A047201 -id:A047201 - cp's OEIS frontend

A064001 Odd abundant numbers not divisible by 5.

81081, 153153, 171171, 189189, 207207, 223839, 243243, 261261, 279279, 297297, 351351, 459459, 513513, 567567, 621621, 671517, 729729, 742203, 783783, 793611, 812889, 837837, 891891, 908523, 960687, 999999, 1024947, 1054053, 1072071

Offset: 1

Harvey P. Dale, Sep 17 2001

nonn

Or, odd abundant numbers that do not end in 5.
All terms below 2000000 are divisible by 21 (so by 3). Moreover, except for a few, most are divisible by 231. - Labos Elemer, Sep 15 2005 [The least term that is not divisible by 21 is a(908) = 28683369. - Amiram Eldar, Jan 27 2025]
An odd abundant number (see A005231) not divisible by 3 nor 5 must have at least 15 distinct prime factors (e.g., 61#/5#*7^2*11*13*17, where # is primorial) and be >= 67#/5#*77 = A047802(3) ~ 2.0*10^25. -- The smallest non-primitive abundant number (cf. A006038) in this sequence is 7*a(1) = 567567 = a(14). - M. F. Hasler, Jul 27 2016
There are 26 terms less than 10^6 and a surprising fact is that 18 of them are doublets (cf. A020338). - Omar E. Pol, Jan 17 2025
The numbers of terms that do not exceed 10^k, for k = 5, 6, ..., are 1, 26, 290, 3071, 31600, 320948, 3174762, 31693948, ... . Apparently, the asymptotic density of this sequence equals 0.000031... . Therefore, the least term not divisible by 3 that was mentioned above is a(~6*10^20) = 20169691981106018776756331. - Amiram Eldar, Jan 27 2025

David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Rev. ed. 1997, p. 169.

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harry J. Smith)
Carlos Rivera, Puzzle 329. Odd abundant numbers not divided by 2 or 3, The Prime Puzzles and Problems Connection.
Jay L. Schiffman, Odd Abundant Numbers, Mathematical Spectrum, Volume 37, Number 2 (January 2005), pp 73-75.

Intersection of A005231 and A047201.
Cf. A006038, A047802, A110585.
Cf. A020338.

Select[ Range[ 1, 10^6, 2 ], DivisorSigma[ 1, # ] - 2# > 0 && Mod[ #, 5 ] != 0 & ]
ta={{0}};Do[g=n;s=DivisorSigma[1, n]-2*n; If[Greater[s, 0]&&!Equal[Mod[n, 2], 0]&& !Equal[Mod[n, 5], 0], Print[n];ta=Append[ta, n]], {n, 1, 2000000}] ta=Delete[ta, 1] (* Labos Elemer, Sep 15 2005 *)

{ n=0; forstep (m=1, 10^9, 2, if (m%5 && sigma(m) > 2*m, write("b064001.txt", n++, " ", m); if (n==1000, break)) ) } \\ Harry J. Smith, Sep 05 2009

More terms from Robert G. Wilson v, Sep 28 2001
Further terms from Labos Elemer, Sep 15 2005
Entry revised by N. J. A. Sloane, Mar 28 2006

A056906 Numbers k such that 36*k^2 + 5 is prime.

Original entry on oeis.org

0, 1, 2, 6, 8, 12, 13, 16, 19, 21, 27, 28, 33, 34, 41, 43, 49, 56, 57, 62, 69, 72, 76, 77, 82, 84, 86, 89, 92, 96, 98, 99, 104, 111, 119, 121, 126, 128, 131, 132, 133, 134, 139, 142, 146, 148, 153, 159, 166, 168, 169, 173, 174

Offset: 1

Henry Bottomley, Jul 07 2000

Except for a(1), a(n) is never a multiple of 5.

			a(3)=2 since 36*2^2 + 5 = 149, which is prime.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

This sequence and formula generate all primes of the form k^2+5, i.e., A056905.
Except for the first term, this sequence is a subsequence of A047201.
Cf. A056900, A056902.

