A064553 -id:A064553 - cp's OEIS frontend

A003963 Fully multiplicative with a(p) = k if p is the k-th prime.

1, 1, 2, 1, 3, 2, 4, 1, 4, 3, 5, 2, 6, 4, 6, 1, 7, 4, 8, 3, 8, 5, 9, 2, 9, 6, 8, 4, 10, 6, 11, 1, 10, 7, 12, 4, 12, 8, 12, 3, 13, 8, 14, 5, 12, 9, 15, 2, 16, 9, 14, 6, 16, 8, 15, 4, 16, 10, 17, 6, 18, 11, 16, 1, 18, 10, 19, 7, 18, 12, 20, 4, 21, 12, 18, 8, 20, 12, 22, 3, 16, 13, 23, 8, 21, 14, 20, 5

Offset: 1

Marc LeBrun

a(n) is the Matula number of the rooted tree obtained from the rooted tree T having Matula number n, by contracting its edges that emanate from the root. Example: a(49) = 16. Indeed, the rooted tree with Matula number 49 is the tree obtained by merging two copies of the tree Y at their roots. Contracting the two edges that emanate from the root, we obtain the star tree with 4 edges having Matula number 16. - Emeric Deutsch, May 01 2015
The Matula (or Matula-Goebel) number of a rooted tree can be defined in the following recursive manner: to the one-vertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the t-th prime number, where t is the Matula-Goebel number of the tree obtained from T by deleting the edge emanating from the root; to a tree T with root degree m>=2 there corresponds the product of the Matula-Goebel numbers of the m branches of T. - Emeric Deutsch, May 01 2015
a(n) is the product of the parts of the partition having Heinz number n. We define the Heinz number of a partition p = [p_1, p_2, ..., p_r] as Product_{j=1..r} (p_j-th prime) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the partition [1, 1, 2, 4, 10] we get 2*2*3*7*29 = 2436. Example: a(75) = 18; indeed, the partition having Heinz number 75 = 3*5*5 is [2,3,3] and 2*3*3 = 18. - Emeric Deutsch, Jun 03 2015
Let T be the free-commutative-monoid monad on the category Set. Then for each set N we have a canonical function m from TTN to TN. If we let N = {1, 2, 3, ...} and enumerate the primes in the usual way (A000040) then unique prime factorization gives a canonical bijection f from N to TN. Then the sequence is given by a(n) = f^-1(m(T(f)(f(n)))). - Oscar Cunningham, Jul 18 2019

T. D. Noe, Table of n, a(n) for n = 1..10000
E. Deutsch, Rooted tree statistics from Matula numbers, Discrete Appl. Math., 160, 2012, 2314-2322.
F. Göbel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.
I. Gutman and Yeong-Nan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.
D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.
Index entries for sequences related to Matula-Göbel numbers
Index entries for sequences computed from indices in prime factorization

Cf. A000720, A001221, A001222, A027748, A049084, A056239, A064553, A124010, A156552, A215366, A227184, A241909, A243354, A243499, A243504.

Product of entries on row n of A112798.

a003963 n = product $
   zipWith (^) (map a049084 $ a027748_row n) (a124010_row n)
-- Reinhard Zumkeller, Jun 30 2012

with(numtheory): a := proc (n) local r, s: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: if n = 1 then 1 elif bigomega(n) = 1 then pi(n) else a(r(n))*a(s(n)) end if end proc: seq(a(n), n = 1 .. 88);
# Alternative:
seq(mul(numtheory:-pi(t[1])^t[2], t=ifactors(n)[2]), n=1..100); # Robert Israel, May 01 2015

a[n_] := Times @@ (PrimePi[ #[[1]] ]^#[[2]]& /@ FactorInteger[n]); a[1] = 1; Table[a[n], {n, 1, 88}]

a(n)=f=factor(n);prod(i=1,#f[,1],primepi(f[i,1])^f[i,2]) \\ Charles R Greathouse IV, Apr 26 2012; corrected by Rémy Sigrist, Jul 18 2019

a(n) = {f = factor(n); for (i=1, #f~, f[i, 1] = primepi(f[i, 1]); ); factorback(f); } \\ Michel Marcus, Feb 08 2015

A003963(n)={n=factor(n); n[,1]=apply(primepi,n[,1]); factorback(n)} \\ M. F. Hasler, May 03 2018

from math import prod
from sympy import primepi, factorint
def A003963(n): return prod(primepi(p)**e for p, e in factorint(n).items()) # Chai Wah Wu, Nov 17 2022

