A003964 -id:A003964 - cp's OEIS frontend

A003961 Completely multiplicative with a(prime(k)) = prime(k+1).

1, 3, 5, 9, 7, 15, 11, 27, 25, 21, 13, 45, 17, 33, 35, 81, 19, 75, 23, 63, 55, 39, 29, 135, 49, 51, 125, 99, 31, 105, 37, 243, 65, 57, 77, 225, 41, 69, 85, 189, 43, 165, 47, 117, 175, 87, 53, 405, 121, 147, 95, 153, 59, 375, 91, 297, 115, 93, 61, 315, 67, 111, 275, 729, 119

Offset: 1

Marc LeBrun

Meyers (see Guy reference) conjectures that for all r >= 1, the least odd number not in the set {a(i): i < prime(r)} is prime(r+1). - N. J. A. Sloane, Jan 08 2021
Meyers' conjecture would be refuted if and only if for some r there were such a large gap between prime(r) and prime(r+1) that there existed a composite c for which prime(r) < c < a(c) < prime(r+1), in which case (by Bertrand's postulate) c would necessarily be a term of A246281. - Antti Karttunen, Mar 29 2021
a(n) is odd for all n and for each odd m there exists a k with a(k) = m (see A064216). a(n) > n for n > 1: bijection between the odd and all numbers. - Reinhard Zumkeller, Sep 26 2001
a(n) and n have the same number of distinct primes with (A001222) and without multiplicity (A001221). - Michel Marcus, Jun 13 2014
From Antti Karttunen, Nov 01 2019: (Start)
More generally, a(n) has the same prime signature as n, A046523(a(n)) = A046523(n). Also A246277(a(n)) = A246277(n) and A287170(a(n)) = A287170(n).
Many permutations and other sequences that employ prime factorization of n to encode either polynomials, partitions (via Heinz numbers) or multisets in general can be easily defined by using this sequence as one of their constituent functions. See the last line in the Crossrefs section for examples.
(End)

			a(12) = a(2^2 * 3) = a(prime(1)^2 * prime(2)) = prime(2)^2 * prime(3) = 3^2 * 5 = 45.
a(A002110(n)) = A002110(n + 1) / 2.

Richard K. Guy, editor, Problems From Western Number Theory Conferences, Labor Day, 1983, Problem 367 (Proposed by Leroy F. Meyers, The Ohio State U.).

Indranil Ghosh, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Index entries for sequences computed from indices in prime factorization.
Index entries for sequences related to Heinz numbers.

See A045965 for another version.
Row 1 of table A242378 (which gives the "k-th powers" of this sequence), row 3 of A297845 and of A306697. See also arrays A066117, A246278, A255483, A308503, A329050.
Cf. A064989 (a left inverse), A064216, A000040, A002110, A000265, A027746, A046523, A048673 (= (a(n)+1)/2), A108228 (= (a(n)-1)/2), A191002 (= a(n)*n), A252748 (= a(n)-2n), A286385 (= a(n)-sigma(n)), A283980 (= a(n)*A006519(n)), A341529 (= a(n)*sigma(n)), A326042, A049084, A001221, A001222, A122111, A225546, A260443, A245606, A244319, A246269 (= A065338(a(n))), A322361 (= gcd(n, a(n))), A305293.
Cf. A191555, A252738.
Cf. A249734, A249735 (bisections).
Cf. A246261 (a(n) is of the form 4k+1), A246263 (of the form 4k+3), A246271, A246272, A246259, A246281 (n such that a(n) < 2n), A246282 (n such that a(n) > 2n), A252742.
Cf. A275717 (a(n) > a(n-1)), A275718 (a(n) < a(n-1)).
Cf. A003972 (Möbius transform), A003973 (Inverse Möbius transform), A318321.
Cf. A300841, A305421, A322991, A250469, A269379 for analogous shift-operators in other factorization and quasi-factorization systems.
Cf. also following permutations and other sequences that can be defined with the help of this sequence: A005940, A163511, A122111, A260443, A206296, A265408, A265750, A275733, A275735, A297845, A091202 & A091203, A250245 & A250246, A302023 & A302024, A302025 & A302026.
A version for partition numbers is A003964, strict A357853.
A permutation of A005408.
Applying the same transformation again gives A357852.
Other multiplicative sequences: A064988, A357977, A357978, A357980, A357983.
A056239 adds up prime indices, row-sums of A112798.
Cf. A000720, A076610, A296150.