[n: n in [0..200]| IsPrime(36*n^2+5)]; // Vincenzo Librandi, Jul 14 2012

Select[Range[0,200],PrimeQ[36#^2+5]&] (* Harvey P. Dale, Jul 25 2011 *)

is(n)=isprime(36*n^2+5) \\ Charles R Greathouse IV, Feb 17 2017

a(n) = sqrt(A056905(n)-5)/6.

Offset corrected by Arkadiusz Wesolowski, Aug 09 2011

A225496 Numbers having no balanced prime factors, cf. A006562.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 21, 22, 23, 24, 26, 27, 28, 29, 31, 32, 33, 34, 36, 37, 38, 39, 41, 42, 43, 44, 46, 47, 48, 49, 51, 52, 54, 56, 57, 58, 59, 61, 62, 63, 64, 66, 67, 68, 69, 71, 72, 73, 74, 76, 77, 78, 79, 81, 82, 83, 84

Offset: 1

Reinhard Zumkeller, May 09 2013

nonn

a(n) = A047201(n) for n <= 42.

			a(40) = 49 = 7^2 = A178943(3)^2;
a(41) = 51 = 3 * 17 = A178943(2) * A178943(6);
a(42) = 52 = 2^2 * 13 = A178943(1)^2 * A178943(5);
a(43) = 54 = 2 * 3^3 = A178943(1) * A178943(2)^3;
a(44) = 56 = 2^3 * 7 = A178943(1)^3 * A178943(3);
a(45) = 57 = 3 * 19 = A178943(2) * A178943(7).

Reinhard Zumkeller, Table of n, a(n) for n = 1..5000

Cf. A225493 (strong), A225494 (balanced), A225495 (weak).

import Data.Set (singleton, fromList, union, deleteFindMin)
a225496 n = a225496_list !! (n-1)
a225496_list = 1 : h (singleton p) ps [p] where
   (p:ps) = a178943_list
   h s xs'@(x:xs) ys
     | m > x     = h (s `union` (fromList $ map (* x) (1 : ys))) xs ys
     | otherwise = m : h (s' `union` (fromList $ map (* m) ys')) xs' ys'
     where ys' = m : ys; (m, s') = deleteFindMin s

Multiplicative closure of A178943; a(n) mod A006562(k) > 0 for all k.

A356858 a(n) is the product of the first n numbers not divisible by 5.

Original entry on oeis.org

1, 1, 2, 6, 24, 144, 1008, 8064, 72576, 798336, 9580032, 124540416, 1743565824, 27897053184, 474249904128, 8536498274304, 162193467211776, 3406062811447296, 74933381851840512, 1723467782592331776, 41363226782215962624, 1075443896337615028224, 29036985201115605762048

Offset: 0

Stefano Spezia, Sep 01 2022

nonn

Unlike the factorial number n!, a(n) does not have trailing zeros.

Harvey P. Dale, Table of n, a(n) for n = 0..434

Cf. A000142, A000351, A002266, A047201.

Cf. A356859 (number of zero digits), A356860 (number of digits), A356861 (number of nonzero digits).

Table[Product[Floor[(5k-1)/4], {k,n}], {n,0,22}] (* or *)
Join[{1}, Table[Floor[(5n-1)/4]!/(Floor[Floor[(5n-1)/4]/5]!*5^Floor[Floor[(5n-1)/4]/5]), {n,22}]]
Join[{1},FoldList[Times,Table[If[Mod[n,5]==0,Nothing,n],{n,30}]]] (* Harvey P. Dale, Nov 03 2024 *)

from math import prod
def a(n): return prod((5*k-1)//4 for k in range(1, n+1))
print([a(n) for n in range(23)]) # Michael S. Branicky, Sep 01 2022

a(n) = Product_{k=1..n} A047201(k).

a(n) = A047201(n)!/(floor(A047201(n)/5)!*5^floor(A047201(n)/5)) for n > 0.

A217562 Even numbers not divisible by 5.

