A117366 -id:A117366 - cp's OEIS frontend

A285639 a(n) = n*A117366(n)/spf(n), where A117366(n) is the smallest prime larger than all prime factors of n, and spf is the smallest prime factor of n (or 1 if n = 1).

2, 3, 5, 6, 7, 15, 11, 12, 15, 35, 13, 30, 17, 77, 35, 24, 19, 45, 23, 70, 77, 143, 29, 60, 35, 221, 45, 154, 31, 105, 37, 48, 143, 323, 77, 90, 41, 437, 221, 140, 43, 231, 47, 286, 105, 667, 53, 120, 77, 175, 323, 442, 59, 135, 143, 308, 437, 899, 61, 210, 67, 1147

Offset: 1

M. F. Hasler, Apr 30 2017

nonn

The smallest prime factor of n is removed, and a prime factor larger than all others is added. This is somewhat in the spirit of A003961 where each of the prime factors is increased to the next larger prime. Therefore a(n) = A003961(n) when n is a prime or a product of consecutive primes.

Leaves invariant A073485, i.e., for all n in A073485, a(n) is again in A073485. More precisely, a(A098012(m,n)) = A098012(m,n+1). - M. F. Hasler, May 03 2017

			a(1) = nextprime(1) = 2.
a(2) = 2 / 2 * nextprime(2) = 3.
a(3) = 3 / 3 * nextprime(3) = 5, and in the same way, a(prime(k))=prime(k+1).
a(4) = 4 / 2 * nextprime(2) = 2*3 = 6.
a(6) = 6 / 2 * nextprime(3) = 3*5 = 15.

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A020639, A003961, A006530, A117366, A151800.

Table[d = FactorInteger[n]; n*NextPrime[d[[-1, 1]]]/d[[1, 1]], {n, 62}] (* Ivan Neretin, Jan 23 2018 *)

a(n,f=factor(n)[,1])={f||f=[1];n\f[1]*nextprime(f[#f]+1)}

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

Original entry on oeis.org

2, 3, 5, 9, 7, 25, 11, 15, 35, 49, 13, 625, 17, 121, 117649, 225, 19, 1225, 23, 2401, 1771561, 169, 29, 875, 717409, 289, 55, 14641, 31, 184877, 37, 21, 4826809, 361, 36226650889, 1500625, 41, 529, 24137569, 77, 43, 143, 47, 28561, 1127357, 841, 53, 1715, 902613283, 514675673281, 47045881, 83521, 59, 3025, 8254129, 214358881, 148035889, 961, 61

Offset: 1

Antti Karttunen, Mar 30 2019

nonn

Cf. A034386, A108951, A117366, A276086, A324887, A324888, A324896, A329047, A342920, A344592, A346106, A346107.
Permutation of A324576.
Cf. also A346105.

With[{b = MixedRadix[Reverse@ Prime@ Range@ 120]}, Array[Function[k, Times @@ Power @@@ # &@ Transpose@ {Prime@ Range@ Length@ k, Reverse@ k}]@ IntegerDigits[Apply[Times, Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Times @@ Prime@ Range@ PrimePi@ p, e}]], b] &, 58]] (* Michael De Vlieger, Nov 18 2019 *)
A276086[n0_] := Module[{m = 1, i = 1, n = n0, p}, While[n > 0, p = Prime[i]; m *= p^Mod[n, p]; n = Quotient[n, p]; i++]; m];
(* b is A108951 *)
b[n_] := b[n] = Module[{pe = FactorInteger[n], p, e}, If[Length[pe] > 1, Times @@ b /@ Power @@@ pe, {{p, e}} = pe; Times @@ (Prime[Range[ PrimePi[p]]]^e)]]; b[1] = 1;
a[n_] := A276086[b[n]];
Array[a, 100] (* Jean-François Alcover, Dec 01 2021, after _Antti Karttunen in A296086 *)

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(i=0,m=1,pr=1,nextpr); while((n>0),i=i+1; nextpr = prime(i)*pr; if((n%nextpr),m*=(prime(i)^((n%nextpr)/pr));n-=(n%nextpr));pr=nextpr); m; };
A324886(n) = A276086(A108951(n));

a(n) = A276086(A108951(n)).
a(n) = A117366(n) * A324896(n).
A001222(a(n)) = A324888(n).
A020639(a(n)) = A117366(n).
A032742(a(n)) = A324896(n).
a(A000040(n)) = A000040(1+n).
From Antti Karttunen, Jul 09 2021: (Start)
For n > 1, a(n) = A003961(A329044(n)).
a(n) = A346091(n) * A344592(n).
a(n) = A346106(n) /  A346107(n).
A003415(a(n)) = A329047(n).
A003557(a(n)) = A344592(n).
A342001(a(n)) = A342920(n) = A329047(n) / A344592(n).
(End)

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).

