A217854 -id:A217854 - cp's OEIS frontend

A363658 Positive numbers m where A217854(m) is positive and increases to a record.

2, 3, 5, 6, 8, 10, 12, 18, 20, 24, 30, 40, 42, 48, 60, 72, 84, 90, 96, 108, 120, 168, 180, 240, 336, 360, 420, 480, 504, 540, 600, 630, 660, 672, 720, 840, 1080, 1260, 1440, 1680, 2160, 2520, 3360, 3780, 3960, 4200, 4320, 4620, 4680, 5040, 7560, 9240, 10080

Offset: 1

Simon Jensen, Jun 13 2023

nonn

(-m)^tau(m) > 0 and (-m)^tau(m) > (-k)^tau(k) for all positive k < m, where tau is the number of divisors function.
There are no squares in this sequence.
It appears that if n > 13, then a(n) = A067128(n). See the link.
Only a finite number of terms in A002093 can also be terms in this sequence. See the link.

			5 is a term since (-5)^tau(5) = (-5)^2 = 25 and 25 > (-k)^tau(k) for k = 1,...,4.

Simon Jensen, Table of n, a(n) for n = 1..135
Simon Jensen, On an extended divisor product summatory function

Cf. A363657, A067128, A000005, A002093, A217854, A224914.

isok(m) = my(x=(-m)^numdiv(m)); if (x>0, for (k=1, m-1, if (x <= (-k)^numdiv(k), return(0))); return(1)); \\ Michel Marcus, Aug 31 2023

A224914 Partial sums of A217854.

Original entry on oeis.org

-1, 3, 12, -52, -27, 1269, 1318, 5414, 4685, 14685, 14806, 3000790, 3000959, 3039375, 3090000, 2041424, 2041713, 36053937, 36054298, 100054298, 100248779, 100483035, 100483564, 110175797740, 110175782115, 110176239091, 110176770532, 110658660836

Offset: 1

Simon Jensen, Apr 19 2013

sign

If there is some n > 47 such that a(n) < 0, then there is some k^2 > 47 such that a(k^2) < 0.
If n > 1 is a square number, then a(n) = a(n-1) - n^tau(n).
If n > 1 is a nonsquare number, then a(n) = a(n-1) + n^tau(n).
If n > 1 is a prime, then a(n) = a(n-1) + n^2.

			a(4) = a(1) + a(2) + a(3) + (-4)^tau(4) = (-1) + 3 + 12 + (-64) = -52.

Simon Jensen, Table of n, a(n) for n = 1..10000
Simon Jensen, On an extended divisor product summatory function

Accumulate@ Table[(-n)^DivisorSigma[0, n], {n, 28}] (* Michael De Vlieger, Mar 18 2016 *)

a(n) = sum(k=1, n, (-k)^numdiv(k)); \\ Michel Marcus, Mar 18 2016

a(n) = Sum_{i=1..n} (-i)^tau(i) = Sum_{i=1..n} (-i)^A000005(i) = Sum_{i=1..n} A217854(i).

A363657 Numbers m where A217854(m) is a record minimum.

Original entry on oeis.org

1, 4, 9, 16, 36, 100, 144, 324, 400, 576, 900, 1764, 2304, 3600, 7056, 8100, 14400, 28224, 32400, 44100, 57600, 108900, 112896, 129600, 176400, 396900, 435600, 518400, 608400, 705600, 1587600, 2822400, 5336100, 6350400, 14288400, 15681600, 17640000, 21344400

Offset: 1

Simon Jensen, Jun 13 2023

nonn

(-m)^tau(m) < 0 and (-m)^tau(m) < (-k)^tau(k) for all positive k < m, where tau is the number of divisors function.
All terms are squares.
It is conjectured that if m is a term, then abs((-m)^tau(m)) <= abs((-k)^tau(k)) for some k < m. See the link.

			9 is a term since (-9)^tau(9) = (-9)^3 = -729 and -729 < (-k)^tau(k) for k = 1,...,8.
25 is not a term since (-25)^tau(5) = (-25)^3 = -15625 > (-16)^tau(16) = (-16)^5 = -1048576 and 16 < 25.

Simon K. Jensen, On an extended divisor product summatory function

Cf. A363658, A000290, A067128, A000005, A217854, A224914.

isok(m) = my(x=(-m)^numdiv(m)); for (k=1, m-1, if (x >= (-k)^numdiv(k), return(0))); return(1); \\ Michel Marcus, Jun 18 2023

A217422 Numbers of the form 2^r*17^s whose decimal representations are such that each digit 0-9 appears a prime number of times.