Original entry on oeis.org

2, 4, 6, 8, 12, 14, 16, 18, 22, 24, 26, 28, 32, 34, 36, 38, 42, 44, 46, 48, 52, 54, 56, 58, 62, 64, 66, 68, 72, 74, 76, 78, 82, 84, 86, 88, 92, 94, 96, 98, 102, 104, 106, 108, 112, 114, 116, 118, 122, 124, 126, 128, 132

Offset: 1

Jeremy Gardiner, Oct 06 2012

Numbers ending with 2,4,6,8 in base 10.
No term is divisible by 10 therefore a subsequence of A067251 (Numbers with no trailing zeros in decimal representation).
Union of this sequence with A005408 (The odd numbers) gives A067251.
Union of this sequence with A045572 (Numbers that are odd but not divisible by 5) gives A047201.
The even numbers divisible by 5 are A008592 (Multiples of 10).

Jeremy Gardiner, Table of n, a(n) for n = 1..4000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A005408, A005843, A045572, A047201, A067251.

for n=1 to 199
if n mod 5 <> 0 and n mod 2 <> 1 then print str$(n)+", ";
next n
print

I:=[2, 4, 6, 8, 12]; [n le 5 select I[n] else Self(n-1) + Self(n-4) - Self(n-5): n in [1..60]]; // Vincenzo Librandi, Dec 28 2012

CoefficientList[Series[2*(1 + x + x^2 + x^3 + x^4)/((1 + x)*(1 + x^2)*(x - 1)^2), {x, 0, 100}], x] (* Vincenzo Librandi, Dec 28 2012 *)

A217562(n)=(n-1)*5\2+2 \\ M. F. Hasler, Oct 07 2012

def A217562(n): return (5*n-1>>1)&-2 # Chai Wah Wu, Apr 21 2025

a(n) = 2*A047201(n).
G.f.: 2*x*(1+x+x^2+x^3+x^4) / ( (1+x)*(1+x^2)*(x-1)^2 ). - R. J. Mathar, Oct 06 2012
a(n) = 2*(n+floor((n-1)/4)). - Aaron J Grech, Sep 28 2024
E.g.f.: (4 - cos(x) + (5*x - 3)*cosh(x) + sin(x) + (5*x - 2)*sinh(x))/2. - Stefano Spezia, Sep 28 2024

A260181 Numbers whose last digit is prime.

Original entry on oeis.org

2, 3, 5, 7, 12, 13, 15, 17, 22, 23, 25, 27, 32, 33, 35, 37, 42, 43, 45, 47, 52, 53, 55, 57, 62, 63, 65, 67, 72, 73, 75, 77, 82, 83, 85, 87, 92, 93, 95, 97, 102, 103, 105, 107, 112, 113, 115, 117, 122, 123, 125, 127, 132, 133, 135, 137, 142, 143, 145, 147

Offset: 1

Wesley Ivan Hurt, Jul 17 2015

Numbers ending in 2, 3, 5 or 7.
The subsequence of primes is A042993. - Michel Marcus, Jul 19 2015
From Wesley Ivan Hurt, Aug 15 2015, Sep 26 2015: (Start)
Ceiling(a(n)/2) = A047201(n).
Complement of (A197652 Union A262389). (End)

Muniru A Asiru, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A042993, A047201, A092620, subset of A118950.
Union of A017293, A017305, A017329 and A017353.
First differences are [1,2,2,5,...] = A002522(A140081(n-1)).
Cf. A197652, A262389.

a:=n->(5*n-4-(-1)^n+((3-(-1)^n)/2)*(-1)^((2*n+5-(-1)^n)/4))/2; List([1..60],n->a(n)); # Muniru A Asiru, Feb 16 2018

[(5*n-4-(-1)^n+((3-(-1)^n) div 2)*(-1)^((2*n+5-(-1)^n) div 4))/2: n in [1..70]]; // Vincenzo Librandi, Jul 18 2015

A260181:=n->(5*n-4-(-1)^n+((3-(-1)^n)/2)*(-1)^((2*n+5-(-1)^n)/4))/2: seq(A260181(n), n=1..100);