A117365 a(n) = largest prime less than the smallest prime dividing n (or 1 if there is no such prime).

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 5, 1, 2, 1, 7, 1, 11, 1, 2, 1, 13, 1, 17, 1, 2, 1, 19, 1, 3, 1, 2, 1, 23, 1, 29, 1, 2, 1, 3, 1, 31, 1, 2, 1, 37, 1, 41, 1, 2, 1, 43, 1, 5, 1, 2, 1, 47, 1, 3, 1, 2, 1, 53, 1, 59, 1, 2, 1, 3, 1, 61, 1, 2, 1, 67, 1, 71, 1, 2, 1, 5, 1, 73, 1, 2, 1, 79, 1, 3, 1, 2, 1, 83, 1, 5, 1, 2, 1, 3, 1

Offset: 1

Leroy Quet, Mar 10 2006

nonn

a(n) = 1 if and only if n is even or if n = 1.

			5 is the smallest prime dividing 35. So a(35) is the largest prime < 5, which is 3.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A117364, A117366, A117367, A117368.

a[n_] := If[EvenQ[n], 1, Prime[PrimePi[FactorInteger[n][[1]][[1]]] - 1]]; Table[a[n], {n, 2, 80}] (* Stefan Steinerberger, Mar 14 2006 *)
Table[NextPrime[FactorInteger[n][[1, 1]], -1] /. -2 -> 1, {n, 96}] (* Michael De Vlieger, Sep 16 2017 *)

More terms from Stefan Steinerberger, Mar 14 2006

More terms from Franklin T. Adams-Watters, Aug 29 2006

A117367 a(n) = smallest prime greater than the smallest prime dividing n.

Original entry on oeis.org

2, 3, 5, 3, 7, 3, 11, 3, 5, 3, 13, 3, 17, 3, 5, 3, 19, 3, 23, 3, 5, 3, 29, 3, 7, 3, 5, 3, 31, 3, 37, 3, 5, 3, 7, 3, 41, 3, 5, 3, 43, 3, 47, 3, 5, 3, 53, 3, 11, 3, 5, 3, 59, 3, 7, 3, 5, 3, 61, 3, 67, 3, 5, 3, 7, 3, 71, 3, 5, 3, 73, 3, 79, 3, 5, 3, 11, 3, 83, 3, 5, 3, 89, 3, 7, 3, 5, 3, 97, 3, 11, 3, 5

Offset: 1

Leroy Quet, Mar 10 2006

nonn

All even-indexed terms are 3.

			5 is the smallest prime dividing 35. So a(35) is the smallest prime > 5, which is 7.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A117364, A117365, A117366.

with(numtheory): a:=proc(n): if n=1 then 2 else nextprime(factorset(n)[1]) fi: end: seq(a(n),n=1..100); # Emeric Deutsch, Apr 22 2006

Table[NextPrime[FactorInteger[n][[1, 1]]], {n, 93}] (* Michael De Vlieger, Sep 16 2017 *)

More terms from Emeric Deutsch, Apr 22 2006

A285993 Largest odd abundant number (A005231) equal to the product of n consecutive primes.

Original entry on oeis.org

15015, 255255, 4849845, 111546435, 33426748355, 1236789689135, 50708377254535, 2180460221945005, 102481630431415235, 5431526412865007455, 320460058359035439845, 19548063559901161830545, 1309720258513377842646515, 1357656019974967471687377449, 107254825578022430263302818471

Offset: 5

M. F. Hasler, Apr 30 2017

nonn

The smallest term is a(5) = 3*5*7*11*13, there is no odd abundant number (A005231) equal to the product of less than 5 consecutive primes.
The smallest odd abundant number (A005231) equal to the product of n consecutive primes is equal (when it exists, i.e., for n >= 5) to the least odd number with n (distinct) prime divisors, equal to the product of the first n odd primes = A070826(n+1) = A002110(n+1)/2.
See A188342 = (945, 3465, 15015, 692835, 22309287, ...) for the least odd primitive abundant number (A006038) with n distinct prime factors, and A275449 for the least odd primitive abundant number with n prime factors counted with multiplicity.
The terms are in general not primitive abundant numbers (A091191), in particular this cannot be the case when a(n) is a multiple of a(n-1), as is the case for most of the terms, for which a(n) = a(n-1)*A117366(a(n-1)). In the other event, spf(a(n)) = nextprime(spf(a(n-1))), and a(n) is in A007741(2,3,4...). These are exactly the primitive terms in this sequence.

