A045967 -id:A045967 - cp's OEIS frontend

A045966 a(1)=3; if n = Product p_i^e_i, n > 1, then a(n) = Product p_{i+2}^e_i.

3, 5, 7, 25, 11, 35, 13, 125, 49, 55, 17, 175, 19, 65, 77, 625, 23, 245, 29, 275, 91, 85, 31, 875, 121, 95, 343, 325, 37, 385, 41, 3125, 119, 115, 143, 1225, 43, 145, 133, 1375, 47, 455, 53, 425, 539, 155, 59, 4375, 169, 605, 161, 475, 61, 1715, 187, 1625, 203, 185, 67

Offset: 1

N. J. A. Sloane

If we had a(1) = 1 (instead of 3), then this would be fully multiplicative with a(prime(k)) = prime(k+2) (see A357852). - Antti Karttunen, Jan 10 2020

From a puzzle proposed by Marc LeBrun.

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

See A027748, A124010 for factorization data for n.
Cf. A000040, A003961, A101300.
Sequences with similar definitions: A045967, A045968, A045970, A126272.
A059896 is used to express relationship between terms of this sequence.
A357852 is a slightly better version. - N. J. A. Sloane, Oct 29 2022

a045966 1 = 3
a045966 n = product $ zipWith (^)
            (map a101300 $ a027748_row n) (a124010_row n)
-- Reinhard Zumkeller, Jun 03 2013, Dec 23 2011

a[1] = 3; a[n_] := With[{f = FactorInteger[n]}, Times @@ (Prime[PrimePi[f[[All, 1]]]+2]^f[[All, 2]])]; Array[a, 60] (* Jean-François Alcover, Jun 19 2015 *)

A045966(n) = if(1==n,3,my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(1+nextprime(1+f[i, 1]))); factorback(f)); \\ Antti Karttunen, Jan 10 2020

From Peter Munn, Dec 27 2019, for n >= 2, k >= 2: (Start)
a(n) = A003961^2(n).
a(n^k) = a(n)^k.
a(A003961(n)) = A003961(a(n)).
a(A059896(n,k)) = A059896(a(n), a(k)).
(End)

More terms from David W. Wilson

A045968 a(1)=5; for n >= 2, if n = Product p_i^e_i, then a(n) = Product p_{i+3}^e_i.

Original entry on oeis.org

5, 7, 11, 49, 13, 77, 17, 343, 121, 91, 19, 539, 23, 119, 143, 2401, 29, 847, 31, 637, 187, 133, 37, 3773, 169, 161, 1331, 833, 41, 1001, 43, 16807, 209, 203, 221, 5929, 47, 217, 253, 4459, 53, 1309, 59, 931, 1573, 259, 61, 26411, 289, 1183, 319, 1127, 67, 9317, 247

Offset: 1

N. J. A. Sloane

			If n = 9 = 3^2, then a(n) = 11^2 = 121 (since 11 is the third prime after 3).

From a puzzle proposed by Marc LeBrun.

Cf. A045965, A045966, A045967, A045969, A045970, A045971, A045972, A045973.

f[p_, e_] := NextPrime[p, 3]^e; a[1] = 5; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 19 2023 *)

More terms from David W. Wilson

Erroneous linear recurrence deleted by Harvey P. Dale, May 07 2018

A045970 a(1)=7; if n = Product p_i^e_i, n > 1, then a(n) = Product p_{i+4}^e_i.

Original entry on oeis.org

7, 11, 13, 121, 17, 143, 19, 1331, 169, 187, 23, 1573, 29, 209, 221, 14641, 31, 1859, 37, 2057, 247, 253, 41, 17303, 289, 319, 2197, 2299, 43, 2431, 47, 161051, 299, 341, 323, 20449, 53, 407, 377, 22627, 59, 2717, 61, 2783, 2873, 451, 67, 190333, 361, 3179, 403

Offset: 1

N. J. A. Sloane

From a puzzle proposed by Marc LeBrun.

Cf. A045965, A045966, A045967, A045968, A045969, A045971, A045972, A045973.

f[p_, e_] := NextPrime[p, 4]^e; a[1] = 7; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 19 2023 *)

More terms from David W. Wilson

A045973 a(1)=10; if n = Product p_i^e_i, n > 1, then a(n) = Product p_{i+1}^e_i * Product p_{i+3}^e_i.