CoefficientList[Series[(2 + x + 2 x^2 + 2 x^3 + 3 x^4)/((x - 1)^2*(1 + x + x^2 + x^3)), {x, 0, 100}], x]
LinearRecurrence[{1, 0, 0, 1, -1}, {2, 3, 5, 7, 12}, 60] (* Vincenzo Librandi, Jul 18 2015 *)
Table[(5n - 4 - (-1)^n + ((3 - (-1)^n)/2)*(-1)^((2*n + 5 - (-1)^n)/4))/2, {n, 100}] (* Wesley Ivan Hurt, Aug 11 2015 *)

is(n)=my(m=digits(n));isprime(m[#m]) \\ Anders Hellström, Jul 19 2015

A260181(n)=(n--)\4*10+prime(n%4+1) \\ is(n)=isprime(n%10) is much more efficient than the above. - M. F. Hasler, Sep 16 2016

G.f.: x*(2+x+2*x^2+2*x^3+3*x^4) / ((x-1)^2*(1+x+x^2+x^3)).
a(n) = a(n-1)+a(n-4)-a(n-5), n>5.
a(n) = (5*n-4-(-1)^n+((3-(-1)^n)/2)*(-1)^((2*n+5-(-1)^n)/4))/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = (2*sqrt(5*sqrt(5+2*sqrt(5))) - 25*log(5) - 40*log(2) + 5*sqrt(5)*arccoth(843/2))/200. - Amiram Eldar, Jul 30 2024

A356860 a(n) is the number of digits in the product of the first n numbers not divisible by 5.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 4, 4, 5, 6, 7, 9, 10, 11, 12, 13, 15, 16, 17, 19, 20, 22, 23, 24, 26, 27, 29, 30, 32, 33, 35, 37, 38, 40, 41, 43, 45, 46, 48, 50, 51, 53, 55, 57, 58, 60, 62, 64, 65, 67, 69, 71, 73, 74, 76, 78, 80, 82, 84, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103

Offset: 0

Stefano Spezia, Sep 01 2022

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A047201, A055642, A356858.

Cf. A356859 (number of zero digits), A356861 (number of nonzero digits).

Table[Length[IntegerDigits[Product[Floor[(5i-1)/4], {i,n}]]], {n,0,68}]
Join[{1},IntegerLength/@FoldList[Times,Table[If[Mod[n,5]==0,Nothing,n],{n,0,100}]]] (* Harvey P. Dale, Jul 20 2025 *)

from math import prod
def a(n): return len(str(prod((5*k-1)//4 for k in range(1, n+1))))
print([a(n) for n in range(69)]) # Michael S. Branicky, Sep 01 2022

a(n) = A055642(A356858(n)).

A174140 Numbers congruent to k mod 25, where 10 <= k <= 24.

Original entry on oeis.org

10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 110, 111, 112, 113, 114, 115, 116

Offset: 1

Rick L. Shepherd, Mar 09 2010

Numbers whose partition into parts of sizes 1, 5, 10, and 25 having a minimal number of parts includes at least one part of size 10.
For each number the partition is unique.
Complement of A174141.
Amounts in cents requiring at least one dime when the minimal number of coins is selected from pennies, nickels, dimes, and quarters (whether usage of bills for whole-dollar amounts is permitted or not).

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for sequences related to making change.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1).

Cf. A174138, A174139, A174141, A047201 (requires at least one part of size 1 (penny)), A008587, A053344 (minimal number of parts), A001299 (number of all such partitions).

