A007954 -id:A007954 - cp's OEIS frontend

A247888 Zeroless numbers n such that n + A007954(n) contains the same digits as n.

239, 326, 364, 497, 563, 598, 613, 637, 695, 819, 1215, 1239, 1326, 1364, 1431, 1497, 1512, 1518, 1563, 1598, 1613, 1637, 1695, 1812, 1815, 1819, 2115, 2139, 2313, 2356, 2369, 2419, 2511, 2594, 2639, 2791, 3126, 3213, 3235, 3238, 3259, 3354, 3365, 3561, 4131, 4219, 4346, 4353, 4395

Offset: 1

Derek Orr, Sep 25 2014

Cf. A007954 (product of digits), A052382 (zeroless numbers).

for(n=0,10^4,d=digits(n);p=prod(i=1,#d,d[i]);dp=digits(n+p);if(p&&vecsort(d,,8)==vecsort(dp,,8),print1(n,", ")))

A061763 Numbers k such that k is divisible by A061762(k) and the product of digits of k (A007954(k)) is not zero.

Original entry on oeis.org

19, 29, 39, 42, 49, 59, 69, 79, 89, 99, 126, 132, 285, 312, 522, 594, 1134, 1144, 1159, 1211, 1275, 1323, 1365, 1573, 1632, 1634, 1674, 1715, 1813, 1815, 1911, 1919, 1932, 1944, 2133, 2139, 2516, 2793, 3132, 3135, 3161, 3211, 3213, 3216, 3321, 3363, 3393

Offset: 1

Amarnath Murthy, May 20 2001

Intersection of A038366 and A052382 (zeroless numbers). - Michel Marcus, Oct 29 2019

			42 is a term as 4+2 + 2*4 = 14 and 42 = 14*3.

S. Parmeswaran, S+P numbers, Mathematics Informatics Quarterly, Vol. 9, No. 3 Sept. 1999, Bulgaria.

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 1001 terms from Harry J. Smith)

Cf. A061762, A038366, A052382.

Select[Range[3400], (y = Times @@ (x = IntegerDigits[#])) != 0 && Divisible[#, Plus @@ x + y] &] (* Jayanta Basu, Jul 14 2013 *)

isok(k) = my(d=digits(k)); vecmin(d) && ((k % (vecprod(d) + vecsum(d))) == 0);

Corrected and extended by Larry Reeves (larryr(AT)acm.org), May 23 2001

Offset corrected by Giovanni Resta, Oct 29 2019

A249516 Numbers k for which the digital product A007954(k) contains the same distinct digits as the number k.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 111, 1111, 1288, 1557, 1575, 1755, 1828, 1882, 2188, 2818, 2881, 3448, 3484, 3577, 3757, 3775, 3844, 4348, 4384, 4438, 4483, 4834, 4843, 5157, 5175, 5377, 5517, 5571, 5715, 5737, 5751, 5773, 7155, 7357, 7375, 7515, 7537, 7551

Offset: 1

Jaroslav Krizek, Oct 31 2014

			1288 is a member since 1288 and its digital product 1*2*8*8 = 128 have the same digit set {1,2,8}.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A007954, A249515, A249517.

[0] cat [n: n in [0..100000] | Set(Intseq(n)) eq Set(Intseq(&*Intseq(n)))];

Select[Range[0,8000],Union[IntegerDigits[Times@@IntegerDigits[#]]] == Union[ IntegerDigits[#]]&] (* Harvey P. Dale, Aug 05 2018 *)

print1(0,", ");for(n=1,10^4,d=digits(n);p=prod(i=1,#d,d[i]);if(vecsort(digits(p),,8)==vecsort(d,,8),print1(n,", "))) \\ Derek Orr, Nov 05 2014

A257554 Numbers k such that A007954(k) divides k and k divides A007954(k)^2.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 36, 128, 175, 384, 735, 1296, 2916, 18432, 34992, 82944, 139968, 442368, 2333772, 4128768, 9289728, 12192768, 13226976, 13395375, 13436928, 27869184, 49787136, 376233984, 429981696, 1269789696, 2633637888, 4161798144, 16728477696, 19999187712, 41479796736, 72236924928

Offset: 1

Max Alekseyev, Apr 29 2015

There are 66 terms below 10^400 with the largest containing 46 digits.

Max Alekseyev, Table of n, a(n) for n = 1..66

Intersection of A007602 and A128606.

for(n=1,10^6, d=digits(n); p=prod(i=1,#d,d[i]); if(p && n%p==0 && p^2%n==0, print1(n,", ") )) \\ Derek Orr, Apr 29 2015

A330880 Numbers m such that m*p is divisible by m-p, where m > p > 0 and p = A007954(m) = the product of digits of m.