a003961 1 = 1
a003961 n = product $ map (a000040 . (+ 1) . a049084) $ a027746_row n
-- Reinhard Zumkeller, Apr 09 2012, Oct 09 2011
(MIT/GNU Scheme, with Aubrey Jaffer's SLIB Scheme library)
(require 'factor)
(define (A003961 n) (apply * (map A000040 (map 1+ (map A049084 (factor n))))))
;; Antti Karttunen, May 20 2014

a:= n-> mul(nextprime(i[1])^i[2], i=ifactors(n)[2]):
seq(a(n), n=1..80);  # Alois P. Heinz, Sep 13 2017

a[p_?PrimeQ] := a[p] = Prime[ PrimePi[p] + 1]; a[1] = 1; a[n_] := a[n] = Times @@ (a[#1]^#2& @@@ FactorInteger[n]); Table[a[n], {n, 1, 65}] (* Jean-François Alcover, Dec 01 2011, updated Sep 20 2019 *)
Table[Times @@ Map[#1^#2 & @@ # &, FactorInteger[n] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}] - Boole[n == 1], {n, 65}] (* Michael De Vlieger, Mar 24 2017 *)

a(n)=local(f); if(n<1,0,f=factor(n); prod(k=1,matsize(f)[1],nextprime(1+f[k,1])^f[k,2]))

a(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ Michel Marcus, May 17 2014

use ntheory ":all";  sub a003961 { vecprod(map { next_prime($) } factor(shift)); }  # _Dana Jacobsen, Mar 06 2016

from sympy import factorint, prime, primepi, prod
def a(n):
    f=factorint(n)
    return 1 if n==1 else prod(prime(primepi(i) + 1)**f[i] for i in f)
[a(n) for n in range(1, 11)] # Indranil Ghosh, May 13 2017

If n = Product p(k)^e(k) then a(n) = Product p(k+1)^e(k).
Multiplicative with a(p^e) = A000040(A000720(p)+1)^e. - David W. Wilson, Aug 01 2001
a(n) = Product_{k=1..A001221(n)} A000040(A049084(A027748(n,k))+1)^A124010(n,k). - Reinhard Zumkeller, Oct 09 2011 [Corrected by Peter Munn, Nov 11 2019]
A064989(a(n)) = n and a(A064989(n)) = A000265(n). - Antti Karttunen, May 20 2014 & Nov 01 2019
A001221(a(n)) = A001221(n) and A001222(a(n)) = A001222(n). - Michel Marcus, Jun 13 2014
From Peter Munn, Oct 31 2019: (Start)
a(n) = A225546((A225546(n))^2).
a(A225546(n)) = A225546(n^2).
(End)
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} ((p^2-p)/(p^2-nextprime(p))) = 2.06399637... . - Amiram Eldar, Nov 18 2022

A299200 Number of twice-partitions whose domain is the integer partition with Heinz number n.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 5, 1, 4, 3, 7, 2, 11, 5, 6, 1, 15, 4, 22, 3, 10, 7, 30, 2, 9, 11, 8, 5, 42, 6, 56, 1, 14, 15, 15, 4, 77, 22, 22, 3, 101, 10, 135, 7, 12, 30, 176, 2, 25, 9, 30, 11, 231, 8, 21, 5, 44, 42, 297, 6, 385, 56, 20, 1, 33, 14, 490, 15, 60, 15, 627, 4

Offset: 1

Gus Wiseman, Feb 05 2018

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The a(15) = 6 twice-partitions: (3)(2), (3)(11), (21)(2), (21)(11), (111)(2), (111)(11).

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A000041, A063834, A112798, A196545, A273873, A281145, A289501, A290261, A296150, A299201, A299202, A299203.

Cf. A000720, A003964.

with(numtheory): with(combinat):
a:= n-> mul(numbpart(pi(i[1]))^i[2], i=ifactors(n)[2]):
seq(a(n), n=1..82);  # Alois P. Heinz, Jan 14 2021

Table[Times@@Cases[FactorInteger[n],{p_,k_}:>PartitionsP[PrimePi[p]]^k],{n,100}]

a(n) = {my(f = factor(n)); for (k=1, #f~, f[k, 1] = numbpart(primepi(f[k, 1]));); factorback(f);} \\ Michel Marcus, Feb 26 2018

Multiplicative with a(prime(n)) = A000041(n).

A357982 Replace prime(k) with A000009(k) in the prime factorization of n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 3, 1, 4, 2, 2, 1, 5, 1, 6, 2, 2, 3, 8, 1, 4, 4, 1, 2, 10, 2, 12, 1, 3, 5, 4, 1, 15, 6, 4, 2, 18, 2, 22, 3, 2, 8, 27, 1, 4, 4, 5, 4, 32, 1, 6, 2, 6, 10, 38, 2, 46, 12, 2, 1, 8, 3, 54, 5, 8, 4, 64, 1, 76, 15, 4, 6, 6, 4, 89, 2, 1

Offset: 1

Gus Wiseman, Oct 25 2022

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. This sequence gives the number of ways to choose a strict partition of each prime index of n.