Flatten[Table[Range[10,24]+25n,{n,0,5}]] (* Harvey P. Dale, Jun 12 2012 *)

Vec(x*(10 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10 + x^11 + x^12 + x^13 + x^14 + x^15) / ((1 - x)^2*(1 + x + x^2)*(1 + x + x^2 + x^3 + x^4)*(1 - x + x^3 - x^4 + x^5 - x^7 + x^8)) + O(x^60)) \\ Colin Barker, Oct 25 2019

a(n+15) = a(n) + 25 for n >= 1.
From Colin Barker, Oct 25 2019: (Start)
G.f.: x*(10 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10 + x^11 + x^12 + x^13 + x^14 + x^15) / ((1 - x)^2*(1 + x + x^2)*(1 + x + x^2 + x^3 + x^4)*(1 - x + x^3 - x^4 + x^5 - x^7 + x^8)).
a(n) = a(n-1) + a(n-15) - a(n-16) for n>16.
(End)

A174141 Numbers congruent to k mod 25, where 0 <= k <= 9.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 150, 151, 152, 153, 154

Offset: 1

Rick L. Shepherd, Mar 09 2010

Numbers whose partition into parts of sizes 1, 5, 10, and 25 having a minimal number of parts does not include a part of size 10.
For each number the partition is unique.
Complement of A174140.
Amounts in cents not including a dime when the minimal number of coins is selected from pennies, nickels, dimes, and quarters (whether usage of bills for whole-dollar amounts is permitted or not).

Index entries for sequences related to making change.
Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1).

Cf. A174138, A174139, A174140, A047201 (requires at least one part of size 1 (penny)), A008587, A053344 (minimal number of parts), A001299 (number of all such partitions).

LinearRecurrence[{1,0,0,0,0,0,0,0,0,1,-1},{0,1,2,3,4,5,6,7,8,9,25},70] (* Harvey P. Dale, May 30 2014 *)

a(n+10) = a(n) + 25 for n >= 1.

a(n)= +a(n-1) +a(n-10) -a(n-11). G.f. x^2*(1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+16*x^9) / ( (1+x)*(1+x+x^2+x^3+x^4)*(x^4-x^3+x^2-x+1)*(x-1)^2 ). - R. J. Mathar, Oct 08 2011

A235933 Numbers coprime to 35.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 11, 12, 13, 16, 17, 18, 19, 22, 23, 24, 26, 27, 29, 31, 32, 33, 34, 36, 37, 38, 39, 41, 43, 44, 46, 47, 48, 51, 52, 53, 54, 57, 58, 59, 61, 62, 64, 66, 67, 68, 69, 71, 72, 73, 74, 76, 78, 79, 81, 82, 83, 86, 87, 88, 89, 92, 93, 94, 96, 97, 99

Offset: 1

Oleg P. Kirillov, Jan 17 2014

The asymptotic density of this sequence is 24/35. - Amiram Eldar, Oct 23 2020

Bruno Berselli, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,1,1,0,-1,-1,0,1,1,0,-1,-1,0,1,1,0,-1,-1,0,1,1,0,-1).

Cf. A160547 (numbers coprime to 31), A229968 (numbers coprime to 33), A204458 (numbers coprime to 34), A007310 (numbers coprime to 36).

Cf. A045572 (numbers not divisible by 5 or 2), A229829 (numbers not divisible by 5 or 3), A047201 (numbers not divisible by 5), A236207 (numbers not divisible by 5 or 11).

a235933 n = a235933_list !! (n-1)
a235933_list = filter ((== 1) . gcd 35) [1..]
-- Reinhard Zumkeller, Mar 27 2014

[n: n in [1..100] | GCD(n,35) eq 1]; // Bruno Berselli, Mar 27 2014

Select[Range[100], GCD[#, 35] == 1 &] (* Bruno Berselli, Mar 27 2014 *)

[i for i in range(100) if gcd(i, 35) == 1] # Bruno Berselli, Mar 27 2014

Signature corrected from Georg Fischer, Feb 07 2021

Erroneous recurrence removed from Bruno Berselli, Feb 08 2021

cp's OEIS Frontend

A064001 Odd abundant numbers not divisible by 5.

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A056906 Numbers k such that 36*k^2 + 5 is prime.

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A225496 Numbers having no balanced prime factors, cf. A006562.

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A356858 a(n) is the product of the first n numbers not divisible by 5.

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A217562 Even numbers not divisible by 5.

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A260181 Numbers whose last digit is prime.

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A356860 a(n) is the number of digits in the product of the first n numbers not divisible by 5.

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