Original entry on oeis.org

24, 36, 45, 48, 144, 384, 624, 672, 798, 816, 3276, 3648, 4864, 5994, 7965, 18816, 56175, 83232, 98496, 177184, 199584, 275772, 344736, 377496, 784896, 879984, 1372896, 1378944, 1635795, 1886976, 2472736, 3364416, 4575375, 6595992, 9289728, 9377424, 28348416, 33247872, 36387792, 58677696

Offset: 1

Scott R. Shannon, May 11 2020

Every term m is the sum of two 7-smooth numbers. Proof: Since (m-p) | m*p, we have m*p = (m - p)*k for some k > 0. Suppose m is not the sum of two 7-smooth numbers. Then m - p is not 7-smooth and so there exists a prime q > 7 such that q | (m - p). Since q doesn't divide p and q | (m - p) but (m - p) | m*p, we have q | m. But since q | m and q | (m - p) we have q | (m - (m - p)) = p, a contradiction. Q.e.d. - David A. Corneth, Jun 15 2020

			24 is a term as p = 2*4 = 8 and 24*8 = 192 is divisible by 24 - 8 = 16.
3648 is a term as p = 3*6*4*8 = 576 and 3648*576 = 2101248 is divisible by 3648-576 = 3072.
1372896 is a term as p = 1*3*7*2*8*9*6 = 18144 and 1372896*18144 = 24909825024 is divisible by 1372896 - 18144 = 1354752.

David A. Corneth, Table of n, a(n) for n = 1..152 (terms <= 10^22; first 82 terms from Giovanni Resta)

Cf. A334679, A334803, A007954, A049102, A085124.

Subsequence of A052382.

npdQ[n_]:=Module[{p=Times@@IntegerDigits[n]},n>p>0&&Divisible[n*p,n-p]]; Select[Range[6*10^7],npdQ] (* Harvey P. Dale, Jun 14 2020 *)

isok(m) = my(p=vecprod(digits(m))); p && (m-p) && !((m*p) % (m-p)); \\ Michel Marcus, May 12 2020

A334542 Numbers m such that m^2 = p^2 + k^2, with p > 0, where p = A007954(m) = the product of digits of m.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 58, 85, 375, 666, 1968, 1998, 3578, 3665, 3891, 4658, 4995, 6675, 7735, 18434, 27475, 28784, 46692, 56763, 58896, 59577, 59949, 76965, 186633, 186673, 795848, 949968, 965667, 1339575, 1587616, 1929798, 2765388, 2989584, 3674195, 4763568, 5762784, 36741656, 58988961, 134369685, 188959392

Offset: 1

Scott R. Shannon, May 05 2020

			58 is a term as p = 5*8 = 40 and 58^2 = 3364 = 40^2 + 42^2.
3891 is a term as p = 3*8*9*1 = 216 and 3891^2 = 15139881 = 216^2 + 3885^2.

Giovanni Resta, Table of n, a(n) for n = 1..140 (terms < 2*10^13)

Cf. A007954, A000404, A078134, A334557, A334558.

Subsequence of A052382 (zeroless numbers).

isok(m) = my(p=vecprod(digits(m))); p && issquare(m^2 - p^2); \\ Michel Marcus, May 06 2020

A334557 Numbers m such that m = p^2 + k^2, with p > 0, where p = A007954(m) = the product of digits of m.

Original entry on oeis.org

1, 13, 41, 61, 125, 212, 281, 613, 1156, 1424, 2225, 3232, 3316, 4113, 11125, 11281, 11525, 12816, 14913, 16317, 16441, 19125, 21284, 21625, 24128, 25216, 27521, 31525, 53125, 56116, 61321, 65161, 71325, 82116, 82217, 83521, 84313, 111812, 113125, 113625, 115336, 115681, 117125, 118372

Offset: 1

Scott R. Shannon, May 06 2020

			13 is a term as p = 1*3 = 3 and 13 = 3^2 + 2^2.
281 is a term as p = 2*8*1 = 16 and 281 = 16^2 + 5^2.
118372 is a term as p = 1*1*8*3*7*2 = 336 and 118372 = 336^2 + 74^2.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A007954, A000404, A078134, A334542, A334558.

isok(m) = my(p=vecprod(digits(m))); p && issquare(m - p^2); \\ Michel Marcus, May 06 2020

A334558 Numbers m such that m^2 + p^2 = k^2, with p > 0, where p = A007954(m) = the product of digits of m.

Original entry on oeis.org

429, 437, 598, 1938, 3584, 3875, 5576, 6864, 16758, 36828, 43778, 47775, 47859, 56637, 56672, 82928, 91798, 129584, 156782, 165688, 165838, 178857, 215985, 379488, 655578, 798847, 1881576, 2893337, 3918768, 4816872, 5439798, 5829795, 7558299, 9675288, 11943887

Offset: 1

Scott R. Shannon, May 06 2020

			429 is a term as p = 4*2*9 = 72 and 429^2 + 72^2 = 189225 = 435^2.
16758 is a term as p = 1*6*7*5*8 = 1680 and 16758^2 + 1680^2 = 283652964 = 16842^2.

Giovanni Resta, Table of n, a(n) for n = 1..147 (terms < 2*10^13)

Cf. A007954, A000404, A078134, A334542, A334557.

isok(m) = my(p=vecprod(digits(m))); p && issquare(m^2 + p^2); \\ Michel Marcus, May 06 2020

A334679 Numbers k such that k*p is divisible by k+p, where p > 0 and p = A007954(k) = the product of digits of k.

