A000420 -id:A000420 - cp's OEIS frontend

A014831 a(1)=1; for n>1, a(n) = 8*a(n-1) + n.

1, 10, 83, 668, 5349, 42798, 342391, 2739136, 21913097, 175304786, 1402438299, 11219506404, 89756051245, 718048409974, 5744387279807, 45955098238472, 367640785907793, 2941126287262362, 23529010298098915, 188232082384791340, 1505856659078330741, 12046853272626645950

Offset: 1

N. J. A. Sloane

			For n=5, a(5) = 1*15 + 7*20 + 7^2*15 + 7^3*6 + 7^4*1 = 5349. [_Bruno Berselli_, Nov 13 2015]

Colin Barker, Table of n, a(n) for n = 1..1000
Dillan Agrawal, Selena Ge, Jate Greene, Tanya Khovanova, Dohun Kim, Rajarshi Mandal, Tanish Parida, Anirudh Pulugurtha, Gordon Redwine, Soham Samanta, and Albert Xu, Chip-Firing on Infinite k-ary Trees, arXiv:2501.06675 [math.CO], 2025. See p. 18.
Index entries for linear recurrences with constant coefficients, signature (10,-17,8).

Cf. A000420, A104712.

a:=n->sum((8^(n-j)-1)/7,j=0..n): seq(a(n), n=1..19); # Zerinvary Lajos, Jan 15 2007
a:= n-> (Matrix ([[1, 0, 1], [1, 1, 1], [0, 0, 8]])^n)[2, 3]: seq (a(n), n=1..25); # Alois P. Heinz, Aug 06 2008

Table[(8^(n + 1) - 7 n - 8)/49, {n, 1, 25}] (* Bruno Berselli, Nov 13 2015 *)
nxt[{n_,a_}]:={n+1,8a+n+1}; NestList[nxt,{1,1},30][[;;,2]] (* Harvey P. Dale, Aug 09 2025 *)

Vec(x/((1 - x)^2*(1 - 8*x)) + O(x^25)) \\ Colin Barker, Jun 03 2020

a(n) = (8^(n+1) - 7*n - 8)/49. - Rolf Pleisch, Oct 21 2010
a(n) = Sum_{i=0..n-1} 7^i*binomial(n+1,n-1-i). - Bruno Berselli, Nov 13 2015
From Colin Barker, Jun 03 2020: (Start)
G.f.: x/((1 - x)^2*(1 - 8*x)).
a(n) = 10*a(n-1) - 17*a(n-2) + 8*a(n-3) for n > 3. (End)
E.g.f.: exp(x)*(8*exp(7*x) - 7*x - 8)/49. - Elmo R. Oliveira, Mar 29 2025

A049305 Numbers k such that k is a substring of 7^k.

Original entry on oeis.org

3, 4, 6, 8, 12, 15, 20, 40, 42, 43, 50, 53, 55, 59, 60, 61, 62, 69, 72, 73, 74, 75, 78, 79, 80, 81, 83, 86, 87, 88, 89, 93, 94, 95, 96, 97, 99, 100, 103, 111, 113, 114, 118, 164, 165, 185, 193, 200, 207, 210, 215, 220, 230, 232, 238, 241, 243, 250, 253, 254, 255

Offset: 1

David W. Wilson

Cf. A000420, A032740, A049301, A049302, A049303, A049304, A049306, A049307.

def ok(n): return str(n) in str(7**n)
print(list(filter(ok, range(256)))) # Michael S. Branicky, Aug 13 2021

A067504 Powers of 7 with digit sum also a power of 7.

Original entry on oeis.org

1, 7, 2401, 1977326743

Offset: 1

Amarnath Murthy, Feb 11 2002

Next terms are 7^4433, 7^30810. - Sascha Kurz, Mar 18 2002

Cf. A067499, A067500, A067501, A067502, A067503.

Subsequence of A000420.