Original entry on oeis.org

10, 21, 55, 441, 91, 1155, 187, 9261, 3025, 1911, 247, 24255, 391, 3927, 5005, 194481, 551, 63525, 713, 40131, 10285, 5187, 1073, 509355, 8281, 8211, 166375, 82467, 1271, 105105, 1591, 4084101, 13585, 11571, 17017, 1334025, 1927, 14973, 21505, 842751

Offset: 1

N. J. A. Sloane

From a puzzle proposed by Marc LeBrun.

Cf. A045965, A045966, A045967, A045968, A045969, A045970, A045971, A045972.

f[p_, e_] := NextPrime[p]^e * NextPrime[p, 3]^e; a[1] = 10; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 19 2023 *)

Sum_{n>=1} 1/a(n) = -9/10 + Product_{k>=1} (1+1/(prime(k)*prime(k+4)-1)) = 0.2602421684... . - Amiram Eldar, Sep 19 2023

More terms from David W. Wilson

A304412 If n = Product (p_j^k_j) then a(n) = Product ((p_j + 1)*(k_j + 1)).

Original entry on oeis.org

1, 6, 8, 9, 12, 48, 16, 12, 12, 72, 24, 72, 28, 96, 96, 15, 36, 72, 40, 108, 128, 144, 48, 96, 18, 168, 16, 144, 60, 576, 64, 18, 192, 216, 192, 108, 76, 240, 224, 144, 84, 768, 88, 216, 144, 288, 96, 120, 24, 108, 288, 252, 108, 96, 288, 192, 320, 360, 120, 864, 124, 384, 192, 21, 336, 1152, 136, 324

Offset: 1

Ilya Gutkovskiy, May 12 2018

			a(36) = a(2^2*3^2) = (2 + 1)*(2 + 1) * (3 + 1)*(2 + 1) = 108.

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Ilya Gutkovskiy, Scatter plot of a(n) up to n=50000.
Vaclav Kotesovec, Graph - the asymptotic ratio (10000 terms)
Index entries for sequences computed from exponents in factorization of n.
Index entries for sequences computed from indices in prime factorization.

Cf. A000005, A000026, A000040, A000203, A000302 (numbers n such that a(n) is odd), A001221, A001615, A003959, A005117, A007947, A008864, A045967, A048250, A064549, A064840, A304407, A304408, A304409, A304411.

Cf. A065463.

a[n_] := Times @@ ((#[[1]] + 1) (#[[2]] + 1) & /@ FactorInteger[n]); a[1] = 1; Table[a[n], {n, 68}]
Table[DivisorSigma[0, n] Total[Select[Divisors[n], SquareFreeQ]], {n, 68}]

a(n)={numdiv(n)*sumdiv(n, d, moebius(d)^2*d)} \\ Andrew Howroyd, Jul 24 2018

a(n) = A000005(n)*A048250(n) = A000005(n)*A000203(A007947(n)).
a(p^k) = (p + 1)*(k + 1) where p is a prime and k > 0.
a(n) = 2^omega(n)*Product_{p|n} (p + 1) if n is a squarefree (A005117), where omega() = A001221.
Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 + 2/p^(s-1) - 1/p^(2*s-1)). - Amiram Eldar, Sep 17 2023
From Vaclav Kotesovec, May 06 2025: (Start)
Let f(s) = Product_{p prime} (p^s-p)^2 * (p^(2*s)+2*p^(s+1)-p) / (p^(2*s) * (p^s-1)^2).
Dirichlet g.f.: zeta(s-1)^2 * f(s).
Sum_{k=1..n} a(k) ~ ((2*log(n) + 4*gamma - 1)*f(2) + 2*f'(2)) * n^2/4, where
f(2) = A065463 = Product_{p prime} (1 - 1/(p*(p+1))) = 0.704442200999165592736603350326637210188586431417098049414226842591...,
f'(2) = f(2) * Sum_{p prime} 2*(2*p^2-1)*log(p) / ((p^2-1)*(p^2+p-1)) = f(2) * 1.799151495460164053607059266860868724519705035904425832307664926571...
and gamma is the Euler-Mascheroni constant A001620. (End)

A045969 a(1)=6; if n = Product p_i^e_i, n>1, then a(n) = Product p_{i+1}^e_i * Product p_{i+2}^e_i.