Original entry on oeis.org

2, 4, 6, 8, 24, 36, 63, 456, 495, 3276, 6624, 7497, 8832, 19728, 23976, 127488, 167328, 273525, 274995, 297675, 576975, 661248, 797769, 853776, 1323648, 1378272, 1491264, 1886976, 3483648, 3679263, 3787749, 4644864, 6386688, 7886592, 7888896, 12841472, 15974784, 16224768

Offset: 1

Scott R. Shannon, May 08 2020

			8 is a term as p = 8 and 8*8 = 64 is divisible by 8+8 = 16.
3276 is a term as p = 3*2*7*6 = 252 and 3276*252 = 825552 is divisible by 3276+252 = 3528.
3787749 is a term as p = 3*7*8*7*7*4*9 = 296352 and 3787749*296352 = 1122506991648 is divisible by 3787749+296352 = 4084101.

Giovanni Resta, Table of n, a(n) for n = 1..95 (terms < 3.5*10^14)

Cf. A330880, A334803, A007954, A049102, A085124.

Subsequence of A052382.

Select[Range[10^6], (p = Times @@ IntegerDigits@ #; p > 0 && Mod[# p, # + p] == 0) &] (* Giovanni Resta, May 08 2020 *)

isok(m) = my(p=vecprod(digits(m))); p && !((m*p) % (m+p)); \\ Michel Marcus, May 08 2020

A340907 Numbers m without zero digits such that pod(q) = pod(k) = pod(m) where q = k + pod(k) and k = m + pod(m) where pod is the product of digits, A007954.

Original entry on oeis.org

262713, 267338, 283628, 342713, 351678, 432713, 451676, 516469, 516657, 516675, 622713, 634838, 651674, 716655, 728364, 851673, 857297, 916465, 1262713, 1267338, 1283628, 1342713, 1351678, 1432713, 1451676, 1516469, 1516657, 1516675, 1622713, 1634838, 1651674

Offset: 1

Bernard Schott, Jan 26 2021

The idea of this sequence comes from a remark of Amiram Eldar in Discussion section of A327750 (m + pod(m) = k with pod(k) = pod(m)) in September 2019.
Question: is it possible to get a longer string of integers with this rule?
From David A. Corneth, Feb 01 2021: (Start)
The product of digits of a(n) is a multiple of 6. In terms below 10^10 however all products of digits of a(n) are a multiple of 36. Is that product a multiple of 36 for all a(n)?
The least term k such that k + 6 is here is k = 56516718. Are there consecutive terms that differ by less than 6? (End)

			262713 + pod(262713) = 262713 + 504 = 263217, whose product of digits is also 504, and 263217 + 504 = 263721 whose product of digits is again 504; hence, m=262713, k=263217, q=263721 and pod(m)=pod(k)=pod(q)=504, so 262713 is a term.

Roman Fedorov, Alexei Belov, Alexander Kovaldzhi, and Ivan Yashchenko, Moscow-Mathematical Olympiads, 2000-2005, Level A, Problem 2, 2003; MSRI, 2011, p. 15 and 97.

David A. Corneth, Table of n, a(n) for n = 1..10000

Subsequence of A327750.

Cf. A007954, A052382, A247888, A340908.

pod[n_] := Times @@ IntegerDigits[n]; seqQ[n_] := Module[{p = pod[n], k, q}, k = n + p; q = k + pod[k]; p > 0 && Equal @@ {p, pod[k], pod[q]}]; Select[Range[2*10^6], seqQ] (* Amiram Eldar, Jan 26 2021 *)

isok(m) = my(pm=vecprod(digits(m)), k=m+pm, pk=vecprod(digits(k)), q=k+pk, pq=vecprod(digits(q))); pm && (pm==pk) && (pk==pq); \\ Michel Marcus, Jan 26 2021

Terms from Amiram Eldar, Jan 26 2021

cp's OEIS Frontend

A247888 Zeroless numbers n such that n + A007954(n) contains the same digits as n.

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A061763 Numbers k such that k is divisible by A061762(k) and the product of digits of k (A007954(k)) is not zero.

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A249516 Numbers k for which the digital product A007954(k) contains the same distinct digits as the number k.

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A257554 Numbers k such that A007954(k) divides k and k divides A007954(k)^2.

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A330880 Numbers m such that m*p is divisible by m-p, where m > p > 0 and p = A007954(m) = the product of digits of m.

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A334542 Numbers m such that m^2 = p^2 + k^2, with p > 0, where p = A007954(m) = the product of digits of m.

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A334557 Numbers m such that m = p^2 + k^2, with p > 0, where p = A007954(m) = the product of digits of m.

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A334558 Numbers m such that m^2 + p^2 = k^2, with p > 0, where p = A007954(m) = the product of digits of m.

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A334679 Numbers k such that k*p is divisible by k+p, where p > 0 and p = A007954(k) = the product of digits of k.