2, 180, 4, 210, 12, 1, 16, 216, 2, 1, 24, 30, 2, 6, 8, 1, 72, 210, 4, 6, 2, 30, 32

Offset: 1

Antti Karttunen, Jan 14 2020

nonn

The primorial inflation of n, A108951(n), divided by its largest squarefree divisor, which is also its largest primorial divisor.

Cf. A002110, A003557, A052126, A108951, A111701, A117366, A329348, A329382, A331292, A331293.

A002110(n) = prod(i=1,n,prime(i));
A331188(n) = if(1==n, n, my(f=factor(n)); prod(i=1, #f~, A002110(primepi(f[i, 1]))^(f[i, 2]-(#f~==i))));

a(n) = A108951(A052126(n)).
a(n) = A003557(A108951(n)).
a(n) = A111701(A108951(n)) = A108951(n) / A002110(A061395(n)).
Other identities. For all >= 1:
A000005(a(n)) = A329382(n) = A005361(A108951(n)).
a(n) mod A117366(n) = A329348(n).

A331292 The next more significant digit after A329348(n) in the primorial base expansion of A108951(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 1, 4, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 4, 0, 0, 0, 1, 0, 1, 0, 0, 5, 0, 0, 3, 6, 8, 0, 0, 0, 0, 2, 0, 0, 0, 0, 3, 0, 0, 3, 0, 1, 0, 0, 0, 0, 5, 0, 2, 0, 0, 3, 0, 16, 0, 0, 2, 1, 0, 0, 2, 1, 0, 0, 0, 0, 10, 12, 0, 0, 0, 1, 6, 0, 12, 2, 6, 0, 0, 0, 0, 3

Offset: 1

Antti Karttunen, Jan 17 2020

nonn

Cf. A007949, A061395, A108951, A117366, A151800, A246277, A324886, A329348, A331188,

A331293.

A331292(n) = { my(h=A061395(n), u = A108951(n) - (A002110(h)*((A331188(n)%prime(1+h))))); ((u/A002110(1+h))%prime(2+h)); };

a(n) = A007949(A246277(A324886(n))).

a(n) = A331293(n) modulo A000040(2+A061395(n)).

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A331294 Positions of records in A331293.

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

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A331188 Primorial inflation of A052126(n), where A052126(n) = n/(largest prime dividing n).

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A331292 The next more significant digit after A329348(n) in the primorial base expansion of A108951(n).

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