Original entry on oeis.org

6, 15, 35, 225, 77, 525, 143, 3375, 1225, 1155, 221, 7875, 323, 2145, 2695, 50625, 437, 18375, 667, 17325, 5005, 3315, 899, 118125, 5929, 4845, 42875, 32175, 1147, 40425, 1517, 759375, 7735, 6555, 11011, 275625, 1763, 10005, 11305, 259875, 2021, 75075

Offset: 1

N. J. A. Sloane

From a puzzle proposed by Marc LeBrun.

Cf. A045965, A045966, A045967, A045968, A045970, A045971, A045972, A045973.

f[p_, e_] := (NextPrime[p] * NextPrime[p, 2])^e; a[1] = 6; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 19 2023 *)

Sum_{n>=1} 1/a(n) = -5/6 + Product_{k>=2} (1+1/(prime(k)*prime(k+1)-1)) = 0.31383788... . - Amiram Eldar, Sep 19 2023

More terms from David W. Wilson

A045971 a(1)=8; if n = Product p_i^e_i, n > 1, then a(n) = Product p_{i+1}^{e_i+2}.

Original entry on oeis.org

8, 27, 125, 81, 343, 3375, 1331, 243, 625, 9261, 2197, 10125, 4913, 35937, 42875, 729, 6859, 16875, 12167, 27783, 166375, 59319, 24389, 30375, 2401, 132651, 3125, 107811, 29791, 1157625, 50653, 2187, 274625, 185193, 456533, 50625, 68921, 328509, 614125

Offset: 1

N. J. A. Sloane

From a puzzle proposed by Marc LeBrun.

Cf. A045965, A045966, A045967, A045968, A045969, A045970, A045972, A045973, A065483.

f[p_, e_] := NextPrime[p]^(e + 2); a[1] = 8; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 19 2023 *)

Sum_{n>=1} 1/a(n) = (4/5) * A065483 - 7/8 = 0.196827322859... . - Amiram Eldar, Sep 19 2023

More terms from David W. Wilson

A045972 a(1)=9; if n = Product p_i^e_i, n > 1, then a(n) = Product p_{i+2}^{e_i+1}.

Original entry on oeis.org

9, 25, 49, 125, 121, 1225, 169, 625, 343, 3025, 289, 6125, 361, 4225, 5929, 3125, 529, 8575, 841, 15125, 8281, 7225, 961, 30625, 1331, 9025, 2401, 21125, 1369, 148225, 1681, 15625, 14161, 13225, 20449, 42875, 1849, 21025, 17689, 75625, 2209, 207025

Offset: 1

N. J. A. Sloane

From a puzzle proposed by Marc LeBrun.

Cf. A045965, A045966, A045967, A045968, A045969, A045970, A045971, A045973, A082695.

f[p_, e_] := NextPrime[p, 2]^(e + 1); a[1] = 9; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 19 2023 *)

Sum_{n>=1} 1/a(n) = (4/7) * (zeta(2)*zeta(3)/zeta(6)) - 8/9 = 0.221737646437... . - Amiram Eldar, Sep 19 2023

More terms from David W. Wilson

A126272 a(1)=27; if n = Product p_i^e_i, n>1, then a(n) = Product p_{i+2}^{e_i+2}.

Original entry on oeis.org

27, 125, 343, 625, 1331, 42875, 2197, 3125, 2401, 166375, 4913, 214375, 6859, 274625, 456533, 15625, 12167, 300125, 24389, 831875, 753571, 614125, 29791, 1071875, 14641, 857375, 16807, 1373125, 50653, 57066625, 68921, 78125, 1685159

Offset: 1

Jonathan Vos Post, Mar 09 2007

Analog of A045967 a(1)=4; if n = Product p_i^e_i, n>1, then a(n) = Product p_{i+1}^{e_i+1}. In a sense, n is the zeroth sequence in a family of sequences, A045967 is the first sequence in a family of sequences and a(n) is the second sequence in a family of sequences.