Original entry on oeis.org

981750581622330147995648, 28801196957834700781586432, 835992910761480393266512789504, 7295132596707416278470844481536, 76976152675689985407324172386304

Offset: 1

James G. Merickel, Oct 05 2012

See formula section for more data. Others in cross-reference are similar and some hold more motivation in comments.

			A217423(1)=47 and A217424(1)=8, so this sequence's first term is 2^47 * 17^8.  It has in its decimal representation two copies each of the digits 0, 2, 3, 4, 6 and 7; and three copies each of 1, 5, 8 and 9.

Cf. A217854, A217423, A217424, A217404, A217407, A217410, A217413, A217416, A217419, A217425, A217428, A217431.

A217422(n) = 2^A217423(n)*17^A217424(n).

A217425 Numbers of the form 5^r*7^s whose decimal representations are such that each digit 0-9 appears a prime number of times.

Original entry on oeis.org

97402668820327149658203125, 81209257154451887573232591061530625, 13375863052949754169544537548117223100875, 4587921027161765680153776379004207523600125, 2478309849684200670569842256516437530517578125

Offset: 1

James G. Merickel, Oct 05 2012

See the formula section for more data, and other sequences in cross-reference for motivation and similar sequences.

			A217426(1)=13 and A217427(1)=20, so this sequence's first term is 5^13 * 7^20.  It has two copies each of the digits 1, 3, 4, 5, 7 and 9; three each of 0's, 6's and 8's; and five 2's.

Cf. A217854, A217426, A217427, A217404, A217407, A217410, A217413, A217416, A217419, A217422, A217428, A217431.

A217425(n) = 5^A217426(n) * 7^A217427(n).

A217431 Numbers of the form 3^r*13^s whose decimal representation has a prime number of copies of each digit 0-9.

Original entry on oeis.org

691159348276025798403, 510798409623548623605717, 5097400863986495932124683149477, 10996481542736751381410324522244489, 915432679064411834115450778445909529

Offset: 1

James G. Merickel, Oct 05 2012

See the formula section for more data, and others in cross-reference for motivation and similar.

a(6), if it exists, is larger than 10^1000. - Giovanni Resta, Jan 16 2014

			a(1) = 3^25 * 13^8 (so A217432(1)=25 and A217432(1)=8). Indeed, it contains two copies of each digit other than 9 and three copies of 9.  No smaller 21-digit number with this general character -- two copies of all but one digit -- and no 20-digit number with two copies of each digit has form 3^a*13^b with a,b > 0.

Cf. A217854, A217432, A217433, A217404, A217407, A217410, A217413, A217416, A217419, A217422, A217425, A217428.

nd = 50; mx = 10^nd; pr = Prime@ Range@ PrimePi@ nd; pQ[n_] := Union[DigitCount@n, pr] == pr; Sort@ Select[ Flatten@ Table[3^p*13^q, {p, Log[3, mx/13]}, {q, Log[13, mx/3^p]}], pQ] (* terms < 10^50, Giovanni Resta, Jan 16 2014 *)

A217431(n) = 3^A217432(n) * 13^A217433(n).

A217413 Numbers of the form 2^r*11^s whose decimal representations are such that each digit 0-9 appears a prime number of times.

Original entry on oeis.org

380270922216319615655477248, 804540132953535715766960128, 808356196450619282354367102976, 3705202243778559670466281109192704, 29084192305393678777275954501779456

Offset: 1

James G. Merickel, Oct 03 2012

See the formula section for more data, and the other sequences in cross-reference for similar sequences and motivation.

			A217414(1)=71 and A217415(1)=5, so this sequence's first term is 2^71 * 11^5.  It has two each of 0's, 3's, 4's, 6's, 8's and 9's; three each of 1's, 5's and 7's; and five 2's.

Cf. A217854, A217414, A217415, A217404, A217407, A217410, A217416, A217419, A217422, A217425, A217428, A217431.

A217413(n) = 2^A217414(n) * 11^A217415(n).

A217416 Numbers of the form 2^r*13^s whose decimal representations are such that each digit 0-9 appears a prime number of times.

Original entry on oeis.org

19461082905477938625332224, 1493825030770756769826391724130304, 1341437769548771819714842333610521088, 1888426137230419620556768936800026624

Offset: 1

James G. Merickel, Oct 03 2012

See the formula section for more data, and the other sequences in cross-reference for similar and motivation.