If n = product prime(k)^e(k) then a(n) = product k^e(k).
Multiplicative with a(p^e) = A000720(p)^e. - David W. Wilson, Aug 01 2001
a(n) = Product_{k=1..A001221(n)} A049084(A027748(n,k))^A124010(n,k). - Reinhard Zumkeller, Jun 30 2012
Rec. eq.: a(1)=1, a(k-th prime) = a(k), a(rs)=a(r)a(s). The Maple program is based on this. - Emeric Deutsch, May 01 2015
a(n) = A243504(A241909(n)) = A243499(A156552(n)) = A227184(A243354(n)) - Antti Karttunen, Mar 07 2017

A181819 Prime shadow of n: a(1) = 1; for n>1, if n = Product prime(i)^e(i), then a(n) = Product prime(e(i)).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 3, 4, 2, 6, 2, 4, 4, 7, 2, 6, 2, 6, 4, 4, 2, 10, 3, 4, 5, 6, 2, 8, 2, 11, 4, 4, 4, 9, 2, 4, 4, 10, 2, 8, 2, 6, 6, 4, 2, 14, 3, 6, 4, 6, 2, 10, 4, 10, 4, 4, 2, 12, 2, 4, 6, 13, 4, 8, 2, 6, 4, 8, 2, 15, 2, 4, 6, 6, 4, 8, 2, 14, 7, 4, 2, 12, 4, 4, 4, 10, 2, 12, 4, 6, 4, 4, 4, 22, 2, 6, 6, 9, 2, 8, 2, 10, 8

Offset: 1

Matthew Vandermast, Dec 07 2010

a(n) depends only on prime signature of n (cf. A025487). a(m) = a(n) iff m and n have the same prime signature, i.e., iff A046523(m) = A046523(n).

Because A046523 (the smallest representative of prime signature of n) and this sequence are functions of each other as A046523(n) = A181821(a(n)) and a(n) = a(A046523(n)), it implies that for all i, j: a(i) = a(j) <=> A046523(i) = A046523(j) <=> A101296(i) = A101296(j), i.e., that equivalence-class-wise this is equal to A101296, and furthermore, applying any function f on this sequence gives us a sequence b(n) = f(a(n)) whose equivalence class partitioning is equal to or coarser than that of A101296, i.e., b is then a sequence that depends only on the prime signature of n (the multiset of exponents of its prime factors), although not necessarily in a very intuitive way. - Antti Karttunen, Apr 28 2022

			20 = 2^2*5 has the exponents (2,1) in its prime factorization. Accordingly, a(20) = prime(2)*prime(1) = A000040(2)*A000040(1) = 3*2 = 6.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n

Column 1 of A353510 (see also A325239 and A325277).
Cf. A000040, A000265, A001511, A001222, A003963, A005361, A007814, A008578, A028234, A046523, A056239, A064553, A064989, A067029, A101296 (restricted growth sequence transform), A108951, A122111, A124010, A124859, A156552, A181820, A181821, A182850, A182855, A182857 (also A323014), A115621, A101296, A238690, A238745, A238747, A238748, A246029, A304465, A304647, A305732, A305733, A320118, A323022, A325501, A325502, A325507, A325508, A325755 (A353566), A325756, A328830 [= a(a(n))], A328835, A351564 (characteristic function of A130091), A351944, A351946, A353379.
Left inverse of A304660.

a181819 = product . map a000040 . a124010_row
-- Reinhard Zumkeller, Mar 26 2012

A181819 := proc(n)
    local a;
    a := 1;
    for pf in ifactors(n)[2] do
        a := a*ithprime(pf[2]) ;
    end do:
    a ;
end proc:
seq(A181819(n),n=1..80) ; # R. J. Mathar, Jan 09 2019
# second Maple program:
a:= n-> mul(ithprime(i[2]), i=ifactors(n)[2]):
seq(a(n), n=1..105);  # Alois P. Heinz, Nov 26 2024

{1}~Join~Table[Times @@ Prime@ Map[Last, FactorInteger@ n], {n, 2, 120}] (* Michael De Vlieger, Feb 07 2016 *)

a(n) = {my(f=factor(n)); prod(k=1, #f~, prime(f[k,2]));} \\ Michel Marcus, Nov 16 2015

;; With memoization-macro definec.
(definec (A181819 n) (cond ((= 1 n) 1) (else (* (A000040 (A067029 n)) (A181819 (A028234 n))))))
;; Antti Karttunen, Feb 05 2016

;; With memoization-macro definec.
(definec (A181819 n) (cond ((= 1 n) 1) ((even? n) (* (A000040 (A007814 n)) (A181819 (A000265 n)))) (else (A181819 (A064989 n)))))
;; Antti Karttunen, Feb 07 2016