A138973 a(n) = 8^n mod 7^n.

Original entry on oeis.org

0, 1, 15, 169, 1695, 15961, 26846, 450066, 5247614, 13156907, 226316077, 680627620, 13354327932, 65310761853, 328708074010, 1951441519231, 15611532153848, 158125187800385, 101848932467045, 7328445851378156, 35829776440278962, 286638211522231696

Offset: 0

N. J. A. Sloane, May 20 2008

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1183

Cf. k^n mod (k-1)^n: A002380 (k=3), A064629 (k=4), A138589 (k=5), A138649 (k=6), A139786 (k=7), this sequence (k=8), A139733 (k=9).

Cf. A000420, A001018.

a[n_]:=PowerMod[8,n,7^n];Array[a,22,0] (* James C. McMahon, Jun 23 2025 *)

a(n) = lift(Mod(8, 7^n)^n); \\ Michel Marcus, Feb 20 2018

[power_mod(8,n,7^n) for n in range(0,22)] # Zerinvary Lajos, Nov 28 2009

A177805 Numbers k such that k divides 15^k - 1.

Original entry on oeis.org

1, 2, 4, 7, 8, 14, 16, 28, 32, 49, 56, 64, 98, 112, 128, 136, 196, 224, 256, 272, 343, 392, 448, 452, 512, 544, 686, 784, 812, 896, 904, 952, 1024, 1088, 1372, 1568, 1624, 1792, 1808, 1904, 2048, 2176, 2312, 2401, 2744, 3136, 3164, 3248, 3584, 3616, 3808, 4096

Offset: 1

Alexander Adamchuk, May 17 2010

nonn

A000420 are the only odd terms of the sequence. - Robert Israel, Feb 25 2020

Robert Israel, Table of n, a(n) for n = 1..2500

Cf. A000420, A014960, A014956, A014951, A014949, A014946, A014945, A067945, A128358, A128360.

Contains A003591.

filter:= n -> 15 &^ n - 1 mod n = 0:
select(filter, [$1..10000]); # Robert Israel, Feb 25 2020

is(n)=Mod(15,n)^n==1 \\ Charles R Greathouse IV, Nov 04 2016

A239012 Exponents m such that the decimal expansion of 7^m exhibits its first zero from the right later than any previous exponent.

Original entry on oeis.org

0, 2, 3, 6, 10, 11, 19, 35, 127, 131, 175, 207, 1235, 2470, 2651, 1241310, 1922910, 471056338, 1001431598, 1720335627, 4203146094, 5353516238, 21838571507, 25770284079, 40822793867

Offset: 1

Charles R Greathouse IV and Robert G. Wilson v, Mar 14 2014

Assume that a zero precedes all decimal expansions. This will take care of those cases in A030703.

Inspired by the seqfan list discussion Re: "possible sequence", beginning with David Wilson 7:57 PM Mar 06 2014 and continued by M. F. Hasler, Allan C. Wechsler and Franklin T. Adams-Watters.

Cf. A000420, A030703, A020665, A031142, A239008, A239009, A239010, A239011, A239013, A239014, A239015.

f[n_] := Position[ Reverse@ Join[{0}, IntegerDigits[ PowerMod[7, n, 10^500]]], 0, 1, 1][[1, 1]]; k = mx = 0; lst = {}; While[k < 500000001, c = f[k]; If[c > mx, mx = c; AppendTo[ lst, k]; Print@ k]; k++]; lst

a(19)-a(22) from Bert Dobbelaere, Jan 21 2019

a(23)-a(25) from Chai Wah Wu, Jan 15 2020

A268354 Highest power of 7 dividing n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 49, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 49, 1, 1, 1, 1, 1, 1, 7

Offset: 1

Tom Edgar, Feb 02 2016

The generalized binomial coefficients produced by this sequence provide an analog to Kummer's Theorem using arithmetic in base 7.