			A217417(1)=10 and A217418(1)=20, so this sequence's first term can be simply written as 338^10 (338=2*13^2).  It has two each of 0's, 1's, 5's, 6's, 7's and 8's; three each of 3's, 4's and 9's; and five 2's.

Cf. A217854, A217417, A217418, A217404, A217407, A217410, A217413, A217419, A217422, A217425, A217428, A217431.

a(n) = 2^A217417(n) * 13^A217418(n).

A217428 Numbers of the form 2^r*19^s whose decimal representation contains each digit 0-9 a prime number of times.

Original entry on oeis.org

611897345462967423035506688, 98653640312161724942079430255181824, 195052744758517664687371356106391552, 69210613747809022824201189555756335104, 401770033097538379880522919696423452672

Offset: 1

James G. Merickel, Oct 05 2012

See the formula section for more data, and others in cross-reference for similar and motivation.

			2^55 * 19^8 has two copies of each of the digits 0, 1, 2, 7 and 9; three copies of each of 3, 4, 5 and 8; and five copies of the digit 6.  All smaller numbers of the required type have at least a digit counted 0, 1, or a composite number of times, so this is a(1).

Cf. A217854, A217429, A217430, A217404, A217407, A217410, A217413, A217416, A217419, A217422, A217425, A217431.

nd = 50; mx = 10^nd; pr = Prime@Range@PrimePi@nd; pQ[n_] := Union[DigitCount@n, pr] == pr; Sort@ Select[Flatten@ Table[2^p 19^q, {p, Log[2, mx/19]}, {q, Log[19, mx/2^p]}], pQ] (* Giovanni Resta, Jan 16 2014 *)

A217431(n) = 2^A217429(n) * 19^A217430(n)

A217419 Numbers factoring 3^r*11^s whose decimal representations are such that each of the digits 0-9 appears a prime number of times.

Original entry on oeis.org

5586701408957811048315412506328906443, 4756351651546448722293215379849833244291829764770574840668685771747

Offset: 1

James G. Merickel, Oct 05 2012

The 3-adic valuations for the terms are 29 and 83, and the 11-adic valuations are 22 and 26. For the other main sequences of trios corresponding to nonsquare semiprimes other than 10 and through 39, including separate prime valuations in companion sequences, see cross-references.
While heuristics imply this sequence is infinite, finding a(3) is difficult: A program counting digits for numbers of the given form restricted to those having an even number of decimal digits (with heuristics and initial data practically ruling out counts of 2 for any digit) produced no result for this sequence through 61 values having no more than one digit counted a nonprime number of times. The last of these values was 3^14650*11^3032 (10148 digits in length). - James G. Merickel, Dec 11 2013
a(3) if it exists has > 10000 digits. - James G. Merickel, Dec 11 2013

			3^29 * 11^22 has two each of 2's, 7's and 9's; three each of 3's and 6's; and five each of 0's, 1's, 4's, 5's and 8's.  No smaller number with only 3 and 11 as prime factors has a prime number of each digit 0-9, so a(1) is this value.

Cf. A217854, A217404, A217407, A217410, A217413, A217416, A217422, A217425, A217428, A217431.

{
k=33;a=[99];t=1;while(1,
v=vectorsmall(10);m=k;
while(m,d=m%10;m\=10;v[d+1];next());
f=1;for(i=1,10,if(isprime(v[i])==0,f=0;break()));
if(f,F=factor(k);print1(3"^"F[1,2]"*"11"^"F[2,2]"="k"\n"));
if(11^(t+1)

cp's OEIS Frontend

A363658 Positive numbers m where A217854(m) is positive and increases to a record.

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A224914 Partial sums of A217854.

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A363657 Numbers m where A217854(m) is a record minimum.

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A217422 Numbers of the form 2^r*17^s whose decimal representations are such that each digit 0-9 appears a prime number of times.

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A217425 Numbers of the form 5^r*7^s whose decimal representations are such that each digit 0-9 appears a prime number of times.

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A217431 Numbers of the form 3^r*13^s whose decimal representation has a prime number of copies of each digit 0-9.

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A217413 Numbers of the form 2^r*11^s whose decimal representations are such that each digit 0-9 appears a prime number of times.

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A217416 Numbers of the form 2^r*13^s whose decimal representations are such that each digit 0-9 appears a prime number of times.

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A217428 Numbers of the form 2^r*19^s whose decimal representation contains each digit 0-9 a prime number of times.

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