A189398 -id:A189398 - cp's OEIS frontend

A061509 Write n in decimal, omit 0's, replace the k-th digit d[k] with the k-th prime, raised to d[k]-th power and multiply.

1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 2, 6, 18, 54, 162, 486, 1458, 4374, 13122, 39366, 4, 12, 36, 108, 324, 972, 2916, 8748, 26244, 78732, 8, 24, 72, 216, 648, 1944, 5832, 17496, 52488, 157464, 16, 48, 144, 432, 1296, 3888, 11664, 34992, 104976, 314928

Offset: 0

Amarnath Murthy, May 06 2001

Not the same as A189398: see formula.

			a(4) = 2^4 = 16, a(123) = (2^1)*(3^2)*(5^3) = 2250.
For n = 0, the list of nonzero digits is empty, and the empty product equals 1.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000 (a(0) = 1 inserted by _M. F. Hasler_, Oct 12 2018)

Cf. A061510, A000040.

a061509 n = product $ zipWith (^)
  a000040_list (map digitToInt $ filter (/= '0') $ show n)
-- Reinhard Zumkeller, May 03 2011

A061509[n_] := If[n == 0, 1, Times @@ (Prime[Range[Length[#]]]^#) & [DeleteCases[IntegerDigits[n], 0]]];
Array[A061509, 100, 0] (* Paolo Xausa, Nov 26 2024 *)

A061509(n)=prod(k=1,#n=select(t->t,digits(n)),prime(k)^n[k]) \\ M. F. Hasler, Aug 16 2014

a(n) = a(n*10^k). a((10^k-1)/9) = primorial(k) = A002110(k).
a(n) = A189398(n) for n <= 100; a(101)=2^1*3^1 = 6 <> A189398(101) = 2^1*3^0*5^1 = 10; a(A052382(n)) = A189398(A052382(n)); a(n) = A000079(A000030(n)) if n has only one nonzero digit; A001221(a(n)) = A055640(n); A001222(a(n)) = A007953(n). - Reinhard Zumkeller, May 03 2011
If n=d[1]d[2]...d[m] in decimal (0M. F. Hasler, Aug 16 2014
A007814(a(n)) = A000030(n). - M. F. Hasler, Aug 18 2014

Corrected and extended by Matthew Conroy, May 13 2001
Offset corrected by Reinhard Zumkeller, May 03 2011
Definition corrected by M. F. Hasler, Aug 16 2014
Extended to a(0) = 1 by M. F. Hasler, Oct 12 2018

A246468 Given a number of k digits x = d_(k)10^(k-1) + d_(k-1)10^(k-2) + … + d_(2)10 + d_(1), consider y = p_(1)^d_(1)p_(2)^d_(2)…p_(k)^d_(k), where p_(i) is the i-th prime. Sequence lists the numbers x such that y / x is an integer.

Original entry on oeis.org

1, 2, 4, 8, 12, 16, 24, 36, 48, 54, 81, 96, 128, 135, 144, 162, 225, 288, 375, 486, 576, 625, 648, 675, 768, 972, 1296, 1575, 1875, 2187, 2268, 2625, 2646, 2688, 3087, 3136, 3375, 3528, 3675, 3888, 3969, 4116, 4374, 4802, 5145, 5292, 5488, 5625, 6048, 6174, 7056

Offset: 1

Paolo P. Lava, Aug 27 2014

a(n) = x such that A054842(x)/x is an integer.

			x = 48 -> y = 2^8*3^4 = 20736 and 20736 / 48 = 432.
x = 972 -> y = 2^2*3^7*5^9 = 17085937500 and 17085937500 / 972 = 17578125.

Paolo P. Lava, Table of n, a(n) for n = 1..1000

Cf. A054842, A189398, A246469.

with(numtheory):P:=proc(q) local a,b,k,n;
for n from 1 to q do a:=n; b:=1;
for k from 1 to ilog10(n)+1 do b:=b*ithprime(k)^(a mod 10); a:=trunc(a/10);
od; if type(b/n,integer) then print(n); fi; od; end: P(10^9);

A246469 Given a number of k digits x = d_(k)10^(k-1) + d_(k-1)10^(k-2) + … + d_(2)10 + d_(1), consider y = p_(1)^d_(k)p_(2)^d_(k-1)…p_(k)^d_(1), where p_(i) is the i-th prime. Sequence lists the numbers x such that y / x is an integer.