			For n < 5, there is no odd abundant number equal to the product of n distinct primes.
For 5 <= n <= 8, the largest odd abundant number equal to the product of n consecutive primes is 3*...*prime(n+1).
For 9 <= n <= 17, the largest odd abundant number equal to the product of n consecutive primes is 5*...*prime(n+2).
For 18 <= n <= 30, the largest odd abundant number equal to the product of n consecutive primes is 7*...*prime(n+3).
For 31 <= n <= 45, the largest odd abundant number equal to the product of n consecutive primes is 11*...*prime(n+4).
For 46 <= n <= 66, the largest odd abundant number equal to the product of n consecutive primes is 13*...*prime(n+5).

Amiram Eldar, Table of n, a(n) for n = 5..341

Cf. A005231, A006038, A007707, A007708, A007741.

A subsequence of A112643 (odd squarefree abundant numbers); see also A108227 (~ A107705) which give indices of primitive terms = those with smallest prime factor larger than that of earlier terms.

a(r,f=vector(r,i,prime(i+1)),o)={ while(sigma(factorback(f),-1)>2, o=f; f=concat(f[^1],nextprime(f[r]+1)));factorback(o)} \\ Intentionally throws an error when n < 5.

a(n) >= a(n-1)*p where p = A117366(a(n-1)) = A151800(A006530(a(n-1))) = nextprime(gpf(a(n-1))), an odd abundant number equal to the product of n consecutive primes. We have strict inequality for n = 9, 18, 31, 46, 67, ..., in which case a(n) = a(n-1)*p*p'/q, where p' = nextprime(p), q = least prime factor of a(n-1). This is the case if a(n) is in A007741.

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)).

A331293 Let h = A061395(n) and u = A108951(n) - (A002110(h)*A329348(n)). Then a(n) = u/A002110(1+h).

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, 7, 0, 0, 1, 0, 5, 0, 0, 2, 4, 0, 0, 0, 1, 0, 1, 0, 0, 5, 0, 0, 3, 19, 8, 0, 0, 0, 14, 2, 0, 0, 0, 0, 3, 0, 0, 3, 10, 1, 0, 0, 0, 0, 5, 0, 9, 0, 0, 25, 0, 16, 0, 0, 2, 43, 0, 0, 2, 1, 0, 0, 0, 0, 10, 12, 0, 0, 0, 1, 6, 0, 38, 2, 17, 0, 0, 0, 0, 16

Offset: 1

Antti Karttunen, Jan 17 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10201
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to primorial numbers

Cf. A002110, A061395, A108951, A329348, A331188, A331292.

Cf. A331294 (positions of records).

A331293(n) = { my(h=A061395(n), u = A108951(n) - (A002110(h)*(A331188(n)%prime(1+h)))); (u/A002110(1+h)); };

a(n) modulo A151800(A117366(n)) = A331292(n).

A119416 n * (smallest prime greater than largest prime factor of n).

Original entry on oeis.org

2, 6, 15, 12, 35, 30, 77, 24, 45, 70, 143, 60, 221, 154, 105, 48, 323, 90, 437, 140, 231, 286, 667, 120, 175, 442, 135, 308, 899, 210, 1147, 96, 429, 646, 385, 180, 1517, 874, 663, 280, 1763, 462, 2021, 572, 315, 1334, 2491, 240, 539, 350, 969, 884, 3127, 270

Offset: 1

Reinhard Zumkeller, May 17 2006

nonn

a(n) = n * A117366(n);

A000005(a(n)) = 2*A000005(n) = A062011(n).

			a(10) = (2*5)*7.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A006530, A087560.

Cf. A151800.

a119416 n = n * (a151800 $ a006530 n) -- Reinhard Zumkeller, Apr 06 2015

Insert[Table[n*Prime[PrimePi[FactorInteger[n][[ -1]][[1]]] + 1],{n,2,100}], 2, 1] (* Stefan Steinerberger, May 18 2006 *)

a(n) = n * A151800(A006530(n)). - Reinhard Zumkeller, Apr 06 2015

cp's OEIS Frontend

A285639 a(n) = n*A117366(n)/spf(n), where A117366(n) is the smallest prime larger than all prime factors of n, and spf is the smallest prime factor of n (or 1 if n = 1).

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

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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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A117365 a(n) = largest prime less than the smallest prime dividing n (or 1 if there is no such prime).

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A117367 a(n) = smallest prime greater than the smallest prime dividing n.

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A285993 Largest odd abundant number (A005231) equal to the product of n consecutive primes.

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