A218013 -id:A218013 - cp's OEIS frontend

A218072 Product of the nonzero digits (in base 10) of n^2.

1, 4, 9, 6, 10, 18, 36, 24, 8, 1, 2, 16, 54, 54, 20, 60, 144, 24, 18, 4, 16, 128, 90, 210, 60, 252, 126, 224, 32, 9, 54, 8, 72, 30, 20, 108, 162, 64, 10, 6, 48, 168, 288, 162, 20, 12, 36, 24, 8, 10, 12, 56, 144, 108, 30, 54, 216, 216, 96, 18, 42, 384, 1458

Offset: 1

Nels Olson, Oct 19 2012

			a(32) = 8 because 32*32 = 1024 and 1*2*4 = 8.

Similar to A053667, which does not exclude zero digits from the product.
Related to A218013.
Cf. A000290, A051801.

Table[Times@@(IntegerDigits[n^2]/.(0->1)),{n,120}] (* Harvey P. Dale, Dec 12 2017 *)

a(n) = {digs = digits(n^2); prod(i=1, #digs, if (digs[i], digs[i], 1));} \\ Michel Marcus, Aug 12 2013

a(n) = vecprod(select(x->(x>1), digits(n^2))); \\ Michel Marcus, Mar 07 2022

a(n) = A051801(n^2). - Michel Marcus, Mar 07 2022

A218014 Location of the n-th prime in its Andrica ranking.

Original entry on oeis.org

27, 6, 13, 1, 31, 4, 54, 8, 3, 100, 5, 25, 155, 28, 9, 16, 243, 19, 49, 288, 21, 62, 24, 12, 75, 422, 81, 444, 84, 2, 112, 37, 580, 11, 634, 47, 53, 150, 57, 60, 788, 20, 840, 183, 872, 10, 14, 218, 1029, 228, 80, 1074, 26, 87, 92, 99, 1237, 103, 281, 1319, 29, 15, 314, 1498, 323

Offset: 1

Marek Wolf and Robert G. Wilson v, Oct 18 2012

nonn

For each consecutive prime pair p < q, d = sqrt(q) - sqrt(p) is unique. Place d in order from greatest to least and specify p.
Last appearance by prime index: 1, 5, 7, 10, 13, 17, 20, 26, 28, 33, 35, 41, 43, 45, 49, ..., .
Last appearance of a minimum prime by Andrica ranking: 2, 11, 17, 29, 41, 59, 71, 101, 107, 137, 149, 179, ..., .
As expected, this sequence is the lesser of the twin primes beginning with the second term, 11. See A001359.

			a(1)=27 since the first prime, 2, does not show up in the ranking until the 27th term. See A218013.
a(4)=1 since the fourth prime, 7, has the maximum A_n value, see A218012; i.e., sqrt(p_n)-sqrt(p_n+1) is at a maximum.

Marek Wolf, A Note on the Andrica Conjecture, arXiv:1010.3945 [math.NT], 2010.

Cf. A079296, A218012, A218015.

lst = {}; p = 2; q = 3; While[p < 1600000, If[ Sqrt[q] - Sqrt[p] > 1/20, AppendTo[lst, {p, Sqrt[q] - Sqrt[p]}]]; p = q; q = NextPrime[q]]; lsu = First@ Transpose@ Sort[lst, #1[[2]] > #2[[2]] &]; Table[ Position[lsu, p, 1, 1], {p, Prime@ Range@ 65}] // Flatten

A218030 Numbers k equal to half of the product of the nonzero (base-10) digits of k^2.

Original entry on oeis.org

2, 5, 54, 648, 2160, 337169025526136832000, 685506275314921762068267522458966662115416623590907309075726336000000, 46641846972427276691124922228108091690332947069125333309512419901440000000000

Offset: 1

Nels Olson, Oct 18 2012

The first 5 terms of the sequence were found by the author around 1980 using his Commodore PET computer. He found the subsequent terms in 1991 by means of an improved program. The author has always referred to these as the "Faithy numbers" after his mother, Faith, who posed the problem.

			For n=5, n^2 is 25; the product of the digits of 25 is 2*5 = 10, which is equal to 2*n.

Michael S. Branicky, Table of n, a(n) for n = 1..12 (all terms < 10^300)
Michael S. Branicky, Python program.
Giovanni Resta, C program for this and related sequences.

Special case of A218013 where the ratio of the digit-product to the original number is 2. Related to A218072.

mx = 2^255; L = {};
p2 = 1; While[p2 < mx, Print["--> 2^", Log[2, p2]];
p3 = p2; While [p3 < mx,
  p5 = p3; While[p5 < mx,
   n = p5; While[n < mx,
    If[2 n == Times @@ Select[IntegerDigits[n^2], # > 0 &],
     AppendTo[L, n]; Print[n]]; n *= 7]; p5 *= 5]; p3 *= 3];
p2 *= 2]; Sort[L] (* Giovanni Resta, Oct 19 2012 *)

is_A218030(n)={my(d=digits(n^2));n*=2;for(i=1,#d,d[i]||next;n%d[i]&return;n\=d[i]);n==1} \\ M. F. Hasler, Oct 19 2012

A218029 Numbers that are equal to the product of the nonzero digits (in base 10) of their square.

Original entry on oeis.org

1, 30240, 60480, 51597803520, 687009790646587541893939200, 2639216811747930700939756830720000000, 399953837788563897626396208106026886244605021117749733129069000654848000000000

Offset: 1

Nels Olson, Oct 18 2012

			For n=30240, n^2 = 914,457,600; the product of the nonzero digits of this square, 9*1*4*4*5*7*6 = 30240 = n.

Special case of A218013 where the ratio of the digit-product to the original number is 1. Related to A218072.

cp's OEIS Frontend

A218072 Product of the nonzero digits (in base 10) of n^2.

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A218014 Location of the n-th prime in its Andrica ranking.

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A218030 Numbers k equal to half of the product of the nonzero (base-10) digits of k^2.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A218029 Numbers that are equal to the product of the nonzero digits (in base 10) of their square.

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Author

Keywords

Examples

Crossrefs