			Since 14 = 7 * 2, a(14) = 7. Likewise, since 7 does not divide 13, a(13) = 1.

Antti Karttunen, Table of n, a(n) for n = 1..20790
Tyler Ball, Tom Edgar, and Daniel Juda, Dominance Orders, Generalized Binomial Coefficients, and Kummer's Theorem, Mathematics Magazine, Vol. 87, No. 2, April 2014, pp. 135-143.
Tom Edgar and Michael Z. Spivey, Multiplicative functions, generalized binomial coefficients, and generalized Catalan numbers, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.6.

Cf. A000420, A001620, A006519, A038500, A234957, A214411, A234959, A060904, A242603, A268357.

[7^Valuation(n,7): n in [1..150]]; // Vincenzo Librandi, Feb 03 2016

7^Table[IntegerExponent[n, 7], {n, 150}] (* Vincenzo Librandi, Feb 03 2016 *)

a(n) = 7^valuation(n, 7) \\ Michel Marcus, Feb 05 2016

Sage
```
[7^valuation(i, 7) for i in [1..100]]
```

a(n) = 7^valuation(n,7).
a(n) = 7^A214411(n).
Completely multiplicative with a(7) = 7, a(p) = 1 for prime p and p <> 7. - Andrew Howroyd, Jul 20 2018
From Peter Bala, Feb 21 2019: (Start)
a(n) = gcd(n,7^n).
a(n) = n/A242603(n).
O.g.f.: x/(1 - x) + 6*Sum_{n >= 1} 7^(n-1)*x^(7^n)/ (1 - x^(7^n)). (End)
Sum_{k=1..n} a(k) ~ (6/(7*log(7)))*n*log(n) + (4/7 + 6*(gamma-1)/(7*log(7)))*n, where gamma is Euler's constant (A001620). - Amiram Eldar, Nov 15 2022
Dirichlet g.f.: zeta(s)*(7^s-1)/(7^s-7). - Amiram Eldar, Jan 03 2023

More terms from Antti Karttunen, Dec 22 2017

A288245 Irregular triangle read by rows of normalized Girard-Waring formula (cf. A210258), for m=7 data values.

Original entry on oeis.org

1, 7, -6, 49, -63, 15, 343, -588, 140, 126, -20, 2401, -5145, 1225, 2205, -175, -525, 15, 16807, -43218, 10290, 27783, -1470, -8820, 126, -2646, 630, 525, -6, 117649, -352947, 84035, 302526, -12005, -108045, 1029, -64827, 10290, 8575, -49, 15435, -441, -1225, 1

Offset: 1

Gregory Gerard Wojnar, Jun 06 2017

Let SM_k = Sum( d_(t_1, t_2, t_3, ..., t_7)* eM_1^t_1 * eM_2^t_2 * ... * eM_7^t_7) summed over all length 7 integer partitions of k, i.e., 1*t_1 + 2*t_2 + 3*t_3 + ... + 7*t_7 = k, where SM_k are the averaged k-th power sum symmetric polynomials in 7 data (i.e., SM_k = S_k/7 where S_k are the k-th power sum symmetric polynomials, and where eM_k are the averaged k-th elementary symmetric polynomials, eM_k = e_k/binomial(7,k) with e_k being the k-th elementary symmetric polynomials. The data d_(t_1, t_2, t_3, ..., t_7) form a triangle, with one row for each k value starting with k=1; the number of terms in successive rows is nondecreasing.

Row sums of positive entries give 1,7,64,609,5846,56161,... Row sums of negative entries are always 1 less than corresponding row sums of positive entries.

			Triangular array begins...
1;
7,-6;
49,-63,15;
343,-588,140,126,-20;
2401,-5145,1225,2205,-175,-525,15;
16807,-43218,10290,27783,-1470,-8820,126,-2646,630,525,-6;
117649,-352947,84035,302526,-12005,-108045,1029,64827,10290,8575,-49,15435,-441,-1225,1;

Gregory Gerard Wojnar, Java program
G. G. Wojnar, D. Sz. Wojnar, and L. Q. Brin, Universal Peculiar Linear Mean Relationships in All Polynomials, pp. 22-24, arXiv:1706.08381 [math.GM], 2107.