Original entry on oeis.org

1, 2, 4, 8, 18, 27, 36, 48, 54, 64, 72, 96, 125, 135, 162, 225, 375, 432, 486, 625, 648, 675, 864, 972, 1225, 1250, 1323, 1350, 1575, 1701, 1715, 1875, 2250, 2646, 2835, 2916, 3375, 3528, 3645, 3675, 3750, 3969, 4116, 4375, 4536, 4725, 4860, 5145, 5488, 5832

Offset: 1

Paolo P. Lava, Aug 27 2014

a(n) = x such that A189398(x) / x is an integer.

			x = 48 -> y = 2^4*3^8 = 104976 and 104976 / 48 = 2187.
x = 972 -> y = 2^9*3^7*5^2 = 27993600 and 27993600 / 972 = 28800.

Paolo P. Lava, Table of n, a(n) for n = 1..1000

Cf. A054842, A189398, A246468.

with(numtheory):P:=proc(q) local a,b,k,n;
for n from 1 to q do a:=n; b:=1;
for k from 1 to ilog10(n)+1 do b:=b*ithprime(ilog10(n)+2-k)^(a mod 10); a:=trunc(a/10);
od; if type(b/n,integer) then print(n); fi; od; end: P(10^9);

A246532 Smallest Meertens number in base n, or -1 if none exists.

Original entry on oeis.org

2, 10, 200, 6, 54, 100, 216, 4199040, 81312000

Offset: 2

David Applegate, Aug 28 2014

A Meertens number in base n is a fixed point of the base n Godel encoding.
The base n Godel encoding of x is 2^d(1) * 3^d(2) * ... * prime(k)^d(k), where d(1)d(2)...d(k) is the base n representation of x.
a(14) = 47250, a(16) = 18.
In a computer search that included all numbers < 10^29 and bases <= 16, the only additional Meertens numbers found were 6 (base 2), 10 (base 2), 49000 (base 5), and 181400 (base 5).
There is no base 11 Meertens number < 11^44 ~= 6.6*10^45.
There is no base 12 Meertens number < 12^40 ~= 1.4*10^43.
There is no base 13 Meertens number < 13^39 ~= 2.7*10^43.
There is no base 15 Meertens number < 15^37 ~= 3.2*10^43.
Other terms: a(17) = 36, a(19) = 96, a(32) = 256, a(51) = 54. - Chai Wah Wu, Aug 28 2014
From Chai Wah Wu, Jul 20 2020: (Start)
All terms are even.
If n > 2 and a(n) != -1, then a(n) > n.
a(2*3^m-m) = 2*3^m for all m >= 0, i.e., a(n) > 0 for an infinite number of values of n.
Other terms: a(64) = a(4096) = 65536, a(71) = 216, a(160) = 324, a(323) = 1296, a(1455) = 2916, a(1942) = 5832, a(7775) = 46656, a(8294) = 82944, a(13118) = 26244.
(End)
Named by Bird (1998) after the Dutch computer scientist Lambert Meertens (b. 1944). - Amiram Eldar, Jun 23 2021
a(10) = 81312000 is the only base-10 Meertens number below 10^50. - Max Alekseyev, Jul 24 2024

			100 is a base 7 Meertens number because 100 = 202_7 = 2^2 * 3^0 * 5^2.
4199040 is a base 9 Meertens number because 4199040 = 7810000_9 = 2^7 * 3^8 * 5^1.

David Applegate, C++ program used to search for Meertens numbers
Richard S. Bird, Functional Pearl: Meertens number, Journal of Functional Programming, Vol. 8, No. 1 (Jan 1998), pp. 83-88.
Wikipedia, Meertens number.
Chai Wah Wu, Meertens Number and Its Variations, IBM Research Report RC25531 (WAT1504-032), April 2015.
Chai Wah Wu, Meertens Number and Its Variations, arXiv:1603.08493 [math.NT], 2016.