If we had a(1) = 1 (instead of 4), then this would be multiplicative and a permutation of A353502. - Amiram Eldar, Aug 11 2022

Cf. A000040, A045967, A065483, A353502.

A126272 := proc(n) local pf,i,p,e,resul ; if n = 1 then 27 ; else pf := ifactors(n)[2] ; resul := 1 ; for i from 1 to nops(pf) do p := op(1,op(i,pf)) ; e := op(2,op(i,pf)) ; resul := resul * nextprime(nextprime(p))^(e+2) ; od ; resul ; fi ; end: for n from 1 to 40 do printf("%d, ",A126272(n)) ; od ; # R. J. Mathar, Apr 20 2007

f[p_, e_] := NextPrime[p, 2]^(e + 2); a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

Sum_{n>=1} 1/a(n) = (72/95)*A065483 - 26/27. - Amiram Eldar, Aug 11 2022

More terms from R. J. Mathar, Apr 20 2007

A328915 If n = Product (p_j^k_j) then a(n) = Product (nextprime(p_j)), where nextprime = A151800.

Original entry on oeis.org

1, 3, 5, 3, 7, 15, 11, 3, 5, 21, 13, 15, 17, 33, 35, 3, 19, 15, 23, 21, 55, 39, 29, 15, 7, 51, 5, 33, 31, 105, 37, 3, 65, 57, 77, 15, 41, 69, 85, 21, 43, 165, 47, 39, 35, 87, 53, 15, 11, 21, 95, 51, 59, 15, 91, 33, 115, 93, 61, 105, 67, 111, 55, 3, 119, 195, 71, 57, 145, 231

Offset: 1

Ilya Gutkovskiy, Oct 30 2019

All terms are odd.

			a(12) = a(2^2 * 3) = a(prime(1)^2 * prime(2)) = prime(2) * prime(3) = 3 * 5 = 15.

Index entries for sequences computed from indices in prime factorization

Cf. A000720, A003961, A007947, A045965, A045967, A048250, A151800, A328878, A328879.

a:= n-> mul(nextprime(i[1]), i=ifactors(n)[2]):
seq(a(n), n=1..100);  # Alois P. Heinz, Oct 30 2019

a[1] = 1; a[n_] := Times @@ (NextPrime[#[[1]]] & /@ FactorInteger[n]); Table[a[n], {n, 1, 70}]

a(n) = my(f=factor(n)); prod(k=1, #f~, nextprime(f[k,1]+1)); \\ Michel Marcus, Oct 30 2019

If n = Product (p_j^k_j) then a(n) = Product (prime(pi(p_j) + 1)), where pi = A000720.

a(n) = A007947(A003961(n)).

cp's OEIS Frontend

A045966 a(1)=3; if n = Product p_i^e_i, n > 1, then a(n) = Product p_{i+2}^e_i.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

Extensions

A045968 a(1)=5; for n >= 2, if n = Product p_i^e_i, then a(n) = Product p_{i+3}^e_i.

Views

Author

Keywords

Examples

References

Crossrefs

Programs

Mathematica

Extensions

A045970 a(1)=7; if n = Product p_i^e_i, n > 1, then a(n) = Product p_{i+4}^e_i.

Views

Author

Keywords

References

Crossrefs

Programs

Mathematica

Extensions

A045973 a(1)=10; if n = Product p_i^e_i, n > 1, then a(n) = Product p_{i+1}^e_i * Product p_{i+3}^e_i.

Views

Author

Keywords

References

Crossrefs

Programs

Mathematica

Formula

Extensions

A304412 If n = Product (p_j^k_j) then a(n) = Product ((p_j + 1)*(k_j + 1)).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A045969 a(1)=6; if n = Product p_i^e_i, n>1, then a(n) = Product p_{i+1}^e_i * Product p_{i+2}^e_i.

Views

Author

Keywords

References

Crossrefs

Programs

Mathematica

Formula

Extensions

A045971 a(1)=8; if n = Product p_i^e_i, n > 1, then a(n) = Product p_{i+1}^{e_i+2}.

Views

Author

Keywords

References

Crossrefs

Programs

Mathematica

Formula

Extensions

A045972 a(1)=9; if n = Product p_i^e_i, n > 1, then a(n) = Product p_{i+2}^{e_i+1}.

Views

Author

Keywords

References