Cf. A028297 (m=2), A287768 (m=3), A288199 (m=4), A288211 (m=5), A288211 (m=6), A288188 (m=8). Also see Girard-Waring A210258.

First entries of each row of triangle are powers of m=7, A000420.

Java
```
// See Wojnar link.
```

A367247 a(n) is the number of n-digit numbers whose difference between the largest and smallest digits is equal to 6.

Original entry on oeis.org

0, 7, 131, 1609, 16415, 150817, 1296191, 10641169, 84520175, 654958177, 4980233951, 37312922929, 276288797135, 2026564724737, 14750977566911, 106695818055889, 767748717541295, 5500729672814497, 39270143125479071, 279511731951144049, 1984459091985376655, 14059238393314971457

Offset: 1

Stefano Spezia, Nov 11 2023

a(n) is the number of n-digit numbers in A366963.

Index entries for linear recurrences with constant coefficients, signature (18,-107,210).

Cf. A366963.
Cf. A000351, A000400, A000420.
Cf. A367243, A367244, A367245, A367246, A367248, A367249, A367250.

LinearRecurrence[{18,-107,210},{0,7,131},22]

a(n) = 27*7^(n-1) - 47*6^(n-1) + 4*5^n.
a(n) = 21*a(n-1) - 146*a(n-2) + 336*a(n-3) for n > 3.
O.g.f.: x^2*(7 + 5*x)/((1 - 5*x)*(1 - 6*x)*(1 - 7*x)).
E.g.f.: (162*exp(7*x) - 329*exp(6*x) + 168*exp(5*x) - 1)/42.

A367248 a(n) is the number of n-digit numbers whose difference between the largest and smallest digits is equal to 7.

Original entry on oeis.org

0, 5, 111, 1601, 19095, 204545, 2045511, 19508081, 179752215, 1613908385, 14202967911, 123028446161, 1052237271735, 8907026785025, 74758478722311, 623053865857841, 5162154289325655, 42558224511290465, 349394287423788711, 2858263098464575121, 23311522539676521975

Offset: 1

Stefano Spezia, Nov 11 2023

a(n) is the number of n-digit numbers in A366964.

Index entries for linear recurrences with constant coefficients, signature (21,-146,336).

Cf. A366964.
Cf. A000400, A000420, A001018.
Cf. A367243, A367244, A367245, A367246, A367247, A367249, A367250.

LinearRecurrence[{21,-146,336},{0,5,111},21]

a(n) = 23*8^(n-1) - 41*7^(n-1) + 3*6^n.
a(n) = 21*a(n-1) - 146*a(n-2) + 336*a(n-3) for n > 3.
O.g.f.: x^2*(5 + 6*x)/((1 - 6*x)*(1 - 7*x)*(1 - 8*x)).
E.g.f.: (161*exp(8*x) - 328*exp(7*x) + 168*exp(6*x) - 1)/56.

cp's OEIS Frontend

A014831 a(1)=1; for n>1, a(n) = 8*a(n-1) + n.

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A049305 Numbers k such that k is a substring of 7^k.

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A067504 Powers of 7 with digit sum also a power of 7.

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A138973 a(n) = 8^n mod 7^n.

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Mathematica

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Sage

A177805 Numbers k such that k divides 15^k - 1.

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Maple

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A239012 Exponents m such that the decimal expansion of 7^m exhibits its first zero from the right later than any previous exponent.

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Mathematica

Extensions

A268354 Highest power of 7 dividing n.

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Magma

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Sage

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A288245 Irregular triangle read by rows of normalized Girard-Waring formula (cf. A210258), for m=7 data values.

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Java