Cf. A189398 (base 10 Godel encoding), A110765 (base 2 Godel encoding).

Conjectural terms removed from the sequence by Max Alekseyev, Jul 22 2024

A270142 a(n) = product of first k composites, with the i-th composite raised to the d-th power, where k = A055642(n) and d is the i-th digit of n.

Original entry on oeis.org

4, 16, 64, 256, 1024, 4096, 16384, 65536, 262144, 4, 24, 144, 864, 5184, 31104, 186624, 1119744, 6718464, 40310784, 16, 96, 576, 3456, 20736, 124416, 746496, 4478976, 26873856, 161243136, 64, 384, 2304, 13824, 82944, 497664, 2985984, 17915904, 107495424

Offset: 1

Felix Fröhlich, Mar 12 2016

All terms are multiples of 4, since A002808(1) = 4 and the most significant digit of n is always nonzero.
Does a term exist such that a(n) = n? Such a number would be the analog of a Meertens number when raising composites to the powers of the digits of n instead of raising primes to the powers of the digits.
From Chai Wah Wu, Dec 15 2022: (Start)
If a(n) is defined using digits of n in base b, then there are bases b and numbers n such that a(n) = n. For instance:
base b             n
------------------------------------------------
2                  4, 24, 36, 24192000, 85155840
3                  2592
4                  4, 103680
6                  20736
8                  16, 256, 13824
12                 1327104
16                 21233664
23                 24
24                 746496
(End)

			a(12) = 144, since A002808(1) = 4, A002808(2) = 6 and 4^1 * 6^2 = 144.

Felix Fröhlich, Table of n, a(n) for n = 1..10000

Cf. A189398, A246532.

composite(n) = my(i=0, c=2); while(1, if(!ispseudoprime(c), i++); if(i==n, return(c)); c++)
compopowerprod(n) = my(d=digits(n)); for(k=1, #d, p=prod(i=1, #d, composite(i)^d[i])); p
a(n) = compopowerprod(n)

from math import prod
from sympy import composite
def A270142(n): return prod(composite(i)**int(d) for i, d in enumerate(str(n),1)) # Chai Wah Wu, Dec 09 2022

cp's OEIS Frontend

A061509 Write n in decimal, omit 0's, replace the k-th digit d[k] with the k-th prime, raised to d[k]-th power and multiply.

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A246468 Given a number of k digits x = d_(k)*10^(k-1) + d_(k-1)*10^(k-2) + … + d_(2)*10 + d_(1), consider y = p_(1)^d_(1)*p_(2)^d_(2)*…*p_(k)^d_(k), where p_(i) is the i-th prime. Sequence lists the numbers x such that y / x is an integer.

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A246469 Given a number of k digits x = d_(k)*10^(k-1) + d_(k-1)*10^(k-2) + … + d_(2)*10 + d_(1), consider y = p_(1)^d_(k)*p_(2)^d_(k-1)*…*p_(k)^d_(1), where p_(i) is the i-th prime. Sequence lists the numbers x such that y / x is an integer.

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A246532 Smallest Meertens number in base n, or -1 if none exists.

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A270142 a(n) = product of first k composites, with the i-th composite raised to the d-th power, where k = A055642(n) and d is the i-th digit of n.

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A246468 Given a number of k digits x = d_(k)10^(k-1) + d_(k-1)10^(k-2) + … + d_(2)10 + d_(1), consider y = p_(1)^d_(1)p_(2)^d_(2)…p_(k)^d_(k), where p_(i) is the i-th prime. Sequence lists the numbers x such that y / x is an integer.

A246469 Given a number of k digits x = d_(k)10^(k-1) + d_(k-1)10^(k-2) + … + d_(2)10 + d_(1), consider y = p_(1)^d_(k)p_(2)^d_(k-1)…p_(k)^d_(1), where p_(i) is the i-th prime. Sequence lists the numbers x such that y / x is an integer.