The indices i, where a(i) = 1, form A003586, and the indices j, where a(j) > 1, form A059485. - Ivan N. Ianakiev, Oct 27 2022

			The a(121) = 9 twice-partitions are: (5)(5), (5)(41), (5)(32), (41)(5), (41)(41), (41)(32), (32)(5), (32)(41), (32)(32).

Other multiplicative sequences: A003961, A357852, A064988, A064989, A357980.
The non-strict version is A299200.
A horizontal version is A357978, non-strict A357977.
A000040 lists the primes.
A056239 adds up prime indices, row-sums of A112798.
Cf. A000041, A000720, A003964, A063834, A076610, A215366, A273873, A296150, A299201-A299203, A357975, A357979, A357983.
Cf. A003586, A059485.

Table[Times@@Cases[FactorInteger[n],{p_,k_}:>PartitionsQ[PrimePi[p]]^k],{n,100}]

f9(n) = polcoeff( prod( k=1, n, 1 + x^k, 1 + x * O(x^n)), n); \\ A000009
a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1] = f9(primepi(f[k,1]))); factorback(f); \\ Michel Marcus, Oct 26 2022

A357977 Replace prime(k) with prime(A000041(k)) in the prime factorization of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 11, 8, 9, 10, 17, 12, 31, 22, 15, 16, 47, 18, 79, 20, 33, 34, 113, 24, 25, 62, 27, 44, 181, 30, 263, 32, 51, 94, 55, 36, 389, 158, 93, 40, 547, 66, 761, 68, 45, 226, 1049, 48, 121, 50, 141, 124, 1453, 54, 85, 88, 237, 362, 1951, 60, 2659, 526

Offset: 1

Gus Wiseman, Oct 23 2022

In the definition, taking A000041(k) instead of prime(A000041(k)) gives A299200.

			We have 35 = prime(3) * prime(4), so a(35) = prime(A000041(3)) * prime(A000041(4)) = prime(3) * prime(5) = 55.

Applying the same transformation again gives A357979.
The strict version is A357978.
Other multiplicative sequences: A003961, A357852, A064988, A064989, A357980.
A000040 lists the primes.
A056239 adds up prime indices, row-sums of A112798.
Cf. A000041, A000720, A003964, A063834, A076610, A215366, A296150, A299201, A299202, A357975, A357983.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
mtf[f_][n_]:=Product[If[f[i]==0,1,Prime[f[i]]],{i,primeMS[n]}];
Array[mtf[PartitionsP],100]

a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1] = prime(numbpart(primepi(f[k,1])))); factorback(f); \\ Michel Marcus, Oct 25 2022

A357852 Replace prime(k) with prime(k+2) in the prime factorization of n.

Original entry on oeis.org

1, 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

Offset: 1

Gus Wiseman, Oct 28 2022

This is the same as A045966 except the first term is 1 instead of 3.

			The terms together with their prime indices begin:
    1: {}
    5: {3}
    7: {4}
   25: {3,3}
   11: {5}
   35: {3,4}
   13: {6}
  125: {3,3,3}
   49: {4,4}
   55: {3,5}
   17: {7}
  175: {3,3,4}
   19: {8}
   65: {3,6}
   77: {4,5}
  625: {3,3,3,3}

Applying the transformation only once gives A003961.
A permutation of A007310.
Other multiplicative sequences: A064988, A064989, A357977, A357980, A357983.
A000040 lists the primes.
A056239 adds up prime indices, row-sums of A112798.
Cf. A000720, A003964, A066207, A076610, A215366, A296150, A299201, A357979.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Product[Prime[i+2],{i,primeMS[n]}],{n,30}]

a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1] = nextprime(nextprime(f[k,1]+1)+1)); factorback(f); \\ Michel Marcus, Oct 28 2022

from math import prod
from sympy import nextprime, factorint
def A357852(n): return prod(nextprime(p,ith=2)**e for p, e in factorint(n).items()) # Chai Wah Wu, Oct 29 2022

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

A357978 Replace prime(k) with prime(A000009(k)) in the prime factorization of n.

Original entry on oeis.org

1, 2, 2, 4, 3, 4, 3, 8, 4, 6, 5, 8, 7, 6, 6, 16, 11, 8, 13, 12, 6, 10, 19, 16, 9, 14, 8, 12, 29, 12, 37, 32, 10, 22, 9, 16, 47, 26, 14, 24, 61, 12, 79, 20, 12, 38, 103, 32, 9, 18, 22, 28, 131, 16, 15, 24, 26, 58, 163, 24, 199, 74, 12, 64, 21, 20, 251, 44, 38

Offset: 1

Gus Wiseman, Oct 24 2022

In the definition, taking A000009(k) instead of prime(A000009(k)) gives A357982.