From Antti Karttunen, Feb 07 2016: (Start)
a(1) = 1; for n > 1, a(n) = A000040(A067029(n)) * a(A028234(n)).
a(1) = 1; for n > 1, a(n) = A008578(A001511(n)) * a(A064989(n)).
Other identities. For all n >= 1:
a(A124859(n)) = A122111(a(n)) = A238745(n). - from Matthew Vandermast's formulas for the latter sequence.
(End)
a(n) = A246029(A156552(n)). - Antti Karttunen, Oct 15 2016
From Antti Karttunen, Apr 28 & Apr 30 2022: (Start)
A181821(a(n)) = A046523(n) and a(A046523(n)) = a(n). [See comments]
a(n) = A329900(A124859(n)) = A319626(A124859(n)).
a(n) = A246029(A156552(n)).
a(a(n)) = A328830(n).
a(A304660(n)) = n.
a(A108951(n)) = A122111(n).
a(A185633(n)) = A322312(n).
a(A025487(n)) = A181820(n).
a(A276076(n)) = A275735(n) and a(A276086(n)) = A328835(n).
As the sequence converts prime exponents to prime indices, it effects the following mappings:
A001221(a(n)) = A071625(n). [Number of distinct indices --> Number of distinct exponents]
A001222(a(n)) = A001221(n). [Number of indices (i.e., the number of prime factors with multiplicity) --> Number of exponents (i.e., the number of distinct prime factors)]
A056239(a(n)) = A001222(n). [Sum of indices --> Sum of exponents]
A066328(a(n)) = A136565(n). [Sum of distinct indices --> Sum of distinct exponents]
A003963(a(n)) = A005361(n). [Product of indices --> Product of exponents]
A290103(a(n)) = A072411(n). [LCM of indices --> LCM of exponents]
A156061(a(n)) = A290107(n). [Product of distinct indices --> Product of distinct exponents]
A257993(a(n)) = A134193(n). [Index of the least prime not dividing n --> The least number not among the exponents]
A055396(a(n)) = A051904(n). [Index of the least prime dividing n --> Minimal exponent]
A061395(a(n)) = A051903(n). [Index of the greatest prime dividing n --> Maximal exponent]
A008966(a(n)) = A351564(n). [All indices are distinct (i.e., n is squarefree) --> All exponents are distinct]
A007814(a(n)) = A056169(n). [Number of occurrences of index 1 (i.e., the 2-adic valuation of n) --> Number of occurrences of exponent 1]
A056169(a(n)) = A136567(n). [Number of unitary prime divisors --> Number of exponents occurring only once]
A064989(a(n)) = a(A003557(n)) = A295879(n). [Indices decremented after <--> Exponents decremented before]
Other mappings:
A007947(a(n)) = a(A328400(n)) = A329601(n).
A181821(A007947(a(n))) = A328400(n).
A064553(a(n)) = A000005(n) and A000005(a(n)) = A182860(n).
A051903(a(n)) = A351946(n).
A003557(a(n)) = A351944(n).
A258851(a(n)) = A353379(n).
A008480(a(n)) = A309004(n).
a(A325501(n)) = A325507(n) and a(A325502(n)) = A038754(n+1).
a(n!) = A325508(n).
(End)

Name "Prime shadow" (coined by Gus Wiseman in A325755) prefixed to the definition by Antti Karttunen, Apr 27 2022

A182860 Number of distinct prime signatures represented among the unitary divisors of n.

Original entry on oeis.org

1, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 4, 2, 3, 3, 2, 2, 4, 2, 4, 3, 3, 2, 4, 2, 3, 2, 4, 2, 4, 2, 2, 3, 3, 3, 3, 2, 3, 3, 4, 2, 4, 2, 4, 4, 3, 2, 4, 2, 4, 3, 4, 2, 4, 3, 4, 3, 3, 2, 6, 2, 3, 4, 2, 3, 4, 2, 4, 3, 4, 2, 4, 2, 3, 4, 4, 3, 4, 2, 4, 2, 3, 2, 6, 3, 3, 3, 4, 2, 6, 3, 4, 3, 3, 3, 4, 2, 4, 4, 3, 2, 4, 2, 4, 4

Offset: 1

Matthew Vandermast, Jan 14 2011

nonn

a(n) = number of members m of A025487 such that d(m^k) divides d(n^k) for all values of k. (Here d(n) represents the number of divisors of n, or A000005(n).)

a(n) depends only on prime signature of n (cf. A025487).