			We have 90 = prime(1) * prime(2)^2 * prime(3), so a(90) = prime(1) * prime(1)^2 * prime(2) = 24.

The non-strict version is A357977.
Other multiplicative sequences: A003961, A357852, A064988, A064989, A357980.
A000040 lists the primes.
A056239 adds up prime indices, row-sums of A112798.
Cf. A000041, A000720, A003964, A063834, A076610, A215366, A296150, A299201, A299202, A357975, A357979, A357983.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
mtf[f_][n_]:=Product[If[f[i]==0,1,Prime[f[i]]],{i,primeMS[n]}];
Array[mtf[PartitionsQ],100]

f9(n) = polcoeff( prod( k=1, n, 1 + x^k, 1 + x * O(x^n)), n); \\ A000009
a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1] = prime(f9(primepi(f[k,1])))); factorback(f); \\ Michel Marcus, Oct 25 2022

A357983 Second MTF-transform of the primes (A000040). Replace prime(k) with prime(A064988(k)) in the prime factorization of n.

Original entry on oeis.org

1, 2, 5, 4, 11, 10, 23, 8, 25, 22, 31, 20, 47, 46, 55, 16, 59, 50, 103, 44, 115, 62, 97, 40, 121, 94, 125, 92, 137, 110, 127, 32, 155, 118, 253, 100, 197, 206, 235, 88, 179, 230, 233, 124, 275, 194, 257, 80, 529, 242, 295, 188, 419, 250, 341, 184, 515, 274

Offset: 1

Gus Wiseman, Oct 24 2022

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. We define the MTF-transform as shifting a number's prime indices along a function; see the Mathematica program.

			First, we have
- 4 = prime(1) * prime(1),
- A000040(1) = 2,
- A064988(4) = prime(2) * prime(2) = 9.
Similarly, A064988(3) = 5. Next,
- 35 = prime(3) * prime(4),
- A064988(3) = 5,
- A064988(4) = 9,
- a(35) = prime(5) * prime(9) = 253.

Other multiplicative sequences: A003961, A357852, A064989, A357977, A357980.
Applying the transformation only once gives A064988.
The union is A076610 (numbers whose prime indices are themselves prime).
For partition numbers instead of primes we have A357979.
A000040 lists the primes.
A056239 adds up prime indices, row-sums of A112798.
Cf. A000720, A003964, A063834, A215366, A296150, A299201, A299202, A357975.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
mtf[f_][n_]:=Product[If[f[i]==0,1,Prime[f[i]]],{i,primeMS[n]}];
Array[mtf[mtf[Prime]],100]

A357979 Second MTF-transform of A000041. Replace prime(k) with prime(A357977(k)) in the prime factorization of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 31, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 59, 32, 33, 62, 35, 36, 37, 38, 39, 40, 127, 42, 79, 44, 45, 46, 47, 48, 49, 50, 93, 52, 53, 54, 55, 56, 57, 58, 211, 60, 61, 118, 63, 64, 65, 66

Offset: 1

Gus Wiseman, Oct 24 2022

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. We define the MTF-transform as applying a function horizontally along a number's prime indices; see the Mathematica program.

			We have:
- 51 = prime(2) * prime(7),
- A357977(2) = 2,
- A357977(7) = 11,
- a(51) = prime(2) * prime(11) = 93.

Other multiplicative sequences: A003961, A357852, A064988, A064989, A357980.
Applying the transformation only once gives A357977, strict A357978.
For primes instead of partition numbers we have A357983.
A000040 lists the primes.
A056239 adds up prime indices, row-sums of A112798.
Cf. A000041, A000720, A003964, A063834, A076610, A215366, A296150, A299201, A299202, A357975.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
mtf[f_][n_]:=Product[If[f[i]==0,1,Prime[f[i]]],{i,primeMS[n]}];
Array[mtf[mtf[PartitionsP]],100]

cp's OEIS Frontend

A003978 Möbius transform of A003964.

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A003979 Inverse Möbius transform of A003964.

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A003961 Completely multiplicative with a(prime(k)) = prime(k+1).

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A299200 Number of twice-partitions whose domain is the integer partition with Heinz number n.

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A357982 Replace prime(k) with A000009(k) in the prime factorization of n.

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A357977 Replace prime(k) with prime(A000041(k)) in the prime factorization of n.

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A357852 Replace prime(k) with prime(k+2) in the prime factorization of n.

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