			60 has 8 unitary divisors (1, 3, 4, 5, 12, 15, 20 and 60). Primes 3 and 5 have the same prime signature, as do 12 (2^2*3) and 20 (2^2*5); each of the other four numbers listed is the only unitary divisor of 60 with its particular prime signature.  Since a total of 6 distinct prime signatures appear among the unitary divisors of 60, a(60) = 6.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Unitary Divisor
Index entries for sequences computed from exponents in factorization of n

Cf. A000005, A034444, A064553, A085082, A146289, A146290, A182861, A182862, A212180, A328830.

Table[Length@ Union@ Map[Sort[FactorInteger[#] /. {p_, e_} /; p > 0 :> If[p == 1, 0, e]] &, Select[Divisors@ n, CoprimeQ[#, n/#] &]], {n, 105}] (* Michael De Vlieger, Jul 19 2017 *)

A181819(n) = {my(f=factor(n)); prod(k=1, #f~, prime(f[k, 2])); }; \\ From A181819
A182860(n) = numdiv(A181819(n)); \\ Antti Karttunen, Jul 19 2017

a(n) = A000005(A181819(n)).
If the canonical factorization of n into prime powers is Product p^e(p), then the formula for d(n^k) is Product_p (ek + 1). (See also A146289, A146290.)
a(n) = A064553(A328830(n)). - Antti Karttunen, Apr 29 2022

A080444 Write n A001055(n) times.

Original entry on oeis.org

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

Offset: 1

Alford Arnold, Mar 21 2003

Consider A001055(24) = 7. The seven ways of factoring 24 can be encoded as 24,30,44,51,55,62 and 83 using A064553.

T(n,k) = A064553(A080688(n,k)) = n for k=1..A001055(n). - Reinhard Zumkeller, Oct 01 2012

			A001055(12) = 4 so a(18) through a(21) = 12,12,12,12

Reinhard Zumkeller, Rows n = 1..1000 of triangle, flattened

a080444 n k = a080444_tabf !! (n-1) !! (k-1)
a080444_row n = a080444_tabf !! (n-1)
a080444_tabf = zipWith replicate a001055_list [1..]
a080444_list = concat a080444_tabf
-- Reinhard Zumkeller, Oct 01 2012

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 06 2003

Keyword tabf added by Reinhard Zumkeller, Oct 01 2012

A128629 A triangular array generated by moving Pascal sequences to prime positions and embedding new sequences at the nonprime locations. (cf. A007318 and A000040).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 4, 9, 10, 5, 1, 1, 6, 10, 16, 15, 6, 1, 1, 5, 18, 20, 25, 21, 7, 1, 1, 8, 15, 40, 35, 36, 28, 8, 1, 1, 9, 27, 35, 75, 56, 49, 36, 9, 1

Offset: 1

Alford Arnold, Mar 29 2007

The array can be constructed by beginning with A007318 (Pascal's triangle) placing each diagonal on a prime row. The other rows are filled in by mapping the prime factorization of the row number to the known sequences on the prime rows and multiplying term by term.

			Row six begins 1 6 18 40 75 126 ... because rows two and three are
1 2 3 4 5 6 ...
1 3 6 10 15 21 ...
The array begins
1 1 1 1 1 1 1 1 1 A000012
1 2 3 4 5 6 7 8 9 A000027
1 3 6 10 15 21 28 36 45 A000217
1 4 9 16 25 36 49 64 81 A000290
1 4 10 20 35 56 84 120 165 A000292
1 6 18 40 75 126 196 288 405 A002411
1 5 15 35 70 126 210 330 495 A000332
1 8 27 64 125 216 343 512 729 A000578
1 9 36 100 225 441 784 1296 2025 A000537
1 8 30 80 175 336 588 960 1485 A002417
1 6 21 56 126 252 462 792 1287 A000389
1 12 54 160 375 756 1372 2304 3645 A019582
1 7 28 84 210 462 924 1716 3003 A000579
1 10 45 140 350 756 1470 2640 4455 A027800
1 12 60 200 525 1176 2352 4320 7425 A004302
1 16 81 256 625 1296 2401 4096 6561 A000583
1 8 36 120 330 792 1716 3432 6435 A000580
1 18 108 400 1125 2646 5488 10368 18225 A019584
1 9 45 165 495 1287 3003 6435 12870 A000581
1 16 90 320 875 2016 4116 7680 13365 A119771
1 15 90 350 1050 2646 5880 11880 22275 A001297
1 12 63 224 630 1512 3234 6336 11583 A027810
1 10 55 220 715 2002 5005 11440 24310 A000582
1 24 162 640 1875 4536 9604 18432 32805 A019583
1 16 100 400 1225 3136 7056 14400 27225 A001249
1 14 84 336 1050 2772 6468 13728 27027 A027818
1 27 216 1000 3375 9261 21952 46656 91125 A059827
1 20 135 560 1750 4536 10290 21120 40095 A085284

Cf. A000040 A007318.

Cf. A064553 (second diagonal), A080688 (second diagonal resorted).

A128629 := proc(n,m) if n = 1 then 1; elif isprime(n) then p := numtheory[pi](n) ; binomial(p+m-1,p) ; else a := 1 ; for p in ifactors(n)[2] do a := a* procname(op(1,p),m)^ op(2,p) ; od: fi; end: # R. J. Mathar, Sep 09 2009

A-number added to each row of the examples by R. J. Mathar, Sep 09 2009

A328879 If n = Product (p_j^k_j) then a(n) = Product (pi(p_j) + 1), where pi = A000720.

Original entry on oeis.org

1, 2, 3, 2, 4, 6, 5, 2, 3, 8, 6, 6, 7, 10, 12, 2, 8, 6, 9, 8, 15, 12, 10, 6, 4, 14, 3, 10, 11, 24, 12, 2, 18, 16, 20, 6, 13, 18, 21, 8, 14, 30, 15, 12, 12, 20, 16, 6, 5, 8, 24, 14, 17, 6, 24, 10, 27, 22, 18, 24, 19, 24, 15, 2, 28, 36, 20, 16, 30, 40, 21, 6, 22, 26, 12

Offset: 1

Ilya Gutkovskiy, Oct 29 2019

a(n) is the product of indices of distinct prime factors of n if 1 is considered as a prime (see A008578).

			a(36) = 6 because 36 = 2^2 * 3^2 = prime(1)^2 * prime(2)^2 and (1 + 1) * (2 + 1) = 6.

Index entries for sequences computed from indices in prime factorization

Cf. A000720, A007947, A008578, A036234, A048250, A064553, A156061.

a[n_] := Times @@ ((PrimePi[#[[1]]] + 1) & /@ FactorInteger[n]); Table[a[n], {n, 1, 75}]

a(n)={my(f=factor(n)[,1]); prod(i=1, #f, 1 + primepi(f[i]))} \\ Andrew Howroyd, Oct 29 2019

A050298 Triangle read by rows: T(n,k) = p(r), where r is the (n-k+1)-th number such that A001222(r+1) = k, and p(r) is the r-th prime.

Original entry on oeis.org

2, 3, 5, 7, 11, 17, 13, 19, 31, 47, 29, 23, 59, 83, 127, 37, 41, 67, 149, 211, 307, 53, 43, 101, 167, 353, 499, 709, 61, 71, 103, 241, 401, 823, 1153, 1613, 79, 73, 109, 257, 587, 937, 1873, 2647, 3659, 107, 89, 179, 277, 607, 1319, 2113, 4201, 5843, 8147

Offset: 1

Alford Arnold, Apr 09 2003

The first column is A055003 and the main diagonal is A051438. When viewed as a sequence, this is a permutation of the prime numbers.

			a(14) = T(5,4) = p(23) = 83 because A001222(23+1) = A001222(24) = 4 since 24 has four prime factors, and this is the (5-4+1) = 2nd number with A001222 = 4.
The table begins:
2
3  5
7  11 17
13 19 31 47
29 23 59 83  127
37 41 67 149 211 307
...

Nathaniel Johnston, Rows n=1..20, flattened

Cf. A000040, A001222, A064553.

with(numtheory): A050298ind := proc(n,k) option remember: local f,m: if(n=k)then return 2^n-1: fi: for m from procname(n-1,k)+1 do if(bigomega(m+1)=k)then return m: fi: od: end: for n from 1 to 6 do seq(ithprime(A050298ind(n,k)),k=1..n);od; # Nathaniel Johnston, May 07 2011

Better name and extended by Nathaniel Johnston, May 07 2011

cp's OEIS Frontend

A003963 Fully multiplicative with a(p) = k if p is the k-th prime.

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A181819 Prime shadow of n: a(1) = 1; for n>1, if n = Product prime(i)^e(i), then a(n) = Product prime(e(i)).

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A182860 Number of distinct prime signatures represented among the unitary divisors of n.

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A080444 Write n A001055(n) times.

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A128629 A triangular array generated by moving Pascal sequences to prime positions and embedding new sequences at the nonprime locations. (cf. A007318 and A000040).

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A328879 If n = Product (p_j^k_j) then a(n) = Product (pi(p_j) + 1), where pi = A000720.

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A050298 Triangle read by rows: T(n,k) = p(r), where r is the (n-k+1)-th number such that A001222(r+1) = k, and p(r) is the r-th prime.

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